id string | solution string | answer string | metadata dict | problem string | tags list | idea string | original_index int64 | candidates list | bma_scores list | jaccard_scores list | domains list | rephrased_problem string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
ours_27905 | Let the distance between \( A \) and \( B \) be \( s \, \text{km} \) and the original speed of the train be \( v \, \text{km/hr} \). During the 6 hours before the halt, the first train covered \( 6v \, \text{km} \). The remaining distance of \( (s - 6v) \, \text{km} \) was covered at a speed of \( 1.2v \, \text{km/hr} ... | 600 \, \text{km} | {
"competition": "misc",
"dataset": "Ours",
"posts": null,
"source": "Problems in Elementary Mathematics - group_21.md"
} | A train left a station \( A \) for \( B \) at 13:00. At 19:00 the train was brought to a halt by a snow drift. Two hours later the railway line was cleared, and to make up for the lost time, the train proceeded at a speed exceeding the original speed by \( 20\% \) and arrived at \( B \) only one hour later. The next da... | [
"/Mathematics/Algebra/AlgebraicEquations/LinearEquation",
"/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations",
"/Mathematics/Algebra/RateProblems"
] | Express total travel time before and after the halt using distance‑speed‑time and set it equal to scheduled time plus observed lateness. | 70,632 | [
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"Algebra"
] | At 13:00 a futuristic hyperloop capsule departs from terminal Alpha en route to terminal Beta. At 19:00 the capsule is forced to stop by a sudden magnetic storm. After a two‑hour interruption the line is cleared, and the capsule resumes its journey, now traveling at a speed that is 20 % greater than its original cruisi... |
numina_10042778 | 3. The trains are moving in the same direction, so they can move in the direction of $A B$ or $B A$. Let's consider each of these cases.
The trains are moving in the direction of $A B$.
1) $40 \times 8=320$ km - the first train traveled;
2) $48 \times 8=384$ km - the second train traveled;
3) $384-320=64$ km - by thi... | 956 | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 3. Two trains leave from two cities at the same time. The first one travels at 40 km/h, while the second one travels at 48 km/h. How far apart will these trains be from each other after 8 hours, if they are moving in the same direction and the distance between the cities is 892 km? | [
"/Mathematics/Algebra/RateProblems/TwoTrainsPuzzle",
"/Mathematics/AppliedMathematics"
] | Use the relative speed (difference of the trains' speeds) multiplied by the time to find the change in separation, then adjust the initial distance. | 19,718 | [
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"Algebra"
] | Two autonomous cargo drones launch simultaneously from opposite launch pads that are 892 km apart along a straight corridor. Drone Alpha, departing from Pad A, flies at a constant speed of 40 km/h, while Drone Beta, departing from Pad B, flies at a constant speed of 48 km/h. Assuming the drones travel in the same direc... |
aops_1184997 | [quote=arqady]
Hence, it remains to prove that $(a+b+c)^4\geq8\sum_{cyc}(a^3b+a^3c)$, which is obvious.[/quote]
Let $x=a^2+b^2+c^2, \ y=ab+bc+ca$.
$(a+b+c)^4=(x+2y)^2\ge 8xy=8\sum_{cyc}(a^3b+a^3c+a^2bc)\ge 8\sum_{cyc}(a^3b+a^3c)$ | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Given $a,b,c\\geq 0$, prove that $$\\sum_{cyc}^{}\\sqrt[3]{\\frac{a}{b+c}}\\geq 2$$",
"content_html": "Given <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/0/f/30fa6... | Given \(a,b,c\ge 0\), prove that
\[
\sum_{\text{cyc}}\sqrt[3]{\frac{a}{b+c}}\ge 2.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/AlgebraicIdentities/PolynomialIdentity",
"/Mathematics/Algebra/Products",
"/Mathematics/Algebra/Sums/PowerSum",
"/Mathematics/Algebra/Sums/Sum"
] | Apply Hölder to turn the sum of cube roots into (a+b+c)^4 ≥ 8∑a^3(b+c) and then use x=∑a^2, y=∑ab so (x+2y)^2 ≥ 8xy. | 147,728 | [
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"Algebra"
] | Three interstellar vessels possess non‑negative fuel amounts $x$, $y$, and $z$. Prove that
\[
\sqrt[3]{\frac{x}{\,y+z\,}}+\sqrt[3]{\frac{y}{\,z+x\,}}+\sqrt[3]{\frac{z}{\,x+y\,}}\ge 2.
\] |
numina_10734145 | Proof: Let \( A=\frac{1}{1+a^{4}}, B=\frac{1}{1+b^{4}}, C=\frac{1}{1+c^{4}}, D=\frac{1}{1+d^{4}} \), then \( a^{4}=\frac{1-A}{A}, b^{4}=\frac{1-B}{B}, c^{4}=\frac{1-C}{C}, d^{4}=\frac{1-D}{D} \).
Using the Arithmetic Mean-Geometric Mean Inequality, we get
\[
\begin{aligned}
& (B+C+D)(C+D+A)(D+A+B)(A+B+C) \\
\geqslant ... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "inequalities"
} | Example 7 Let $a, b, c, d>0$. When $\frac{1}{1+a^{4}}+\frac{1}{1+b^{4}}+\frac{1}{1+c^{4}}+\frac{1}{1+d^{4}}=1$, prove: $a b c d \geqslant 3$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | [
"/Mathematics/Algebra/GeneralAlgebra/Algebra"
] | Express a⁴ as (1‑A)/A and apply AM‑GM to the four triple‑sums (B+C+D), (C+D+A), (D+A+B), (A+B+C) to bound ∏(1‑A)/A. | 137,416 | [
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"Algebra"
] | Example 7.
Let $u,\;v,\;w,\;t$ be positive real numbers. If
\[
\frac{1}{1+u^{4}}+\frac{1}{1+v^{4}}+\frac{1}{1+w^{4}}+\frac{1}{1+t^{4}}=1,
\]
prove that
\[
u\,v\,w\,t\;\ge\;3.
\] |
numina_10117640 | Because $0<x_{k} \leqslant x_{n}<1 \quad(k=1,2, \cdots, n)$
So $0<\frac{1-x_{n}}{1-x_{k}} \leqslant 1 \quad \cdot(k-n$ when taking “=” sign $)$
So $\left(1-x_{n}\right)^{2} \frac{x_{k}^{k}}{\left(1-x_{k}^{k} 1\right)^{2}}=\frac{\left(1-x_{n}\right)^{2}}{\left(1-x_{k}\right)^{2}} \cdot \frac{x_{k}^{k}}{\left(1+x_{k}... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | II. (50 points)
Let $0<x_{1}<x_{2}<\cdots<x_{n}<1$, prove that:
$$
\left(1-x_{n}\right)^{2}\left[\frac{x_{1}}{\left(1-x_{1}^{2}\right)^{2}}+\frac{x_{2}^{2}}{\left(1-x_{2}^{3}\right)^{2}}+\cdots+\frac{x_{n}^{n}}{\left(1-x_{n}^{n+1}\right)^{2}}\right]<1
$$ | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Products",
"/Mathematics/Algebra/Sums/Sum"
] | Factor $1-x^{k+1}$ and use AM‑GM on $1+x+⋯+x^k$ to bound each term by $1/(k+1)^2$, then sum the resulting series. | 128,224 | [
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"Algebra"
] | Suppose a sequence of $n$ optical filters has transmission coefficients $0<\tau_1<\tau_2<\dots<\tau_n<1$. Prove that
\[
\bigl(1-\tau_n\bigr)^2\Biggl[\frac{\tau_1}{\bigl(1-\tau_1^{\,2}\bigr)^{2}}
+\frac{\tau_2^{\,2}}{\bigl(1-\tau_2^{\,3}\bigr)^{2}}
+\cdots
+\frac{\tau_n^{\,n}}{\bigl(1-\tau_n^{\,n+1}\bigr)^{2}}\Biggr]... |
numina_10049625 | ### 4.30 Method I. We have
$\sin 10 \alpha \sin 8 \alpha + \sin 8 \alpha \sin 6 \alpha - \sin 4 \alpha \sin 2 \alpha = \sin 8 \alpha \times$ $\times (\sin 10 \alpha + \sin 6 \alpha) - 2 \sin^2 2 \alpha \cos 2 \alpha = \sin 8 \alpha \cdot 2 \sin 8 \alpha \cdot$ $\cdot \cos 2 \alpha - 2 \sin^2 2 \alpha \cos 2 \alpha = 2... | 2\cos2\alpha\sin6\alpha\sin10\alpha | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | $4.30 \sin 10 \alpha \sin 8 \alpha + \sin 8 \alpha \sin 6 \alpha - \sin 4 \alpha \sin 2 \alpha$. | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity"
] | Convert the sine sums using sum‑to‑product identities to factor the expression. | 83,090 | [
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"Algebra"
] | Problem 12.5.
In a futuristic starship, the navigation console records the orientation angles of the warp drive as integer multiples of a fundamental angle $\beta$.
Compute the simplified value of the trigonometric expression
\[
\sin(10\beta)\,\sin(8\beta)\;+\;\sin(8\beta)\,\sin(6\beta)\;-\;\sin(4\beta)\,\sin(2\... |
ours_17896 | The given inequality can be transformed into:
\[
x^{3}(x+1) + y^{3}(y+1) + z^{3}(z+1) \geq \frac{3}{4}(x+1)(y+1)(z+1)
\]
By the AM-GM inequality, it suffices to prove a stronger inequality:
\[
x^{4} + x^{3} + y^{4} + y^{3} + z^{4} + z^{3} \geq \frac{1}{4} \left[(x+1)^{3} + (y+1)^{3} + (z+1)^{3}\right]
\]
Define \( ... | null | {
"competition": "imo",
"dataset": "Ours",
"posts": null,
"source": "IMO1998SL.md"
} | Let \( x, y, \) and \( z \) be positive real numbers such that \( x y z = 1 \). Prove that
\[
\frac{x^{3}}{(1+y)(1+z)}+\frac{y^{3}}{(1+z)(1+x)}+\frac{z^{3}}{(1+x)(1+y)} \geq \frac{3}{4}
\] | [
"/Mathematics/Algebra/GeneralAlgebra/Algebra",
"/Mathematics/Algebra/Products/Product",
"/Mathematics/Algebra/Sums/PowerSum",
"/Mathematics/Algebra/Sums/Sum"
] | Homogenize the sum by writing each term as x³(x+1)/(x+1)(y+1)(z+1) and apply AM‑GM (or Chebyshev) to compare higher and lower power sums | 64,372 | [
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"Algebra"
] | In a cutting‑edge spacecraft, three propulsion units operate with dimensionless performance coefficients \(a\), \(b\), and \(c\). These coefficients are positive real numbers and satisfy the balance condition \(a\,b\,c = 1\). Prove that
\[
\frac{a^{3}}{(1+b)(1+c)}\;+\;\frac{b^{3}}{(1+c)(1+a)}\;+\;\frac{c^{3}}{(1+a)(1+... |
aops_576294 | [quote="oldbeginner"]If $a, b, c>0$ prove that
\[\frac{(a+b)(2b-c)}{a^2-ab+b^2}+\frac{(b+c)(2c-a)}{b^2-bc+c^2}+\frac{(c+a)(2a-b)}{c^2-ca+a^2}\le 6\][/quote]
After expanding we need to prove that
$\sum_{cyc}(a^5b+a^5c+3a^4b^2+a^4c^2-8a^4bc-4a^3b^3+2a^3b^2c+8a^3c^2b-4a^2b^2c^2)\geq0$,
which follows from two obvious ineq... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "If $a, b, c>0$ prove that\n\\[\\frac{(a+b)(2b-c)}{a^2-ab+b^2}+\\frac{(b+c)(2c-a)}{b^2-bc+c^2}+\\frac{(c+a)(2a-b)}{c^2-ca+a^2}\\le 6\\]",
"content_html": "If <img src=\"//latex.artofproblemsolving.com... | If \(a,b,c>0\), prove that
\[
\frac{(a+b)(2b-c)}{a^2-ab+b^2}+\frac{(b+c)(2c-a)}{b^2-bc+c^2}+\frac{(c+a)(2a-b)}{c^2-ca+a^2}\le 6.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/AlgebraicIdentities/PolynomialIdentity",
"/Mathematics/Algebra/Polynomials/HomogeneousPolynomial",
"/Mathematics/Algebra/Polynomials/MultivariatePolynomial",
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Alg... | Expand the inequality, clear denominators and rewrite the resulting homogeneous polynomial as a sum of clearly non‑negative symmetric terms. | 272,120 | [
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"Algebra"
] | Let $x,\;y,\;z>0$. Prove that
\[
\frac{(x+y)(2y-z)}{x^{2}-xy+y^{2}}+\frac{(y+z)(2z-x)}{y^{2}-yz+z^{2}}+\frac{(z+x)(2x-y)}{z^{2}-zx+x^{2}}\le 6\,.
\] |
numina_10731840 | Assume without loss of generality that $x \geqslant y \geqslant z$, it is easy to see that
$$\begin{array}{l}
\frac{x^{k+1}}{x^{k+1}+y^{k}+z^{k}} \geqslant \frac{y^{k+1}}{y^{k+1}+z^{k}+x^{k}} \geqslant \frac{z^{k+1}}{z^{k+1}+x^{k}+y^{k}} \\
z^{k+1}+x^{k}+y^{k} \geqslant y^{k+1}+z^{k}+x^{k} \geqslant x^{k+1}+y^{k}+z^{k}... | \frac{1}{7} | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "inequalities"
} | Example 6.49 (2007 Serbia Mathematical Olympiad) $x, y, z>0, x+y+z=1$, prove that
$$\frac{x^{k+2}}{x^{k+1}+y^{k}+z^{k}}+\frac{y^{k+2}}{y^{k+1}+z^{k}+x^{k}}+\frac{z^{k+2}}{z^{k+1}+x^{k}+y^{k}} \geqslant \frac{1}{7}$$ | [
"/Mathematics/Algebra/Sums/PowerSum",
"/Mathematics/Algebra/Sums/Sum"
] | Order the variables and apply Chebyshev's inequality to the paired sequences to bound the sum from below | 54,249 | [
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"Algebra"
] | Three explorers—Ada, Ben, and Cara—divide a whole unit of treasure into positive shares $p$, $q$, $r$ satisfying $p+q+r=1$. Let $m$ be a fixed non‑negative integer. Prove that
\[
\frac{p^{\,m+2}}{p^{\,m+1}+q^{\,m}+r^{\,m}}+\frac{q^{\,m+2}}{q^{\,m+1}+r^{\,m}+p^{\,m}}+\frac{r^{\,m+2}}{r^{\,m+1}+p^{\,m}+q^{\,m}}\ge\fra... |
aops_2813730 | Let me try.
$x_1+ x_2+ x_3=9 ... \Large{ \textcircled{1}} $
$ x_1 x_2 x_3=15 ... \Large{ \textcircled{2}} $
$ x_1- x_2= x_2- x_3 ... \Large{ \textcircled{3}} $
Insert $ \Large{ \textcircled{3}}$ into $ \Large{ ... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "【National College Entrance Exam， old China】A$\\Sigma$-1\nSolve：\n$x^3-9x^2+23x-15=0$\nand, $ x_1- x_2=x_2-x_3$",
"content_html": "【National College Entrance Exam， old China】<span style=\"white-spac... | 【National College Entrance Exam， old China】A$\Sigma$-1
Solve：
$x^3-9x^2+23x-15=0$
and, $ x_1- x_2=x_2-x_3$ | [
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation",
"/Mathematics/Algebra/AlgebraicEquations/CubicEquation",
"/Mathematics/Algebra/Polynomials/CubicEquation",
"/Mathematics/Algebra/Polynomials/CubicPolynomial",
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Algebra/Polynomials/Polyn... | Recognize the roots form an arithmetic progression, then use Vieta’s sum to determine the middle root. | 216,619 | [
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"Algebra"
] | A team of astronomers has discovered three exoplanets whose orbital periods (in Earth‑days) are denoted by \(p\), \(q\) and \(r\). These periods satisfy the cubic equation
\[
z^{3}-9z^{2}+23z-15=0,
\]
and they are evenly spaced, i.e. the difference between the longest and the intermediate period equals the differen... |
numina_10051913 | ## Solution.
$\sin ^{6} \alpha+\cos ^{6} \alpha=\sin ^{4} \alpha-\sin ^{2} \alpha \cdot \cos ^{2} \alpha+\cos ^{4} \alpha=1-3 \sin ^{2} \alpha \cdot \cos ^{2} \alpha=$ $=1-\frac{3}{4} \sin ^{2} 2 \alpha=\frac{1+3 \cos ^{2} 2 \alpha}{4} \Rightarrow A=\frac{4}{1+3 \cos ^{2} 2 \alpha}$.
From this, it is clear that $A$ t... | 4 | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | ### 3.485 Find the maximum value of the expression
$$
A=\frac{1}{\sin ^{6} \alpha+\cos ^{6} \alpha} \text { for } 0 \leq \alpha \leq \frac{\pi}{2}
$$ | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity"
] | Rewrite sin⁶α+cos⁶α via double-angle identities to get (1+3·cos²2α)/4 and maximize the reciprocal by setting cos2α=0. | 21,284 | [
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"Algebra"
] | In a photonics lab, the combined signal strength $S$ of two coherent laser beams intersecting at an angle $\varphi$ is described by
\[
S=\sin^{6}\!\varphi+\cos^{6}\!\varphi .
\]
The engineer defines the inverse‑signal factor
\[
R=\frac{1}{\sin^{6}\!\varphi+\cos^{6}\!\varphi}
\]
and wants to know its largest pos... |
numina_10050744 | Solution.
Rewrite this system in the form
$\left\{\begin{array}{l}(x-y)(x-y)(x+y)=45, \\ x+y=5\end{array} \Leftrightarrow\left\{\begin{array}{l}(x-y)^{2}(x+y)=45, \\ x+y=5 .\end{array} \Rightarrow\right.\right.$
$\Rightarrow(x-y)^{2}=9$, from which $x-y=-3$ or $x-y=3$. We obtain a combination of two systems:
1) $\l... | (4,1),(1,4) | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 6.086. $\left\{\begin{array}{l}(x-y)\left(x^{2}-y^{2}\right)=45, \\ x+y=5\end{array}\right.$
Solve the system of equations:
\[
\left\{\begin{array}{l}
(x-y)\left(x^{2}-y^{2}\right)=45, \\
x+y=5
\end{array}\right.
\] | [
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation",
"/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations",
"/Mathematics/Algebra/Polynomials/Factorization",
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Algebra/Polynomials/PolynomialEquation"
] | Factor x^2-y^2 as (x-y)(x+y) and substitute the given sum to obtain a simple equation for (x-y)^2. | 20,989 | [
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"Algebra"
] | In a mystical quest, two treasure hunters, Aria and Brin, each collect a pile of enchanted stones. Let $a$ be the number of ruby stones Aria finds and let $b$ be the number of sapphire stones Brin discovers. Their hauls satisfy the following conditions:
\[
\begin{cases}
(a-b)\bigl(a^{2}-b^{2}\bigr)=45,\\[4pt]
a+b=5.
\... |
ours_4129 | Let \( a+b+c=ab+bc+ca=k \). Since \((a+b+c)^{2} \geq 3(ab+bc+ca)\), we have \( k^{2} \geq 3k \). Since \( k>0 \), it follows that \( k \geq 3 \).
We have \( bc \geq ca \geq ab \), so from the above relation, we deduce that \( bc \geq 1 \).
By AM-GM, \( b+c \geq 2\sqrt{bc} \) and consequently \( b+c \geq 2 \). The equa... | null | {
"competition": "bmo",
"dataset": "Ours",
"posts": null,
"source": "2019_bmo_shortlist-2.md"
} | Let \( a, b, c \) be real numbers such that \( 0 \leq a \leq b \leq c \). Prove that if
\[
a+b+c=ab+bc+ca>0,
\]
then \(\sqrt{bc}(a+1) \geq 2\). When does the equality hold? | [
"/Mathematics/Algebra/AlgebraicIdentities"
] | Rewrite the given equality to express a in terms of b and c, then bound bc and b+c using ordering and AM‑GM to obtain sqrt(bc)(a+1)≥2. | 56,770 | [
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"Algebra"
] | In a laboratory experiment three reagents have concentrations \(x, y, z\) (real numbers) that satisfy
\[
0\le x\le y\le z .
\]
If these concentrations obey
\[
x+y+z = xy+yz+zx \;>\;0,
\]
prove that
\[
\sqrt{yz}\,(x+1)\ge 2 .
\]
Determine all triples \((x,y,z)\) for which equality occurs. |
numina_10094681 | Analyzing the given two expressions, which are symmetric with respect to $x, y, z$, we can assume $x \leqslant y \leqslant z$.
Solution: Without loss of generality, let $x \leqslant y \leqslant z$, then
$$
\begin{aligned}
& x^{3}+y^{3}+z^{3}-x^{2}(y+z)-y^{2}(z+x)-z^{2}(x+y)+3 x y z \\
= & x^{3}-x^{2}(y+z)+x y z+y^{3}+z... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | Example 7 Let $x, y, z \in \mathbf{R}^{\prime}$, compare $x^{3}+y^{3}+z^{3}+3 x y z$ with $x^{2}(y+z)+y^{2}(z+x)+z^{2}(x+y)$. | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/AlgebraicIdentities/PolynomialIdentity",
"/Mathematics/Algebra/Polynomials/HomogeneousPolynomial",
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Algebra/Polynomials/SymmetricPolynomial"
] | Rewrite the difference using Schur’s inequality, factoring it into a sum of non‑negative terms such as x(x−y)(x−z)+(y−z)^2(y+z−x). | 125,311 | [
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"Algebra"
] | Let $U$, $V$, $W$ be three positive real numbers representing the thrust (in newtons) of three rockets. Compare the magnitudes of the two expressions
\[
U^{3}+V^{3}+W^{3}+3UVW
\qquad\text{and}\qquad
U^{2}(V+W)+V^{2}(W+U)+W^{2}(U+V).
\] |
numina_10180045 | ## Solution:
Applying the inequality of means, we have:
$a^{2}+b c \geq 2 \sqrt{a^{2} b c} \quad \frac{1}{a^{2}+b c} \leq \frac{1}{2 \sqrt{a^{2} b c}} \quad \frac{a}{a^{2}+b c} \leq \frac{1}{2 \sqrt{b c}}$
$b^{2}+a c \geq 2 \sqrt{b^{2} a c} \quad \frac{1}{b^{2}+a c} \leq \frac{1}{2 \sqrt{b^{2} a c}} \quad \frac{b}{b... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | ## Problem 1
Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be strictly positive real numbers. Show that:
$\frac{\mathrm{a}}{\mathrm{a}^{2}+\mathrm{bc}}+\frac{\mathrm{b}}{\mathrm{b}^{2}+\mathrm{ac}}+\frac{\mathrm{c}}{\mathrm{c}^{2}+\mathrm{ab}} \leq \frac{1}{2}\left(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{~b}}+\frac{1}{\math... | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity"
] | Apply AM‑GM to bound a/(a^2+bc) by 1/(2√{bc}) and then use AM‑GM again to replace √{bc} terms with (1/2)(1/b+1/c). | 135,700 | [
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"Algebra"
] | Imagine three interstellar fuel tanks aboard a star‑cruiser, each holding a strictly positive real amount of energy denoted by $a$, $b$, and $c$. Demonstrate that
\[
\frac{a}{a^{2}+bc}+\frac{b}{b^{2}+ac}+\frac{c}{c^{2}+ab}\le\frac12\left(\frac1a+\frac1b+\frac1c\right).
\]
(Problem adapted from the 2025 Galactic Mathe... |
aops_436106 | [hide="Solution"]
If a quadratic has real roots, then its discriminant must be nonnegative. [hide="Reasoning"]
The quadratic formula is $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$. Its discriminant, $b^2-4ac$, is located inside the square root. What happens if the discriminant is negative?[/hide]
The discriminant of $x^2+bx+... | null | {
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"dataset": "AOPS",
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{
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"content_bbcode": "If $x^2+bx+16$ has real roots, find all possible values of $b$. Express your answer in interval notation.",
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"/Mathematics/... | Require the discriminant to be nonnegative for a quadratic to have real roots. | 253,948 | [
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"Algebra"
] | An engineer models the bending stress \(S\) (in MPa) of a beam as a quadratic function of a load parameter \(t\):
\[
S(t)=t^{2}+mt+16 .
\]
For the beam to experience zero stress at some real load values, the equation \(S(t)=0\) must have real solutions. Determine all real numbers \(m\) that make this possible, and ... |
numina_10048390 | ## Solution.
Domain of definition: $x^{2}-4 \geq 0 \Leftrightarrow x \in(-\infty ;-2] \cup[2 ; \infty)$.
Let's write the equation in the form $2^{x+\sqrt{x^{2}-4}}-\frac{5}{2} \cdot 2^{\frac{x+\sqrt{x^{2}-4}}{2}}-6=0$. Solving it as a quadratic equation in terms of $2^{\frac{x+\sqrt{x^{2}-4}}{2}}$, we get $2^{\frac{x... | \frac{5}{2} | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 7.215. $2^{x+\sqrt{x^{2}-4}}-5 \cdot(\sqrt{2})^{x-2+\sqrt{x^{2}-4}}-6=0$. | [
"/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation",
"/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula"
] | Rewrite everything as powers of 2 and let y = 2^{(x+√(x^2−4))/2}, turning the equation into a quadratic in y. | 20,482 | [
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"Algebra"
] | A sorceress named Selene is adjusting the power level $p$ of a protective rune. The rune’s stability is governed by the equation
\[
2^{\,p+\sqrt{p^{2}-4}}\;-\;5\cdot(\sqrt{2})^{\,p-2+\sqrt{p^{2}-4}}\;-\;6\;=\;0.
\]
Determine every real value of $p$ for which the rune remains stable. |
aops_80134 | [quote="Werwulff"][quote="nthd"]Prove that$\forall x,y,z \ge 0$
$\sum(\sqrt{x^2+yz}-\sqrt{y^2+zx})^2\le\sum(x-y)^2$
:) :)[/quote]
It is trivial and thus boring to show that this inequality is equivalent to
\[ \sum 2xy\leq\sum\sqrt{(x^2+yz)(y^2+zx)} \]
which can easily be proved using cauc... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Prove that$\\forall x,y,z \\ge 0$\r\n $\\sum(\\sqrt{x^2+yz}-\\sqrt{y^2+zx})^2\\le\\sum(x-y)^2$\r\n :) :)",
"content_html": "Prove tha<span style=\"white-space:nowrap;\">t<... | Prove that for all nonnegative real numbers \(x,y,z\),
\[
\sum_{\text{cyc}}\bigl(\sqrt{x^2+yz}-\sqrt{y^2+zx}\bigr)^2 \le \sum_{\text{cyc}}(x-y)^2.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Sums/Sum"
] | Rewrite the inequality as a sum of square‑root products and then bound those terms using Cauchy–Schwarz together with AM‑GM. | 279,208 | [
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"Algebra"
] | Consider three energy reactors that produce non‑negative real power levels \(p\), \(q\) and \(r\). Prove that
\[
\bigl(\sqrt{p^{2}+qr}-\sqrt{q^{2}+rp}\bigr)^{2}
+\bigl(\sqrt{q^{2}+rp}-\sqrt{r^{2}+pq}\bigr)^{2}
+\bigl(\sqrt{r^{2}+pq}-\sqrt{p^{2}+qr}\bigr)^{2}
\le (p-q)^{2}+(q-r)^{2}+(r-p)^{2}.
\] |
numina_10234387 | 1. **Denote the sums for convenience:**
\[
\sum_{j=1}^{m} a_{ij} = x_i \quad \text{and} \quad \sum_{i=1}^{n} a_{ij} = y_j
\]
Then, the function \( f \) can be rewritten as:
\[
f = \frac{n \sum_{i=1}^{n} x_i^2 + m \sum_{j=1}^{m} y_j^2}{\left( \sum_{i=1}^{n} \sum_{j=1}^{m} a_{ij} \right)^2 + mn \sum_{i=... | \frac{m+n}{mn+n} | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "aops_forum"
} | For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where
\[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\su... | [
"/Mathematics/Algebra/Sums/PowerSum",
"/Mathematics/Algebra/Sums/Sum"
] | Apply Karamata's inequality (majorization of the convex square function) to bound the numerator, and use Cauchy-Schwarz for the lower bound. | 106,988 | [
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"Algebra"
] | Let \(p\) and \(q\) be integers greater than \(1\). In a cloud‑storage network there are \(q\) data centers, each holding data of \(p\) different categories. Denote by \(d_{ij}\ge 0\) the amount of data of category \(j\) stored in data center \(i\) \((i=1,\dots ,q;\;j=1,\dots ,p)\), and assume that not all \(d_{ij}\) a... |
aops_575416 | [quote="hungkg"]Let $a,b,c$ be nonnegative real numbers such as $ab+bc+ca=1$. Prove that
\[\frac{1}{{\sqrt {{a^2} + {b^2}} }} + \frac{1}{{\sqrt {{b^2} + {c^2}} }} + \frac{1}{{\sqrt {{c^2} + {a^2}} }} \ge 2 + \frac{1}{{\sqrt 2 }}.\][/quote]
[b]This is my proof for problem[/b]
Without of generality, we can assume that $... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let $a,b,c$ be nonnegative real numbers such as $ab+bc+ca=1$. Prove that \n\\[\\frac{1}{{\\sqrt {{a^2} + {b^2}} }} + \\frac{1}{{\\sqrt {{b^2} + {c^2}} }} + \\frac{1}{{\\sqrt {{c^2} + {a^2}} }} \\ge 2 + \\f... | Let \(a,b,c\) be nonnegative real numbers such that \(ab+bc+ca=1\). Prove that
\[
\frac{1}{\sqrt{a^2+b^2}}+\frac{1}{\sqrt{b^2+c^2}}+\frac{1}{\sqrt{c^2+a^2}}\ge 2+\frac{1}{\sqrt2}.
\] | [
"/Mathematics/Algebra/GeneralAlgebra/Algebra"
] | Rewrite each term using ab+bc+ca=1 as √((ab+bc+ca)/(a²+b²)), bound it by ratios of a+c and b+c, and reduce to a single‑variable inequality in u. | 272,051 | [
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"Algebra"
] | In a next‑generation energy station, three non‑negative real readings $p$, $q$, and $r$ (measured in arbitrary units) satisfy the relation $pq+qr+rp=1$. Prove that
\[
\frac{1}{\sqrt{p^{2}+q^{2}}}\;+\;\frac{1}{\sqrt{q^{2}+r^{2}}}\;+\;\frac{1}{\sqrt{r^{2}+p^{2}}}\;\ge\;2\;+\;\frac{1}{\sqrt{2}}\,.
\] |
aops_277949 | [quote="quykhtn-qa1"]Let $ a,b,c$ be positive numbers such that: $ a \plus{} b \plus{} c \equal{} 1$.Prove that:
$ \frac {a}{b^2 \plus{} b} \plus{} \frac {b}{c^2 \plus{} c} \plus{} \frac {c}{a^2 \plus{} a} \ge \frac {36(a^2 \plus{} b^2 \plus{} c^2)}{ab \plus{} bc \plus{} ca \plus{} 5}$[/quote]
hello,i can prove your p... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let $ a,b,c$ be positive numbers such that: $ a\\plus{}b\\plus{}c\\equal{}1$.Prove that:\r\n$ \\frac{a}{b^2\\plus{}b}\\plus{}\\frac{b}{c^2\\plus{}c}\\plus{}\\frac{c}{a^2\\plus{}a} \\ge \\frac{36(a^2\\plus{... | Let \(a,b,c\) be positive numbers such that \(a+b+c=1\). Prove that
\[
\frac{a}{b^2+b}+\frac{b}{c^2+c}+\frac{c}{a^2+a}\ge \frac{36(a^2+b^2+c^2)}{ab+bc+ca+5}.
\] | [
"/Mathematics/Algebra"
] | Apply Cauchy–Schwarz (Titu’s lemma) to bound the sum by (a+b+c)^2/(∑ab+∑b^2a) and then use the inequality a^2b+b^2c+c^2a+abc ≤ 4/27 to estimate the denominator. | 215,484 | [
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"Algebra"
] | Let $x,\;y,\;z$ be positive real numbers satisfying $x+y+z=1$. Show that
\[
\frac{x}{y^{2}+y}+\frac{y}{z^{2}+z}+\frac{z}{x^{2}+x}\ge
\frac{36\bigl(x^{2}+y^{2}+z^{2}\bigr)}{xy+yz+zx+5}\,.
\] |
numina_10103728 | [Proof] Let $d=u_{k}-u_{k-1}$ be the common difference of the arithmetic sequence, then
$$
\begin{aligned}
t_{n} & =\frac{\sqrt{u_{2}}-\sqrt{u_{1}}}{u_{2}-u_{1}}+\frac{\sqrt{u_{3}}-\sqrt{u_{2}}}{u_{3}-u_{2}}+\cdots+\frac{\sqrt{u_{n}}-\sqrt{u_{n-1}}}{u_{n}-u_{n-1}} \\
& =\frac{\sqrt{u_{n}}-\sqrt{u_{1}}}{d} . \\
& =\frac... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | $3 \cdot 49$ Positive numbers $u_{1}, u_{2}, \cdots, u_{n}$ form an arithmetic sequence, prove:
$$
\begin{aligned}
t_{n} & =\frac{1}{\sqrt{u_{1}}+\sqrt{u_{2}}}+\frac{1}{\sqrt{u_{2}}+\sqrt{u_{3}}}+\cdots+\frac{1}{\sqrt{u_{n-1}}+\sqrt{u_{n}}} \\
& =\frac{n-1}{\sqrt{u_{1}}+\sqrt{u_{n}}} .
\end{aligned}
$$ | [
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"/Mathematics/Algebra/Sums/Sum",
"/Mathematics/Algebra/Sums/TelescopingSeries",
"/Mathematics/Algebra/Sums/TelescopingSum"
] | Rationalize each term to a difference of square roots over the constant difference, creating a telescoping sum. | 126,483 | [
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"Algebra"
] | **Problem 7.** An astrophysicist measures the luminosities of $n$ consecutive stars in a straight‑line cluster. Let the recorded luminosities be positive real numbers $L_1, L_2, \dots, L_n$, and assume they increase by a constant amount, i.e. $\{L_k\}_{k=1}^n$ forms an arithmetic progression. Prove that
\[
S_n \;=\; \... |
aops_3202306 | [quote=williamxiao][hide=sol?]Subtract equation 1 from equation 2 to get $(y+x)(y-x) + (y-x)z = 1, (x+y+z)(y-x) = 1$
Same with 3 and 2, $(y+z)(y-z) + (y-z)x = 1$, $(y+z+x)(y-z) = 1$
So $(x+y+z)(y-x) = (x+y+z)(y-z)$. That means $(y-x) = (y-z)$, so $x = z$.
We then have $(2x+y)(y-x) = 1$. Next, we have by substituting x... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Find $x^{2}+y^{2}+z^{2}$ given\n$$\\begin{cases}\nx^{2}-yz=1 & \\\\\ny^{2}-xz=2 & \\\\\nz^{2}-xy=1\n\\end{cases}$$\n\n\n",
"content_html": "Find <img src=\"//latex.artofproblemsolving.com/e/e/f/eefec... | Find \(x^{2}+y^{2}+z^{2}\) given
\[
\begin{cases}
x^{2}-yz=1,\\[4pt]
y^{2}-xz=2,\\[4pt]
z^{2}-xy=1.
\end{cases}
\] | [
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation",
"/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations",
"/Mathematics/Algebra/GeneralAlgebra/Algebra",
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Algebra/Polynomials/PolynomialEquation"
] | Subtract the equations and factor each as (x+y+z)·(difference of variables) to conclude x = z. | 229,849 | [
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"Algebra"
] | In a secret laboratory, three experimental reactors have power levels denoted by the real numbers \(R\), \(S\), and \(T\). Their performance parameters satisfy the system
\[
\begin{cases}
R^{2} - S\,T = 1,\\[4pt]
S^{2} - R\,T = 2,\\[4pt]
T^{2} - R\,S = 1.
\end{cases}
\]
Determine the total squared power output \(R^... |
aops_1202710 | [quote=daisyxixi]
Let a, b and c be positive real such that $ab+bc+ca=2$.
Prove $abc(a+b+c+9abc)\le 4$.
[/quote]
Or
$abc(a+b+c)\le \frac{(ab+bc+ca)^2}{3} \ \ \Rightarrow \ \ abc(a+b+c)\le \frac{4}{3}$
$3\sqrt[3]{a^2b^2c^2}\le ab+bc+ca \ \ \Rightarrow \ \ 9a^2b^2c^2\le \frac{8}{3}$
$abc(a+b+c+9abc)= abc(a+b+c)+9a... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Hi all,\n\nHere is another inequality that I failed to prove...\n\nI want to ask if it is true that we have to try the problem by expressing the left side of the inequality as only the variable $abc$?\n\nP... | Let \(a,b,c>0\) satisfy \(ab+bc+ca=2\). Prove
\[
abc\bigl(a+b+c+9abc\bigr)\le 4.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/GeneralAlgebra/Algebra"
] | Separate the expression into abc(a+b+c) and 9a^2b^2c^2 and bound each term using the condition ab+bc+ca=2 with AM‑GM (or Cauchy‑Schwarz) inequalities. | 148,940 | [
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"Algebra"
] | Three autonomous drones have positive performance indices \(p,\;q,\;r\). Their calibration satisfies the relation
\[
pq+qr+rp=2 .
\]
Show that the combined metric
\[
pqr\bigl(p+q+r+9pqr\bigr)
\]
does not exceed \(4\). |
ours_18647 | Solution: We use the identities \(\sin^2 x + \cos^2 x = 1\) and \(\cot^2 x + 1 = \csc^2 x\) to simplify the equation. Substituting these identities, the equation becomes:
\[
1 + \csc^2 x = \csc^2 x + \sec^2 x
\]
This simplifies to:
\[
1 = \sec^2 x
\]
Thus, \(\cos^2 x = \frac{1}{2}\), which implies \(\cos x = \pm \f... | \frac{\pi}{4}, \frac{3\pi}{4} | {
"competition": "jhmt",
"dataset": "Ours",
"posts": null,
"source": "AlgebraKey2006.md"
} | Find all \(x\) in \([0, \pi]\) inclusive so that
\[
\sin^2 x + \csc^2 x + \cos^2 x = \cot^2 x + \sec^2 x
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity"
] | Use the Pythagorean identities to rewrite the equation, replace cot² with csc²−1 and apply sin²+cos²=1 to isolate sec²=2. | 6,585 | [
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"Algebra"
] | In a spacecraft navigation system, the pitch angle \(\varphi\) (measured in radians) is restricted to the closed interval \(0\le\varphi\le\pi\). For the system to operate correctly, this angle must satisfy
\[
\sin^{2}\varphi+\csc^{2}\varphi+\cos^{2}\varphi=\cot^{2}\varphi+\sec^{2}\varphi .
\]
Find all values of \(\... |
numina_10089611 | ## Solution.
Note that $x$ must be different from zero for all expressions to make sense. 1 point
By moving all addends to the same side, we have
$$
\frac{x-m}{x^{2}}-2 \cdot \frac{x-m}{x}+x-m \geqslant 0
$$
1 point
Further simplification yields
$$
(x-m)\left(\frac{1}{x^{2}}-\frac{2}{x}+1\right) \geqslant 0
$$
o... | x\in\begin{cases}{1}\cup[,+\infty)& | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | ## Task A-1.1.
Depending on the real parameter $m$, determine for which real numbers $x$ the following inequality holds:
$$
\frac{x-m}{x^{2}}+x \geqslant 2\left(1-\frac{m}{x}\right)+m
$$ | [
"/Mathematics/Algebra"
] | Factor the inequality into (x-m)*(1/x-1)^2 ≥ 0 and use the nonnegativity of the square. | 87,497 | [
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"Algebra"
] | A railway engineer is analyzing the performance of a locomotive. The speed $v$ (in km/h) of the train and a design constant $k$ are related by the inequality
\[
\frac{v-k}{v^{2}}+v \ge 2\!\left(1-\frac{k}{v}\right)+k .
\]
For a given real constant $k$, determine all real speeds $v$ (with $v\neq0$) for which the ine... |
aops_281664 | [quote="ifai"]Suppose that $ \{a_n\}$ and $ \{b_n\}$ are two arithmetic sequences such that
$ \frac {S_n}{T_n} \equal{} \frac {3n \plus{} 2}{4n \minus{} 5},$
where $ S_n \equal{} a_1 \plus{} \cdots \plus{} a_n$ and $ T_n \equal{} b_1 \plus{} \cdots \plus{} b_n$. Find $ \frac {a_{2009}}{b_{2009}}$.[/quote]
$ S_{401... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Suppose that $ \\{a_n\\}$ and $ \\{b_n\\}$ are two arithmetic sequences such that\r\n\r\n$ \\frac{S_n}{T_n}\\equal{}\\frac{3n\\plus{}2}{4n\\minus{}5},$\r\n\r\nwhere $ S_n\\equal{}a_1\\plus{}\\cdots\\plus{}... | Suppose that \( \{a_n\}\) and \( \{b_n\}\) are two arithmetic sequences such that
\[
\frac{S_n}{T_n}=\frac{3n+2}{4n-5},
\]
where \(S_n=a_1+\cdots+a_n\) and \(T_n=b_1+\cdots+b_n\). Find \(\dfrac{a_{2009}}{b_{2009}}\). | [
"/Mathematics/Algebra/Sums/CumulativeSum",
"/Mathematics/Algebra/Sums/PartialSum",
"/Mathematics/Algebra/Sums/RunningSum",
"/Mathematics/Algebra/Sums/Sum"
] | Use the fact that the sum of an odd number of terms of an arithmetic progression equals the number of terms times its middle term. | 216,738 | [
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"Algebra"
] | Two explorers, Maya and Leo, are logging the number of ancient coins they uncover in successive dig levels.
The counts Maya records form an arithmetic progression \(\{m_n\}\), and Leo’s counts form another arithmetic progression \(\{l_n\}\).
Define \(M_n=m_1+m_2+\dots+m_n\) as Maya’s cumulative total after \(n\) l... |
aops_174069 | [quote="ElChapin"]For positive $ x_i$ such that $ \sum_{i \equal{} 1}^{n}{\sqrt {x_i}} \equal{} 1$
Prove that the following holds:
$ \sum_{i \equal{} 1}^{n}{x_i^2}\ge(\sum_{i \equal{} 1}^{n}{x_1})^3$
:)[/quote]
I think:(Using Holder)
$ (\sum_{i \equal{} 1}^{n}{x_i^2})(\sum_{i \equal{} 1}^{n}{\sqrt{x... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "For positive $ x_i$ such that $ \\sum_{i \\equal{} 1}^{n}{\\sqrt {x_i}} \\equal{} 1$\r\n\r\n\r\nProve that the following holds:\r\n\r\n $ \\sum_{i \\equal{} 1}^{n}{x_i^2}\\ge(\\sum_{i \\equal{} ... | For positive real numbers \(x_i\) such that \(\sum_{i=1}^n \sqrt{x_i}=1\), prove that
\[
\sum_{i=1}^n x_i^2 \ge \biggl(\sum_{i=1}^n x_i\biggr)^3.
\] | [
"/Mathematics/Algebra/Sums/PowerSum",
"/Mathematics/Algebra/Sums/Sum"
] | Apply Hölder's inequality to the three sequences x_i^2, sqrt(x_i), sqrt(x_i) and use that the sum of the square roots equals 1. | 176,766 | [
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"Algebra"
] | A squadron consists of $N$ starships. Let $a_i>0$ denote the quantity of exotic fuel stored on the $i$‑th starship (measured in arbitrary units). It is known that the square roots of these quantities add up to one:
\[
\sum_{i=1}^{N}\sqrt{a_i}=1.
\]
Prove that the sum of the squares of the fuel quantities is at least ... |
aops_120113 | [quote="Beat"]$x,y,z>0$, $x+y+z=1$ : $\sum\frac{\sqrt{xy}}{z(\frac{1}{z}+1)}\le \frac{3}{4}$[/quote]
Let $x=a^{2},$ $y=b^{2}$ and $z=c^{2},$ where $a,$ $b$ and $c$ are positive numbers.
Hence, $\sum\frac{\sqrt{xy}}{z(\frac{1}{z}+1)}\le \frac{3}{4}\Leftrightarrow\sum_{cyc}\left(\frac{1}{4}-\frac{ab}{c^{2}+1}\right)\ge... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "$x,y,z>0$, $x+y+z=1$ : $\\sum\\frac{\\sqrt{xy}}{z(\\frac{1}{z}+1)}\\le \\frac{3}{4}$",
"content_html": "<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/e/5/5e5616447e... | Let \(x,y,z>0\) with \(x+y+z=1\). Prove that
\[
\sum_{\text{cyc}}\frac{\sqrt{xy}}{z\bigl(\frac{1}{z}+1\bigr)}\le \frac{3}{4}.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Sums/Sum"
] | Set x=a^2, y=b^2, z=c^2 so each denominator becomes c^2+1, then rewrite the inequality as a sum of squares using a^2+b^2+c^2≥ab+bc+ca. | 148,849 | [
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"Algebra"
] | Three distinct power modules—Alpha, Beta, and Gamma—are loaded onto a spacecraft with positive masses \(a, b, g\) satisfying \(a+b+g=1\). Show that
\[
\frac{\sqrt{ab}}{g\!\left(\frac{1}{g}+1\right)}\;+\;\frac{\sqrt{bg}}{a\!\left(\frac{1}{a}+1\right)}\;+\;\frac{\sqrt{ga}}{b\!\left(\frac{1}{b}+1\right)}\;\le\;\frac{3}... |
numina_10085103 | ## Solution:
$$
\begin{aligned}
& a+b c=a(a+b+c)+b c=(a+b)(a+c) . \text { Similarly, } b+c a=(b+c)(b+a) \text { and } c+a b= \\
& (c+a)(c+b) . \text { Then: } \sqrt{a+b c}+\sqrt{b+c a}+\sqrt{c+a b}=\sqrt{(a+b)(a+c)}+\sqrt{(b+c)(b+a)} \\
& +\sqrt{(c+a)(c+b)} \leq \frac{a+b+a+c}{2}+\frac{b+c+b+a}{2}+\frac{c+a+c+b}{2}=\f... | =b==\frac{1}{3} | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 8.8. Let $a, b, c$ be positive real numbers such that $a+b+c=1$. Prove the inequality
$$
\sqrt{a+b c}+\sqrt{b+c a}+\sqrt{c+a b} \leq 2
$$
Find the numbers $a, b, c$ for which equality holds. | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/AlgebraicOperations/GeneralAlgebraicOperations/BinaryOperation",
"/Mathematics/Algebra/AlgebraicOperations/GeneralAlgebraicOperations/BinaryOperator",
"/Mathematics/Algebra/GeneralAlgebra/Algebra"
] | Rewrite each radicand as (a+b)(a+c) using a+b+c=1, then apply √(xy) ≤ (x+y)/2 to each term. | 86,775 | [
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"Algebra"
] | Let $u, v, w$ be positive real numbers representing the fractions of a day allocated to three different activities, and suppose $u+v+w=1$. Prove that
\[
\sqrt{\,u+vw\,}+\sqrt{\,v+wu\,}+\sqrt{\,w+uv\,}\le 2 .
\]
Determine all triples $(u,v,w)$ for which equality occurs. |
numina_10078672 | First, we use the arithmetic-geometric mean inequality on each term to simplify the overly dubious denominator of our fractions:
$$
\frac{a^{2}}{\frac{b+c}{2}+\sqrt{b c}}+\frac{b^{2}}{\frac{c+a}{2}+\sqrt{c a}}+\frac{c^{2}}{\frac{a+b}{2}+\sqrt{a b}} \geq \frac{a^{2}}{b+c}+\frac{b^{2}}{c+a}+\frac{c^{2}}{a+b}
$$
It rema... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | Let $a, b, c$ be three positive real numbers such that $a+b+c=1$.
Show that
$$
\frac{a^{2}}{\frac{b+c}{2}+\sqrt{b c}}+\frac{b^{2}}{\frac{c+a}{2}+\sqrt{c a}}+\frac{c^{2}}{\frac{a+b}{2}+\sqrt{a b}} \geq \frac{1}{2}
$$
## 2 Solution | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/AlgebraicOperations",
"/Mathematics/Algebra/Sums/Sum"
] | Use AM‑GM to bound (b+c)/2+√bc by b+c, then apply Cauchy–Schwarz (Engel/Titu) to the sum of a²/(b+c). | 122,556 | [
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"Algebra"
] | Three astronauts—Alex, Blake, and Casey—share one unit of anti‑gravity fuel among the three propulsion pods of their spacecraft. Let $x$, $y$, and $z$ (all positive) denote the fractions of fuel allocated to Alex's, Blake's, and Casey's pods, respectively, so that $x+y+z=1$. Prove that
\[
\frac{x^{2}}{\dfrac{y+z}{2}... |
aops_1702426 | By AM-GM,
\begin{align*}
\frac{4-a}{b}+\frac{4-b}{c}+\frac{4-c}{d}+\frac{4-d}{a}&=\frac{b+c+d}b+\frac{c+d+a}c+\frac{d+a+b}d+\frac{a+b+c}a\\
&=4+\frac cb+\frac db+\frac dc+\frac ac+\frac ad+\frac bd+\frac ba+\frac ca\\
&\ge4+8\\
&=12
\end{align*} | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "[b]Easy version:[/b]\nLet $a,b,c,d>0$ such that $a+b+c+d = 4$. Prove that:\n\\[ \\frac{4-a}{b}+\\frac{4-b}{c}+\\frac{4-c}{d}+\\frac{4-d}{a} \\geq 12 \\]\n[b]Stronger version:[/b]\nLet $a,b,c,d>0$ such that... | Easy version:
Let \(a,b,c,d>0\) such that \(a+b+c+d = 4\). Prove that
\[
\frac{4-a}{b}+\frac{4-b}{c}+\frac{4-c}{d}+\frac{4-d}{a} \ge 12.
\]
Stronger version:
Let \(a,b,c,d>0\) such that \(a+b+c+d = 4\). Prove that
\[
\frac{4-a}{b}+\frac{4-b}{c}+\frac{4-c}{d}+\frac{4-d}{a} +4abcd \ge 16.
\] | [
"/Mathematics/Algebra/Sums/Sum"
] | Rewrite each term using a+b+c+d=4 to get sums of reciprocal fractions, then apply AM‑GM (x/y + y/x ≥ 2) to bound them. | 174,565 | [
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"Algebra"
] | **Basic version.**
Four interstellar outposts—Alpha, Beta, Gamma, and Delta—receive positive amounts of antimatter, denoted respectively by \(p,\;q,\;r,\;s\). Their total antimatter supply equals four units, i.e. \(p+q+r+s=4\). Prove that
\[
\frac{4-p}{q}+\frac{4-q}{r}+\frac{4-r}{s}+\frac{4-s}{p}\ge 12 .
\]
**Str... |
ours_17179 | Using the AM-GM inequality, we have:
\[
\begin{aligned}
(n+k-1) x_{1}^{n} x_{2} \cdots x_{k} &\leq n x_{1}^{n+k-1} + x_{2}^{n+k-1} + \cdots + x_{k}^{n+k-1}, \\
(n+k-1) x_{1} x_{2}^{n} \cdots x_{k} &\leq x_{1}^{n+k-1} + n x_{2}^{n+k-1} + \cdots + x_{k}^{n+k-1}, \\
&\vdots \\
(n+k-1) x_{1} x_{2} \cdots x_{k}^{n} &\leq x... | null | {
"competition": "imo",
"dataset": "Ours",
"posts": null,
"source": "IMO1967SL.md"
} | Prove the inequality
\[ x_{1} x_{2} \cdots x_{k}\left(x_{1}^{n-1}+x_{2}^{n-1}+\cdots+x_{k}^{n-1}\right) \leq x_{1}^{n+k-1}+x_{2}^{n+k-1}+\cdots+x_{k}^{n+k-1}, \]
where \( x_{i} > 0 \) for \( i = 1, 2, \ldots, k \), \( k \in \mathbb{N} \), \( n \in \mathbb{N} \). | [
"/Mathematics/Algebra/Products/Product",
"/Mathematics/Algebra/Sums/PowerSum",
"/Mathematics/Algebra/Sums/Sum"
] | Apply the AM‑GM inequality to each term where one variable is raised to power n and the others to power 1, then sum the resulting inequalities. | 63,751 | [
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"Algebra"
] | Let \(P_{1},P_{2},\dots ,P_{m}\) be positive real numbers and let \(m,q\in\mathbb{N}\). Prove that
\[
P_{1}P_{2}\cdots P_{m}\Bigl(P_{1}^{\,q-1}+P_{2}^{\,q-1}+\cdots +P_{m}^{\,q-1}\Bigr)\le
P_{1}^{\,q+m-1}+P_{2}^{\,q+m-1}+\cdots +P_{m}^{\,q+m-1}\,.
\] |
ours_3184 | Solution: Use the fact that the summand can be expressed as a telescoping series:
\[
\cot ^{-1}(1+1 / k) - \cot ^{-1}(1+1 /(k-1))
\]
This telescoping nature simplifies the sum to:
\[
\cot ^{-1}(1+1 / n)
\]
Thus, the answer is \(\cot ^{-1}(1+1 / n)\). | \cot ^{-1}(1+1 / n) | {
"competition": "alg_misc",
"dataset": "Ours",
"posts": null,
"source": "Telescoping Sums and Products - Po-Shen Loh - MOP 2003.md"
} | Evaluate:
$$
\sum_{k=1}^{n} \cot ^{-1}\left(2 k^{2}\right)=\cot ^{-1}(1+1 / n)
$$ | [
"/Mathematics/Algebra/Sums/PartialSum",
"/Mathematics/Algebra/Sums/Sum",
"/Mathematics/Algebra/Sums/TelescopingSeries",
"/Mathematics/Algebra/Sums/TelescopingSum"
] | Rewrite each term as a difference of two consecutive arccot expressions, producing a telescoping sum. | 56,175 | [
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"Algebra"
] | Captain Vega of the interstellar fleet records the angular offset to each navigation beacon. For the $r$‑th beacon (with $r=1,2,\dots ,p$) the recorded offset is $\cot^{-1}\!\bigl(2\,r^{2}\bigr)$ radians. Determine the total accumulated offset after the $p$‑th beacon and express it as a single inverse‑cotangent express... |
aops_2156012 | We use the formula $d=rt.$ The distance that you travel is equal to your rate (basically your speed) times the time.
First, let's say that Sunny runs at a speed $r,$ so Moonbeam runs at a speed of $mr.$ Also, let's say that Sunny runs $d$ meters before Moonbeam catches up to him, which means that Moonbeam runs $d+h$ m... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Sunny runs at a steady rate, and Moonbeam runs $m$ times as fast, where $m$ is a number greater than $1$. If Moonbeam gives Sunny a headstart of $h$ meters, how many meters must Moonbeam run to overtake Su... | Sunny runs at a steady rate, and Moonbeam runs \(m\) times as fast, where \(m>1\). If Moonbeam gives Sunny a head start of \(h\) meters, how many meters must Moonbeam run to overtake Sunny? Express your answer in terms of \(h\) and \(m\). | [
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation",
"/Mathematics/Algebra/AlgebraicEquations/LinearEquation",
"/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations",
"/Mathematics/Algebra/RateProblems"
] | Equate the common time for both runners using distance = speed × time to relate their distances and speeds. | 194,983 | [
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"Algebra"
] | A jogger named Felix maintains a constant speed \(v\). His friend Greta runs \(k\) times as fast as Felix, where \(k>1\). Greta allows Felix a head start of \(a\) meters. How many meters must Greta run to catch up to Felix? Express your answer in terms of \(a\) and \(k\). |
ours_6920 | Let \( f(0)=a \), where \( a \in \mathbb{R} \). Setting \( x=0 \) in the condition, we obtain \( f(a)=0 \). For \( y=a \), we have
\[
f(f(x))=x f(a+1).
\]
Assume first that \( f(a+1) \neq 0 \). Then \( f \) is injective. Indeed, if we assume that \( f(x_1)=f(x_2) \) for \( x_1 \neq x_2 \) and substitute these values ... | null | {
"competition": "bulgarian_tsts",
"dataset": "Ours",
"posts": null,
"source": "KBOM-All-2024-9-12 кл-sol (1).md"
} | Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) for which the following condition holds:
\[
f(f(x)+x f(y))=x f(y+1), \forall x, y \in \mathbb{R}.
\] | [
"/Mathematics/Algebra/GeneralAlgebra/Algebra"
] | Plug y = f(0) to obtain f(f(x)) = c·x and split into the cases c ≠ 0 (injectivity forces f(x)=x) and c = 0 (gives the zero function). | 58,630 | [
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"Algebra"
] | In a clandestine research facility a scientist has devised a protocol $P:\mathbb R\to\mathbb R$ that assigns a real number to every real input. The protocol is required to obey the rule
\[
P\bigl(P(u)+u\,P(v)\bigr)=u\,P(v+1)\qquad\text{for all }u,v\in\mathbb R.
\]
Determine all possible protocols $P$ that satisfy thi... |
aops_1915627 | [quote=MessingWithMath]How about the other way around, flipping the inequality sign?[/quote]
well by AMGM $1=\sum x_i \geq 5\sqrt[5]{\prod x_i}\implies \prod x_i \leq \left(\frac{1}{5}\right )^5\leq \left(\frac{3}{10}\right)^{10}$ | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "If $x_1,x_2,x_3,x_4,x_5\\in\\mathbb R^+$ such that $\\sum_{i=1}^5x_i=1$. Then prove or disprove that\n\n$$\\prod_{i=1}^{5}x_i\\geq(\\frac{3}{5})^{10}$$\n\n@below that's why I said Prove or [b]disprove[/b] ... | If \(x_1,x_2,x_3,x_4,x_5\in\mathbb{R}^+\) satisfy \(\sum_{i=1}^5 x_i=1\). Prove or disprove that
\[
\prod_{i=1}^{5} x_i \ge \left(\frac{3}{5}\right)^{10}.
\] | [
"/Mathematics/Algebra/Products/Product",
"/Mathematics/Algebra/Sums/Sum"
] | Apply AM‑GM to the sum constraint to obtain an upper bound for the product, which contradicts the claimed lower bound. | 185,995 | [
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"Algebra"
] | Suppose five positive real numbers \(u, v, w, x, y\) (for instance, the quantities of a rare mineral extracted from five distinct mines) satisfy
\[
u+v+w+x+y = 1 .
\]
Determine whether the following statement holds:
\[
u\,v\,w\,x\,y \;\ge\; \Bigl(\frac{3}{5}\Bigr)^{10}.
\] |
aops_1773503 | $a^2+b^2+2 =\frac{(a^2+1)+(b^2+1)+(a^2+b^2)+2}{2}\geq a+b+ab+1=(a+1)(b+1)$
$\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2}+\sqrt{c^2+a^2+2}\geq \sqrt{(a+1)(b+1)}+\sqrt{(b+1)(c+1)}+\sqrt{(c+1)(a+1)} \geq 3\sqrt[6]{8(a+1)(b+1)(c+1)}=6$
| null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "prove\n√⟨a²+b²+2⟩+√⟨b²+c²+2⟩+√⟨c²+a²+2⟩≥6\nif\na+b+c+ab+bc+ca+abc=7",
"content_html": "prove<br>\n√⟨a²+b²+2⟩+√⟨b²+c²+2⟩+√⟨c²+a²+2⟩≥6<br>\nif<br>\na+b+c+ab+bc+ca+abc=7",
"post_id": 11650399,
... | prove
√⟨a²+b²+2⟩+√⟨b²+c²+2⟩+√⟨c²+a²+2⟩≥6
if
a+b+c+ab+bc+ca+abc=7 | [
"/Mathematics/Algebra/GeneralAlgebra/Algebra"
] | Rewrite the condition as (a+1)(b+1)(c+1)=8, bound each √(a²+b²+2) by √((a+1)(b+1)), then apply AM‑GM to the three terms. | 178,783 | [
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"Algebra"
] | In a far‑off star cluster, three planets have interaction parameters $p$, $q$, and $r$. These parameters satisfy
\[
p+q+r+pq+qr+rp+pqr = 7 .
\]
Prove that
\[
\sqrt{p^{2}+q^{2}+2}\;+\;\sqrt{q^{2}+r^{2}+2}\;+\;\sqrt{r^{2}+p^{2}+2}\;\ge\;6 .
\] |
aops_1137661 | [quote=checkmatetang][quote]Julius has some spare change consisting of quarters, dimes and nickels. If the ratio of quarters to dimes is 3:4 and the ratio of quarters to nickels is 4:5, what is the ratio of dimes to nickels? Express your answer as a common fraction. I don't really know how to do this problem, so can I ... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Julius has some spare change consisting of quarters, dimes and nickels. If the ratio of\n\nquarters to dimes is 3:4 and the ratio of quarters to nickels is 4:5, what is the ratio of dimes \n\nto nickels? E... | Julius has some spare change consisting of quarters, dimes, and nickels. If the ratio of quarters to dimes is \(3:4\) and the ratio of quarters to nickels is \(4:5\), what is the ratio of dimes to nickels? Express your answer as a common fraction.
When Mr. Tesla drives 60 mi/h, the commute from home to his office take... | [
"/Mathematics/Algebra/AlgebraicEquations/LinearEquation",
"/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations",
"/Mathematics/Algebra/GeneralAlgebra/Algebra",
"/Mathematics/Algebra/RateProblems"
] | Translate each given relationship into algebraic equations and solve the resulting system. | 144,337 | [
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"Algebra"
] | A. Sir Caldor has a stash of three kinds of ancient coins: gold, silver, and copper. The ratio of gold coins to silver coins is \(3:4\), and the ratio of gold coins to copper coins is \(4:5\). What is the ratio of silver coins to copper coins? Express your answer as a common fraction.
B. When Captain Mira pilots her h... |
numina_10100305 | 13
$$
\begin{aligned}
\frac{2 x y}{x+y-1} & =\frac{(x+y)^{2}-\left(x^{2}+y^{2}\right)}{x+y-1} \\
& =\frac{(x+y)^{2}-1}{x+y-1} \\
& =x+y+1 .
\end{aligned}
$$
Since
$$
\left(\frac{x+y}{2}\right)^{2} \leqslant \frac{x^{2}+y^{2}}{2}=\frac{1}{2},
$$
we have
$$
-\sqrt{2} \leqslant x+y \leqslant \sqrt{2}
$$
Thus,
$$
\frac{... | 1-\sqrt{2} | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 13 If the real numbers $x, y$ satisfy $x^{2}+y^{2}=1$, then the minimum value of $\frac{2 x y}{x+y-1}$ is $\qquad$
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity"
] | Rewrite the fraction using (x+y)^2 - (x^2+y^2) to simplify to x+y+1, then bound x+y via the given circle constraint. | 29,005 | [
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"Algebra"
] | Problem 8. In a robotics calibration task, the two real parameters $u$ and $v$ satisfy $u^{2}+v^{2}=1$. Determine the smallest possible value of \(\displaystyle\frac{2uv}{u+v-1}\), which is \(\qquad\). |
aops_3328316 | [hide=sol] Plugging in $x=y=0$ into the second property, we have $f(0)=\dfrac{2f(0)}{1+(f(0))^2}.$ Note $f(0)\neq 0,$ so $1=\dfrac{2}{1+(f(0))^2},$ and $(f(0))^2=1.$ It then follows that $f(0)=1$ or $f(0)=-1.$ If $f(0)=1,$ then plugging in $x=0$ yields $f(y)=\dfrac{1+f(y)}{1+f(y)}=1,$ and if $f(0)=-1,$ then plugging in... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Find all functions $f$ that map the set of real numbers into the set of real numbers, satisfying the following conditions: \n\n1) $|f(x)|\\ge 1$,\n\n2) $f(x+y)=\\frac{f(x)+f(y)}{1+f(x)f(y)}$ of all real va... | Find all functions \(f:\mathbb{R}\to\mathbb{R}\) satisfying:
1. \(|f(x)|\ge 1\) for all real \(x\).
2. \(f(x+y)=\dfrac{f(x)+f(y)}{1+f(x)f(y)}\) for all real \(x,y\). | [
"/Mathematics/Algebra"
] | Plug zeros into the functional equation to determine f(0) and then use x=0 to force the function to be constant ±1. | 234,348 | [
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"Algebra"
] | A certain alien instrument assigns to every real number \(a\) a real “energy reading’’ \(E(a)\). The readings must satisfy two conditions:
1. Each reading stays at least one unit away from zero, i.e. \(|E(a)|\ge 1\) for all real \(a\).
2. For any two real inputs \(a\) and \(b\) the instrument obeys the superposition... |
aops_539329 | $\max(\sqrt{x}+\sqrt{y}+\sqrt{z})=1+\sqrt2+\sqrt3$.
Let $\sqrt{x}+\sqrt{y}+\sqrt{z}=A$ and $\sqrt{xy}+\sqrt{yz}+\sqrt{zx}=B$.
Hence, $A^2=x+y+z+2B$ and $B^2=xy+xz+yz+2\sqrt{xyz}A$.
Id est, $A^2=x+y+z+2\sqrt{xy+xz+yz+2\sqrt{xyz}A}\leq6+2\sqrt{11+2\sqrt{6}A}$,
which gives $A\leq1+\sqrt2+\sqrt3$.
The equality occurs, when... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "let $x,y,z$ be non negative reals such that $x+y+z \\leq 6, xy+xz+yz \\leq 11, xyz \\leq 6$. Determine the maximum value of $\\sqrt{x}+\\sqrt{y}+\\sqrt{z}$.",
"content_html": "let <img src=\"//latex... | Let \(x,y,z\) be nonnegative real numbers such that
\[
x+y+z \le 6,\qquad xy+xz+yz \le 11,\qquad xyz \le 6.
\]
Determine the maximum value of \(\sqrt{x}+\sqrt{y}+\sqrt{z}\). | [
"/Mathematics/Algebra/Sums/Sum"
] | Introduce a=√x, b=√y, c=√z and use (a+b+c)^2 = a^2+b^2+c^2 + 2(ab+bc+ca) together with the given symmetric bounds to bound a+b+c. | 267,975 | [
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"Algebra"
] | Three mystical elixirs – Aurora, Boreal, and Celestia – are brewed in amounts \(a\), \(b\), and \(c\) (measured in non‑negative litres). The quantities satisfy
\[
a+b+c\le 6,\qquad ab+ac+bc\le 11,\qquad abc\le 6.
\]
What is the largest possible value of \(\sqrt{a}+\sqrt{b}+\sqrt{c}\)? |
numina_10034709 | 4. It will not catch up, as the dog can only catch the hare after 50 seconds. In this time, the hare can run 700 m, while the bushes are located 520 m away. | Itwillnotcatchup | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 4. A dog is chasing a hare at a speed of 17 m/s, the hare is running at a speed of $14 \boldsymbol{m} /$ s. The distance between them before the chase was 150 m. Will the dog catch the hare if there are bushes $520 \mathcal{M}$ from the hare where he can hide? | [
"/Mathematics/Algebra/RateProblems",
"/Mathematics/AppliedMathematics"
] | Use relative speed to compute catch-up time and compare with hare's time to reach the bushes. | 80,185 | [
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"Algebra"
] | 4. The Model X1 security drone travels at a speed of 17 m/s, while the Model Z3 nanobot retreats at a speed of 14 m/s. When the pursuit begins, the drone is 150 m behind the nanobot. A secured containment vault is located 520 m ahead of the nanobot’s starting position, where the nanobot can hide. Will the drone interce... |
aops_214281 | $ \sum_{cyc} {\frac {a}{b \plus{} cd}}\geq \sum_{cyc} {\frac {a}{b \plus{} 2c}} \equal{} \sum_{cyc} {\frac {a^2}{ab \plus{} 2ac}}\geq \frac {(a \plus{} b \plus{} c \plus{} d)^2}{ab \plus{} bc \plus{} cd \plus{} da \plus{} 2ac \plus{} 2bd}\geq\frac {4}{3}$ | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "For every $ 1\\leq a,b,c,d\\leq 2$, prove that\r\n\r\n$ \\frac{4}{3}\\leq \\frac{a}{b\\plus{}cd}\\plus{}\\frac{b}{c\\plus{}da}\\plus{}\\frac{c}{d\\plus{}ab}\\plus{}\\frac{d}{a\\plus{}bc}\\leq 2$.",
"... | For every real numbers \(a,b,c,d\) with \(1\le a,b,c,d\le 2\), prove that
\[
\frac{4}{3}\le \frac{a}{b+cd}+\frac{b}{c+da}+\frac{c}{d+ab}+\frac{d}{a+bc}\le 2.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Sums/Sum"
] | Replace each product cd by at most 2c using d≤2, then apply Cauchy–Schwarz (Titu’s lemma) to obtain a lower bound for the sum. | 194,630 | [
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"Algebra"
] | In a distant galaxy, the fuel capacities of four starships—denoted by real numbers \(p, q, r, s\)—are each constrained to the interval \([1,2]\). Prove that the combined efficiency index
\[
\frac{p}{\,q+r s\,}+\frac{q}{\,r+s p\,}+\frac{r}{\,s+p q\,}+\frac{s}{\,p+q r\,}
\]
always satisfies
\[
\frac{4}{3}\;\le\;\frac{p... |
numina_10190841 | Solution. The number $x$ is a solution to the above equation if and only if $x$ is a solution to the system
$$
\begin{aligned}
& \left(\frac{1}{\sin x}-\frac{1}{\cos x}\right)^{2}=8 \\
& \frac{1}{\sin x}-\frac{1}{\cos x} \geq 0
\end{aligned}
$$
From equation (1) we have
$$
\begin{aligned}
& \frac{1}{\sin ^{2} x}+\fr... | \frac{3\pi}{4},\frac{\pi}{12} | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 2. In the interval $0 \leq x \leq \pi$ find the solutions to the equation
$$
\frac{1}{\sin x}-\frac{1}{\cos x}=2 \sqrt{2}
$$ | [
"/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation",
"/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula",
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Polynomials/PolynomialEquation",
"/Mathematics/Algebra/Polynomials/QuadraticPolynomial"
] | Rewrite the equation as a quadratic in sin 2x using the double‑angle identity after squaring. | 47,474 | [
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"Algebra"
] | A spacecraft’s guidance computer records the orientation angle $\theta$ (in radians) of the vessel relative to a reference direction. The admissible range for $\theta$ is $0\le\theta\le\pi$. The computer imposes the requirement that the difference between the reciprocal of the sine of $\theta$ and the reciprocal of the... |
aops_606029 | Alternatively we can substitute $x=\frac{a}{b+c}$, $y=\frac{b}{c+a}$, $z=\frac{c}{a+b}$
Then clearly the condition gets satisfied
$(i)$Lower Bound
$\frac{3}{4} \leq \sum \frac{ab}{(b+c)(c+a)} \iff 6abc \le \sum ab^2 +\sum a^2b$
This is obvious
Upper Bound
$ \frac{ab}{(b+c)(c+a)} <1 \iff \sum ab(a+b) \le (a+b)(b+... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Suppose that $x,y ,z $ be positive real numbers such that $yz+zx+xy+2xyz=1$ .Prove that\n\\[(i) \\ \\frac{3}{4}\\le yz+zx+xy <1,\\]\n\\[(ii) \\ xyz\\le \\frac{1}{8}.\\]",
"content_html": "Suppose th... | Suppose that \(x,y,z\) are positive real numbers such that
\[
yz+zx+xy+2xyz=1.
\]
Prove that
\[
\text{(i)}\quad \frac{3}{4}\le yz+zx+xy<1,
\qquad
\text{(ii)}\quad xyz\le\frac{1}{8}.
\] | [
"/Mathematics/Algebra/GeneralAlgebra/Algebra",
"/Mathematics/Algebra/Products/Product",
"/Mathematics/Algebra/Sums/Sum"
] | Introduce a,b,c with x = a/(b+c), y = b/(c+a), z = c/(a+b) to satisfy the condition identically and then apply AM‑GM. | 274,993 | [
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"Algebra"
] | A space‑faring federation monitors three newly discovered exoplanets, assigning each a positive real stability coefficient—$\alpha$, $\beta$, and $\gamma$. These coefficients are linked by the relation
\[
\alpha\beta+\beta\gamma+\gamma\alpha+2\alpha\beta\gamma=1.
\]
Prove that
\[
\text{(i)}\quad \frac34\le \alph... |
aops_1297228 | [hide=Solution]We approach this by contradiction, suppose both are $0$. If $a+b+c=0$ then $a=-b-c$. WLOG suppose $|b|\ge |c|$. This means the expression $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ is essentially $$\frac{1}{b}+\frac{1}{c}-\frac{1}{b+c}$$ $$=\frac{b^2+c^2+bc}{bc(b+c)}.$$ We are approaching this by contradictio... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let $a$, $b$, and $c$ be nonzero real numbers. Prove that $a+b+c$ and $\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}$ cannot both be $0$.",
"content_html": "Let <span style=\"white-space:nowrap;\"><img src=... | Let \(a\), \(b\), and \(c\) be nonzero real numbers. Prove that \(a+b+c\) and \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\) cannot both be \(0\). | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Sums/Sum"
] | Substitute a = -b - c, rewrite 1/a+1/b+1/c as (b^2+bc+c^2)/(bc(b+c)), and note b^2+bc+c^2>0, giving a contradiction. | 154,265 | [
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"Algebra"
] | In the bustling market of Eldoria, three merchants assign non‑zero real price tags $p$, $q$, and $r$ (in gold coins) to their wares. Prove that it is impossible for the total price $p+q+r$ and the sum of the reciprocals of the prices $\displaystyle\frac1p+\frac1q+\frac1r$ to both be zero simultaneously. |
aops_1944928 | [hide=Solution]Replace $2x=t$, and we will use \[sin3t=3sint-4si{{n}^{3}}t,\cos 3t=4{{\cos }^{3}}t-3\cos t\]. So we get: ${{\sin }^{3}}2x\cos 6x+{{\cos }^{3}}2x\sin 6x=3\sin t\cos t({{\cos }^{2}}t-{{\sin }^{2}})t=\frac{3}{2}\sin 2t\cos 2t=\frac{3}{4}\sin 4t=\frac{3}{4}\sin 8x$[/hide] | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "The expression\n\\[\\sin^3 2x \\cos 6x + \\cos^3 2x \\sin 6x\\]\ncan be written in the equivalent form $a \\sin bx$ for some positive constants $a$ and $b.$ Find $a + b.$",
"content_html": "The expr... | The expression
\[
\sin^3(2x)\cos(6x)+\cos^3(2x)\sin(6x)
\]
can be written in the equivalent form \(a\sin(bx)\) for some positive constants \(a\) and \(b\). Find \(a+b\). | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity"
] | Apply the triple-angle identities to rewrite the terms and factor them into a single sine of a multiple angle. | 187,329 | [
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"Algebra"
] | A star‑ship’s coolant system exhibits a periodic vibration described by
\[
F(\tau)=\sin^{3}(5\tau)\,\cos(15\tau)+\cos^{3}(5\tau)\,\sin(15\tau).
\]
It is known that this function can be written in the form \(P\sin(Q\tau)\) for some positive constants \(P\) and \(Q\). Find the value of \(P+Q\). |
numina_10189851 | Solution. With simple trigonometric transformations, we obtain
$$
\begin{aligned}
& y_{1}=\sin ^{4} x+\cos ^{4} x=\frac{3+\cos 4 x}{4} \\
& y_{2}=\sin ^{6} x+\cos ^{6} x=\frac{5+3 \cos 4 x}{8}
\end{aligned}
$$
Therefore, these functions have a fundamental period equal to $\frac{2 \pi}{4}=\frac{\pi}{2}$, and it holds ... | \\frac{\pi}{12}+k\frac{\pi}{2},k\in\mathbb{Z} | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 2. Given the functions $y_{1}=\sin ^{4} x+\cos ^{4} x$ and $y_{2}=\sin ^{6} x+\cos ^{6} x$. Prove that these functions have a fundamental period of $\frac{\pi}{2}$ and that $3 y_{1}-2 y_{2}=1$. For which values of $x$ is the first function $\frac{1}{16}$ greater than the second? | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity"
] | Rewrite sin⁴x+cos⁴x and sin⁶x+cos⁶x using power‑reduction identities to obtain expressions in cos 4x. | 100,479 | [
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"Algebra"
] | In a laboratory experiment a rotating polarizer is turned through an angle $\alpha$.
The measured normalized intensities of two orthogonal polarization components are modeled by
\[
P(\alpha)=\sin^{4}\alpha+\cos^{4}\alpha ,\qquad
Q(\alpha)=\sin^{6}\alpha+\cos^{6}\alpha .
\]
\begin{enumerate}
\item Show that both ... |
numina_10156231 | SOLUTION
$\mathbf{E}$
From the information given in the question we see that $x$ satisfies the equation $x-\frac{1}{10}=\frac{x}{10}$. Multiplying both sides by 10 gives $10 x-1=x$, so $9 x=1$. So $x=\frac{1}{9}$. | \frac{1}{9} | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 3. The number $x$ has the following property: subtracting $\frac{1}{10}$ from $x$ gives the same result as multiplying $x$ by $\frac{1}{10}$. What is the number $x$ ?
A $\frac{1}{100}$
B $\frac{1}{11}$
C $\frac{1}{10}$
D $\frac{11}{100}$
E $\frac{1}{9}$ | [
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation",
"/Mathematics/Algebra/AlgebraicEquations/LinearEquation"
] | Translate the condition into the equation x − 1/10 = x/10 and solve the resulting linear equation. | 42,195 | [
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"Algebra"
] | A botanist discovers a curious growth factor $y$ in a rare plant. The observation notes that subtracting $\frac{1}{10}$ from $y$ yields exactly the same value as taking $\frac{1}{10}$ of $y$. What is the value of $y$?
A $\displaystyle \frac{1}{100}$
B $\displaystyle \frac{1}{11}$
C $\displaystyle \frac{1}{10}$ ... |
numina_10105627 | (15) When $a=b=c=1$, we can get $k \geqslant 2$. Now we prove the inequality
$$
a b+b c+c a+2\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geqslant 9
$$
holds for all positive real numbers $a, b, c$.
By the AM-GM inequality, we have
$$
a b+\frac{1}{a}+\frac{1}{b} \geqslant 3 \sqrt[3]{a b \cdot \frac{1}{a} \cdot \f... | 2 | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | (15) Find the smallest positive real number $k$, such that the inequality
$$
a b+b c+c a+k\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geqslant 9,
$$
holds for all positive real numbers $a, b, c$. | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/AlgebraicOperations",
"/Mathematics/Algebra/AlgebraicProperties"
] | Apply AM‑GM to each triple (ab, 1/a, 1/b) to obtain ab+1/a+1/b≥3 and sum the three resulting inequalities. | 30,374 | [
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"Algebra"
] | In an interstellar engineering contest, each participant must configure three independent power generators whose output levels are represented by the positive real numbers $x$, $y$ and $z$. The overall efficiency of the trio is evaluated by the sum of the pairwise products of the outputs together with a safety buffer t... |
numina_10149319 | It is sufficient to show that (∑ a i ) 2 /(n - 1) - ∑ a i 2 ≤ 2a 1 a 2 . But this follows immediately from the Cauchy inequality for the two n-1 tuples: a 1 + a 2 , a 3 , a 4 , ... , a n ; and 1, 1, ... , 1: (∑ a i ) 2 <= (n - 1)( (a 1 + a 2 ) 2 + a 3 2 + ... + a n 2 ) = (n - 1) ∑ a i 2 + (n - 1) 2a 1 a 2 . 38th Putnam... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 38th Putnam 1977 Problem B5 a 1 , a 2 , ... , a n are real and b < (∑ a i ) 2 /(n - 1) - ∑ a i 2 . Show that b < 2a i a j for all distinct i, j. | [
"/Mathematics/Algebra/LinearAlgebra",
"/Mathematics/Algebra/Sums/Sum"
] | Apply Cauchy–Schwarz to the vectors (a_i+a_j, a_k, …) and (1, …, 1) to bound (∑a_i)^2 by (n−1)∑a_i^2 plus 2a_i a_j. | 132,277 | [
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"Algebra"
] | Consider $n$ enchanted crystals $C_1,\dots,C_n$, each possessing a real magical intensity $m_1,m_2,\dots,m_n$. Let $\Delta$ be a real number such that
\[
\Delta \;<\; \frac{\bigl(\displaystyle\sum_{k=1}^{n} m_k\bigr)^{2}}{\,n-1\,}\;-\;\sum_{k=1}^{n} m_k^{2}.
\]
Prove that for every pair of distinct crystals $i\neq ... |
aops_126769 | It's not hard. By the [b]AM-GM Inequality[/b] we have:
$\sum\frac{x^{2}}{(2y+3z)(2z+3y)}\ge\sum\frac{x^{2}}{\left[\frac{5(y+z)}{2}\right]^{2}}=\frac{4}{25}\sum\left(\frac{x}{y+z}\right)^{2}\ge\frac{4}{25}\cdot\frac{\left(\sum\frac{x}{y+z}\right)^{2}}{3}\ge\frac{4}{25}\cdot\frac{\left(\frac{3}{2}\right)^{2}}{3}=\frac{3... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "For $a,b,c$ are positive numbers. Find minimum of expression:\r\n$A=\\frac{x^{2}}{(2y+3z)(2z+3y)}+\\frac{y^{2}}{(2x+3z)(2z+3x)}+\\frac{z^{2}}{(2x+3y)(2y+3x)}$",
"content_html": "For <img src=\"//late... | For positive numbers \(x,y,z\), find the minimum of
\[
A=\frac{x^{2}}{(2y+3z)(2z+3y)}+\frac{y^{2}}{(2x+3z)(2z+3x)}+\frac{z^{2}}{(2x+3y)(2y+3x)}.
\] | [
"/Mathematics/Algebra"
] | Apply AM‑GM to bound each denominator by a symmetric expression in y+z, reducing the sum to a form amenable to Cauchy‑Schwarz and Nesbitt. | 152,600 | [
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"Algebra"
] | Let \(p,q,r\) be three positive real numbers representing the power outputs of three space‑engine thrusters. Determine the smallest possible value of
\[
E=\frac{p^{2}}{(2q+3r)(2r+3q)}+\frac{q^{2}}{(2p+3r)(2r+3p)}+\frac{r^{2}}{(2p+3q)(2q+3p)}.
\] |
numina_10192098 | $\mathrm{Al}$ and Bert must arrive at a town $22.5 \mathrm{~km}$ away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at $8 \mathrm{~km} / \mathrm{h}$, leaves the bicycle and then walks at $5 \mathrm{~km} / \mathrm{h}$. Al walks at $4 \mathrm{~km} / \mathrm{h}$, reaches the bi... | 75 | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | $\mathrm{Al}$ and Bert must arrive at a town $22.5 \mathrm{~km}$ away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at $8 \mathrm{~km} / \mathrm{h}$, leaves the bicycle and then walks at $5 \mathrm{~km} / \mathrm{h}$. Al walks at $4 \mathrm{~km} / \mathrm{h}$, reaches the bi... | [
"/Mathematics/Algebra/RateProblems"
] | Let the distance the bike travels before being left be a variable, write each person’s total time, and set the times equal. | 47,923 | [
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"Algebra"
] | Cassie and Dylan need to reach a remote outpost that lies 22.5 km from their starting point. They have only one fold‑up electric scooter to share and they must arrive together. Dylan starts by riding the scooter at 8 km/h, then abandons it and continues on foot at 5 km/h. Cassie walks from the start at 4 km/h, picks up... |
numina_10180447 |
Solution: We use the trivial inequalities $a^{2}+1 \geq 2 a, b^{2}+1 \geq 2 b$ and $c^{2}+1 \geq 2 c$. Hence we obtain
$$
\frac{a^{2}+1}{b+c}+\frac{b^{2}+1}{c+a}+\frac{c^{2}+1}{a+b} \geq \frac{2 a}{b+c}+\frac{2 b}{c+a}+\frac{2 c}{a+b}
$$
$$
\frac{2 a}{b+c}+\frac{2 b}{c+a}+\frac{2 c}{a+b} \geq 3
$$
Adding 6 both sid... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} |
3. If $a, b, c$ are three positive real numbers, prove that
$$
\frac{a^{2}+1}{b+c}+\frac{b^{2}+1}{c+a}+\frac{c^{2}+1}{a+b} \geq 3
$$
| [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/AlgebraicProperties",
"/Mathematics/Algebra/Sums/Sum"
] | Apply a²+1≥2a (AM‑GM) to reduce the sum, then use (x+y+z)(1/x+1/y+1/z)≥9 with x=b+c, y=c+a, z=a+b. | 135,767 | [
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"Algebra"
] | Three interstellar probes have normalized power outputs given by the positive real numbers $p$, $q$, and $r$. Show that
\[
\frac{p^{2}+1}{\,q+r\,}+\frac{q^{2}+1}{\,r+p\,}+\frac{r^{2}+1}{\,p+q\,}\ge 3\,.
\] |
ours_12154 | Let the speed of the sports plane be \(x \, \text{km/h}\). Then the speed of the fighter jet is \(3x \, \text{km/h}\). Since the sports plane flew \(x \, \text{km}\) in one hour, the fighter jet flew \((x + 200) \, \text{km}\) in half an hour, which means it flew \(2 \cdot (x + 200) \, \text{km}\) in one hour. Therefor... | null | {
"competition": "german_mo",
"dataset": "Ours",
"posts": null,
"source": "Loesungen_MaOlympiade_94.md"
} | A fighter jet flew \(200 \, \text{km}\) further in half an hour than a sports plane did in one hour. What was the speed of each of these two planes if the speed of the fighter jet was three times that of the sports plane? | [
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation",
"/Mathematics/Algebra/AlgebraicEquations/LinearEquation",
"/Mathematics/Algebra/RateProblems"
] | Convert the half‑hour distance difference into a per‑hour equation using distance = speed × time and the given speed ratio. | 62,102 | [
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"Algebra"
] | In a speed trial, a futuristic hovercraft traveled 200 kilometres farther in 30 minutes than a conventional electric scooter did in 60 minutes. If the hovercraft’s speed is three times the scooter’s speed, what are the speeds of the hovercraft and the scooter (in km/h)? |
aops_2157404 | [quote=NaPrai]By [b]AM-GM[/b], we have \begin{align*}\sum_{cyc}\frac{\sqrt{x_1x_2-1}}{x_2+x_3} &\le \frac{1}{2}\sum_{cyc}\sqrt{\frac{x_1x_2-1}{x_2x_3}} \\&= \frac{1}{2}\sum_{cyc}\sqrt{\frac{1}{x_3}\left(x_1-\frac{1}{x_2}\right)} \\&\le \frac{1}{4}\sum_{cyc}\left(x_1-\frac{1}{x_2}+\frac{1}{x_3}\right) \\&= \frac{1}{4}\s... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "For(n>2), $x_1,x_2,..,x_n$ positive real numbers not less than $k>0$ , prove that $$ \\frac{1}{4}\\sum_{cyc}x_1+\\frac{1-k^2}{4}\\sum_{cyc}\\frac{1}{x_1} \\geq \\sum_{cyc}\\frac{\\sqrt{x_1x_2-k^2}}{x_2+x_3... | For n > 2, let x_1, x_2, ..., x_n be positive real numbers with x_i ≥ k > 0 for all i. Prove that
\[
\frac{1}{4}\sum_{i=1}^n x_i+\frac{1-k^2}{4}\sum_{i=1}^n\frac{1}{x_i} \ge \sum_{i=1}^n\frac{\sqrt{x_i x_{i+1}-k^2}}{x_{i+1}+x_{i+2}},
\]
where indices are taken modulo n. | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Sums/Sum"
] | Apply AM‑GM to bound each term via denominator ≥ 2√(x_{i+1}x_{i+2}) then use √(ab) ≤ (a+b)/2 to linearize the sum. | 195,017 | [
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"Algebra"
] | Let $n>2$ be an integer. Consider $n$ enchanted vials containing positive amounts of elixir, and denote the amount in the $i$‑th vial by $e_i$. Assume there exists a fixed positive constant $\tau$ such that $e_i\ge \tau>0$ for every $i$. Prove that
\[
\frac14\sum_{i=1}^{n} e_i \;+\; \frac{1-\tau^{2}}{4}\sum_{i=1}^{n... |
aops_1202117 | Let $P(x,y)$ denote $f(xy)f(\frac{f(y)}{x})=1$ for all $x,y\in \mathbb{R}^+$
By $P(\frac{1}{x},f(y))$ give us $f(xf(f(y)))f(\frac{f(y)}{x})=1$ for all $x,y\in \mathbb{R}^+$
So $f(xy)=f(xf(f(y)))$ for all $x,y \in \mathbb{R}^+$
By $2)$ we get that if $f(a)=f(b),a<b$ then $f(x)=f(a)$ for all $x\in [a,b]$
If there exist $... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Find all functions $f:\\mathbb{R^{+}}\\rightarrow \\mathbb{R^{+}}$ such that\n$i/ f(xy)f(\\frac{f(y)}{x})=1$ $ \\forall x, y\\in \\mathbb{R^{+}}$\n$ii/ \\forall x\\geq y> 0$ then $f(x)\\leq f(y)$\n\n",
... | Find all functions \(f:\mathbb{R}^{+}\to\mathbb{R}^{+}\) such that
(i) \(f(xy)\,f\!\left(\dfrac{f(y)}{x}\right)=1\) for all \(x,y\in\mathbb{R}^{+}\).
(ii) For all \(x\ge y>0\) we have \(f(x)\le f(y)\). | [
"/Mathematics/Algebra/GeneralAlgebra/Algebra"
] | Substitute (1/x, f(y)) to get f(xy)=f(x f(f(y))) and apply the monotonicity condition to conclude f(f(y))=y or that f is constant. | 148,915 | [
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"Algebra"
] | Let \(g:\mathbb{R}^{+}\to\mathbb{R}^{+}\) be a mapping from the set of positive real numbers to itself. Determine all such functions \(g\) that satisfy the following two properties:
\begin{enumerate}
\item[(i)] For every pair of positive real numbers \(u\) and \(v\),
\[
g(uv)\;g\!\left(\frac{g(v)}{u}\right)=1.
\]
\it... |
aops_3255223 | [quote=hangb6pbc]If $a \in R$, only $a=0$ work
$2=\left(\dfrac{sinx}{\sqrt{1-a+a^2}}+\dfrac{cosx}{\sqrt{1+a+a^2}} \right)^2 \leq (sin^2x+cos^2x)\left(\dfrac{1}{1-a+a^2}+\dfrac{1}{1+a+a^2} \right)=\dfrac{2(a^2+1)}{a^4+a^2+1}$
$\Rightarrow a^4 \leq 0 \Rightarrow a=0$[/quote]
Nice solution! ;-D | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Solve the equation and discuss:\n$ \\frac{ \\sin x} {\\sqrt{1 - a + a^2} } + \\frac{ \\cos x} {\\sqrt{1 + a + a^2} } = \\sqrt{2}$",
"content_html": "Solve the equation and discuss:<br>\n<img src=\"... | Solve the equation and discuss:
$ \frac{ \sin x} {\sqrt{1 - a + a^2} } + \frac{ \cos x} {\sqrt{1 + a + a^2} } = \sqrt{2}$ | [
"/Mathematics/Algebra"
] | Apply Cauchy‑Schwarz to bound the weighted sin x + cos x sum, yielding a condition on a. | 231,822 | [
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"Algebra"
] | A researcher models the normalized intensity of a combined wave by
\[
\frac{\sin\varphi}{\sqrt{\,1-q+q^{2}\,}} \;+\; \frac{\cos\varphi}{\sqrt{\,1+q+q^{2}\,}} \;=\; \sqrt{2}.
\]
Determine all real values of the parameter $q$ for which there exists at least one phase $\varphi$ satisfying the equation, and discuss the... |
aops_1971996 | [quote=NTNT]:read: $ Let \enskip a,b,c>0 \enskip and \enskip (a+b)(b+c)(c+a)=1 \enskip. Prove \enskip that:$ \\
$ii. \enskip \left(\frac{2a}{b+c}+\frac{2b}{c+a}+\frac{2c}{a+b}+3\right)^2 \geq \frac{32}{3}(a+b+c)^3 \enskip $ :pilot: \\[/quote]
$RHS = \frac{32(a+b+c)^3}{3(a+b)(b+c)(c+a)} \leq 12 \frac{(a+b+c)^2}{... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "\n :read: $ Let \\enskip a,b,c>0 \\enskip and \\enskip (a+b)(b+c)(c+a)=1 \\enskip. Prove \\enskip that:$ \\\\ \n $i. \\enskip \\left(\\frac{2a}{b+c}+\\frac{2b}{c+a}+\\frac{2c}{a+b}+3\\right)^2 \\ge... | Let \(a,b,c>0\) and \((a+b)(b+c)(c+a)=1\). Prove that:
i. \[
\left(\frac{2a}{b+c}+\frac{2b}{c+a}+\frac{2c}{a+b}+3\right)^2 \ge 24\bigl(a^2+b^2+c^2+ab+bc+ca\bigr).
\]
ii. \[
\left(\frac{2a}{b+c}+\frac{2b}{c+a}+\frac{2c}{a+b}+3\right)^2 \ge \frac{32}{3}(a+b+c)^3.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/AlgebraicIdentities/CompletingtheSquare"
] | Apply Cauchy–Schwarz to obtain a lower bound ( (a+b+c)^2/(ab+bc+ca) + 3 )^2, then reduce the inequality to (x+3)^2 ≥ 12x using the condition (a+b)(b+c)(c+a)=1. | 188,616 | [
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"Algebra"
] | Let \(x,\;y,\;z>0\) be three positive numbers satisfying
\[
(x+y)(y+z)(z+x)=1.
\]
Prove the following two inequalities:
\[
\left(\frac{2x}{y+z}+\frac{2y}{z+x}+\frac{2z}{x+y}+3\right)^{2}\ge 24\bigl(x^{2}+y^{2}+z^{2}+xy+yz+zx\bigr),
\]
\[
\left(\frac{2x}{y+z}+\frac{2y}{z+x}+\frac{2z}{x+y}+3\right)^{2}\ge\frac{32}{3... |
numina_10101122 | 6. D Let $x=0, f(y)+f(-y)=0, f(-y)=-f(y)$, so $f(y)$ is an odd function. Also, by substituting $-y$ for $y$, we have $f(x-y)+f(x+y)=2 f(x) g(-y)$, thus $g(-y)=g(y)$. | D | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 6. Let $f(x), g(x)$ be two functions defined on $(-\infty,+\infty)$, for any real numbers $x, y$, satisfying $f(x+y) +$ $f(x-y)=2 f(x) \cdot g(y)$. If $f(0)=0$, but $f(x)$ is not identically zero, then
A. $f(x), g(x)$ are both odd functions
B. $f(x), g(x)$ are both even functions
C. $f(x)$ is an even function, $g(x)$ i... | [
"/Mathematics/Algebra"
] | Plug x=0 to obtain f(y)+f(-y)=0 (f odd), then replace y by -y to deduce g(-y)=g(y) (g even). | 89,159 | [
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"Algebra"
] | 6. In a laboratory experiment, two real‑valued measurement devices produce outputs $S(t)$ and $T(t)$ for any real time $t$. Their readings satisfy
\[
S(t+u)+S(t-u)=2\,S(t)\,T(u)\qquad\text{for all real numbers }t,u .
\]
It is given that $S(0)=0$, but $S$ is not the zero function. Which of the following statements a... |
aops_2032931 | [quote=BestChoice123]For $x,y,z>0$ such that $\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}=2$. Show that $$(x+y)(y+z)(z+x)\ge 1$$[/quote]
There exists $a,b,c>0$ such that $ x=\frac{a}{b+c}, y=\frac{b}{c+a},z=\frac{c}{a+b}.$ So we need to [url=https://artofproblemsolving.com/community/c6h487722p2732787]prove that[/url] \[... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "For $x,y,z>0$ such that $\\frac{1}{x+1}+\\frac{1}{y+1}+\\frac{1}{z+1}=2$. Show that $$(x+y)(y+z)(z+x)\\ge 1$$",
"content_html": "For <img src=\"//latex.artofproblemsolving.com/5/e/5/5e5616447e444b385... | For \(x,y,z>0\) such that
\[
\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}=2,
\]
show that
\[
(x+y)(y+z)(z+x)\ge 1.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/GeneralAlgebra/Algebra"
] | Introduce positive a,b,c with x = a/(b+c), y = b/(c+a), z = c/(a+b) to turn the given sum condition into an identity and simplify the product inequality. | 191,183 | [
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"Algebra"
] | At a deep‑space outpost, three engineers—Nova, Vega, and Orion—record the power levels of their three reactors as positive real numbers \(p,\;q,\) and \(r\). They find that
\[
\frac{1}{p+1}+\frac{1}{q+1}+\frac{1}{r+1}=2.
\]
Show that the product of the sums of each pair of power levels satisfies
\[
(p+q)(q+r)(r+... |
aops_1887566 | [quote=White-Wolf]If $a,b,c\geq 0$ and $ab+bc+ac=3$ then prove that:
$4abc+9(a+b+c)\geq 31$[/quote]
The following inequality is also true :
If $a,b,c\geq 0$ and $ab+bc+ac=3$, for all $k \le 2 \sqrt{3}-3$ we have :
$$a+b+c+k\times abc\geq 3+k$$
(In your case $k=\frac49 \le 2 \sqrt{3}-3$) | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "If $a,b,c\\geq 0$ and $ab+bc+ac=3$ then prove that:\n\n$4abc+9(a+b+c)\\geq 31$",
"content_html": "If <img src=\"//latex.artofproblemsolving.com/3/0/f/30fa6a15477ea7489e3fe6af59cf06dc5d88be81.png\" cl... | If \(a,b,c\ge 0\) and \(ab+bc+ca=3\), prove that
\[
4abc+9(a+b+c)\ge 31.
\] | [
"/Mathematics/Algebra/Products/Product",
"/Mathematics/Algebra/Sums/Sum"
] | Apply the inequality a+b+c+k·abc ≥ 3+k valid for k ≤ 2√3−3, taking k = 4/9. | 184,513 | [
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"Algebra"
] | Three exploratory drones, Alpha, Beta, and Gamma, each carry a non‑negative amount of power, denoted by \(p_{\alpha}, p_{\beta}, p_{\gamma}\). Their pairwise power interactions are given by the products of their powers, and the sum of these three interactions equals three:
\[
p_{\alpha}p_{\beta}+p_{\beta}p_{\gamma}+p_... |
numina_10039181 | 34.19. The point is that if equality (3) holds for all real $x, y, x \neq y$, then
$$
\varphi\left(\frac{x+y}{2}\right)=\frac{\varphi(x)+\varphi(y)}{2}
$$
To prove this, replace $x$ with $x+y$ and $y$ with $x-y$ in (3). As a result, we get
$$
\frac{f(x+y)-g(x-y)}{2 y}=\varphi(x)
$$
for all real $x, y, y \neq 0$. Se... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 34.19. Prove that the functional equation
$$
\frac{f(x)-g(y)}{x-y}=\varphi\left(\frac{x+y}{2}\right)
$$
can be reduced to the functional equation (2) | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity"
] | Apply linear substitutions to eliminate f,g and obtain ϕ(x)+ϕ(y)=2ϕ((x+y)/2), i.e. the Jensen functional equation for ϕ. | 117,865 | [
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"Algebra"
] | In the telemetry network of the interstellar vessel *Nimbus*, let $f(\alpha)$ denote the reading from the forward‑facing sensor at the real‑valued time $\alpha$, and let $g(\beta)$ denote the reading from the rear‑facing sensor at the real‑valued time $\beta$. A calibration function $\varphi(\gamma)$ supplies the expec... |
aops_1149761 | [quote=loopback]Givn $a+b+c=0$ and $ a^2+b^2+c^2=1$ find maximum value of ${(9abc)}^2$[/quote]
Because If $a,\,b,\,c$ are real number such that $a+b+c=0$ then \[(a^2+b^2+c^2)^3 \geqslant 54a^2b^2c^2.\] | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Givn $a+b+c=0$ and $ a^2+b^2+c^2=1$ find maximum value of ${(9abc)}^2$",
"content_html": "Givn <img src=\"//latex.artofproblemsolving.com/3/7/e/37e4c3e85710cff4f69bfec2adaff5ea38111a3a.png\" class=\"... | Given \(a+b+c=0\) and \(a^2+b^2+c^2=1\). Find the maximum value of \((9abc)^2\). | [
"/Mathematics/Algebra"
] | Use the inequality (a^2+b^2+c^2)^3 ≥ 54 a^2 b^2 c^2 that holds when a+b+c=0 to bound |abc|. | 145,231 | [
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"Algebra"
] | In a futuristic research laboratory three quantum oscillators—named Orion, Vega, and Sirius—have dimensionless amplitudes \(x\), \(y\), and \(z\). Their amplitudes satisfy
\[
x+y+z=0\qquad\text{and}\qquad x^{2}+y^{2}+z^{2}=1.
\]
Determine the greatest possible value of the square of nine times the product of the th... |
numina_10160476 | 4. Answer: (D)
If $x \leq 0$, then $|x|=-x$, and we obtain from $|x|+x+5 y=2$ that $y=\frac{2}{5}$. Thus $y$ is positive, so $|y|-y+x=7$ gives $x=7$, which is a contradiction since $x \leq 0$. Therefore we must have $x>0$. Consequently, $|x|+x+5 y=2$ gives the equation
$$
2 x+5 y=2 \text {. }
$$
If $y \geq 0$, then $|... | 3 | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 4. If $x$ and $y$ are real numbers for which $|x|+x+5 y=2$ and $|y|-y+x=7$, find the value of $x+y$.
(A) -3
(B) -1
(C) 1
(D) 3
(E) 5 | [
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"/Mathematics/Algebra/AlgebraicEquations/LinearEquation",
"/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations",
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] | Analyze the signs of x and y to replace absolute values, discard contradictory cases, and solve the resulting linear system. | 42,941 | [
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"Algebra"
] | 4. In a video‑game tournament, the skill ratings of two competitors, Maya and Noah, are represented by real numbers $s$ and $t$, respectively. Their ratings satisfy
\[
|s|+s+5t=2\qquad\text{and}\qquad |t|-t+s=7.
\]
What is the value of $s+t$?
\begin{enumerate}[(A)]
\item $-3$
\item $-1$
\item $1$
\item $3$
\item $... |
numina_10126023 | [Solution] Rewrite the original system of equations as
$$
\left\{\begin{array}{l}
6=x^{2}-(y-z)^{2}=(x-y+z)(x+y-z), \\
2=y^{2}-(z-x)^{2}=(y-z+x)(y+z-x), \\
3=z^{2}-(x-y)^{2}=(z-x+y)(z+x-y) .
\end{array}\right.
$$
(1) $\times$ (2) $\times$ (3) gives
$$
36=(x-y+z)^{2}(y-z+x)^{2}(z-x+y)^{2} \text {, }
$$
Thus, $\quad(x-y... | \begin{pmatrix}x_{1}=\frac{5}{2},y_{1}=\frac{3}{2},z_{1}=2\\x_{2}=\frac{-5}{2},y_{2}=\frac{-3}{2},z_{2}=-2\end{pmatrix} | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 4・129 Solve the system of equations $\left\{\begin{array}{l}x^{2}=6+(y-z)^{2}, \\ y^{2}=2+(z-x)^{2}, \\ z^{2}=3+(x-y)^{2} .\end{array}\right.$ | [
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"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Polynomials/Factorization",
"/Mathe... | Factor each equation as a difference of squares to obtain linear factors, then multiply the three equations to solve for the linear combinations of x, y, z. | 92,524 | [
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"Algebra"
] | In a distant kingdom, three wizards—Merlin, Morgana, and Uther—each possess a mystical power level, denoted by the real numbers $p$, $q$, and $r$, respectively. The ancient tome of spells imposes the following constraints on their powers:
\[
\begin{cases}
p^{2}=6+(q-r)^{2},\\[4pt]
q^{2}=2+(r-p)^{2},\\[4pt]
r^{2}=3+(p-... |
aops_535042 | Set $a=\dfrac{2x}{y+z},b=\dfrac{2y}{x+z},c=\dfrac{2z}{x+y}=>ab+bc+ac+abc=4$ (you can check it).
We can prove that $a+b+c+abc\geq 4<=>\sum x^{3}+3xyz\geq \sum xy(x+y)$
Which is true by Schur inequality. | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "find the max & min of $a+b+c+abc$ where $a,b,c\\in\\mathbb{R}^{+}$ and $ab+bc+ca+abc =4$.",
"content_html": "find the max & min of <img src=\"//latex.artofproblemsolving.com/1/a/9/1a9a299c16f3f11... | Find the maximum and minimum of
\[
a+b+c+abc
\]
subject to \(a,b,c\in\mathbb{R}^+\) and
\[
ab+bc+ca+abc=4.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities"
] | Introduce a=2x/(y+z), b=2y/(x+z), c=2z/(x+y) to satisfy the constraint identically, then apply Schur inequality to bound a+b+c+abc. | 267,459 | [
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"Algebra"
] | Three interstellar vessels have propulsion coefficients $p$, $q$, $r$, each a positive real number. Their mutual interaction satisfies
\[
pq+qr+rp+pqr=4.
\]
Determine the greatest and the least possible values of the total performance measure
\[
p+q+r+pqr.
\] |
numina_10734119 | Prove $\left(a^{2}+b c\right)\left(b^{2}+c a\right)\left(c^{2}+a b\right) \leqslant 1$
$$\begin{array}{l}
\Leftrightarrow(a+b)^{2}(a+c)^{2}(b+c)^{2} \geqslant 4\left(a^{2}+b c\right)\left(b^{2}+c a\right)\left(c^{2}+a b\right) \\
\Leftrightarrow(a-b)^{2}(a-c)^{2}(b-c)^{2}+\sum_{\text {sym }}\left(4 a^{3} b^{2} c+\frac{... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "inequalities"
} | Example 49 Non-negative real numbers $a, b, c$ satisfy $(a+b)(b+c)(c+a)=2$. Prove:
$$\left(a^{2}+b c\right)\left(b^{2}+c a\right)\left(c^{2}+a b\right) \leqslant 1$$ | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Polynomials/MultivariatePolynomial",
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Algebra/Polynomials/SymmetricPolynomial",
"/Mathematics/Algebra/Products/Product",
"/Mathematics/Algebra/Sums/Sum"
] | Rewrite the inequality as (a+b)^2(b+c)^2(c+a)^2 ≥4∏(a^2+bc) and show the difference is a sum of squares. | 137,413 | [
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"Algebra"
] | In a covert planetary mining operation, the non‑negative real ore quantities extracted from three sites are denoted by $p$, $q$, and $r$. They satisfy the relation
\[
(p+q)(q+r)(r+p)=2.
\]
Show that
\[
\bigl(p^{2}+qr\bigr)\bigl(q^{2}+rp\bigr)\bigl(r^{2}+pq\bigr)\le 1.
\] |
numina_10103851 | 7. $[-2,+\infty)$.
Let $f(x)=\sin ^{2} x+a \sin x+a+3$, then
$$
f(x)=\left(\sin x+\frac{a}{2}\right)^2+a+3-\frac{a^{2}}{4} \geqslant 0 .
$$
Since $x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right]$, thus, $\sin x \in\left[-\frac{1}{2}, 1\right]$.
Let $t=\sin x$, then $-\frac{1}{2} \leqslant t \leqslant 1$. Let
$$
g(t)=... | [-2,+\infty) | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 7. For any $x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right]$, the inequality $\sin ^{2} x+a \sin x+a+3 \geqslant 0$ always holds. Then the range of the real number $a$ is $\qquad$ . | [
"/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation",
"/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula",
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Algebra/Polynomials/QuadraticPolynomial"
] | Complete the square and require the quadratic's minimum on sin x∈[–½,1] be nonnegative. | 89,544 | [
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"Algebra"
] | A spacecraft’s attitude control system defines a stability index \(S(\phi)\) as a function of the spacecraft’s orientation angle \(\phi\) (in radians) by
\[
S(\phi)=\sin^{2}\phi+\kappa\,\sin\phi+\kappa+3 .
\]
The orientation angle may be set to any value in the interval \(\displaystyle\Big[-\frac{\pi}{6},\;\frac{\p... |
aops_336316 | first u write the given expressions as 1/x+1/y=5/6
1/y+1/z=7/10
this means 1/x-1/z=5/6-7/10
also from last equation , 1/x+1/z=8/15
from these get 1/x. then u will get 1... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "I am having a nightmare trying to solve this, yet it seems so simple\r\n\r\n[quote]Find x,y and z when....\n\n5xy/(x+y) =6\n\n7yz/(y+z) =10\n\n8zx/(z+x) =15[/quote]\r\n\r\nThank you",
"content_html... | Find \(x,y,z\) satisfying
\[
\frac{5xy}{x+y}=6,\qquad \frac{7yz}{y+z}=10,\qquad \frac{8zx}{z+x}=15.
\] | [
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation",
"/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations",
"/Mathematics/Algebra/GeneralAlgebra/Algebra"
] | Replace each fraction by a linear equation in the reciprocals 1/x, 1/y, 1/z. | 235,419 | [
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"Algebra"
] | A circuit contains three unknown resistors with resistances \(R_{A},R_{B},R_{C}\) (in ohms). When \(R_{A}\) and \(R_{B}\) are connected in parallel, the measured equivalent resistance multiplied by five equals six ohms:
\[
\frac{5\,R_{A}R_{B}}{R_{A}+R_{B}}=6.
\]
When \(R_{B}\) and \(R_{C}\) are connected in parallel,... |
aops_215205 | [quote="stergiu"]If $ a , b , c$ are positive numbers with sum $ 1$ , prove that :
$ \frac {ab}{1 \plus{} c} \plus{} \frac {bc}{1 \plus{} a} \plus{} \frac {ca}{1 \plus{} b} \leq \frac {1}{4}$
[/quote]
[quote="arqady"]
$ \sum_{cyc}\frac {ab}{1 \plus{} c}\leq\sum_{cyc}\frac {\sqrt {ab}}{1 \plus{} c}\leq\frac {1}{4}.$ ... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "If $ a , b , c$ are positive numbers with sum $ 1$ , prove that :\r\n\r\n $ \\frac {ab}{1 \\plus{} c} \\plus{} \\frac {bc}{1 \\plus{} a} \\plus{} \\frac {ca}{1 \\plus{} b} \\leq \\frac {1}{4}$\r\n\r\n Babi... | If \(a,b,c\) are positive numbers with \(a+b+c=1\), prove that
\[
\frac{ab}{1+c}+\frac{bc}{1+a}+\frac{ca}{1+b}\le\frac{1}{4}.
\] | [
"/Mathematics/Algebra/GeneralAlgebra/Algebra",
"/Mathematics/Algebra/Products/Product",
"/Mathematics/Algebra/Sums/Sum"
] | Replace each product ab by its larger square root √(ab) (since variables ≤1) and then bound the resulting sum via Cauchy–Schwarz using a+b+c=1. | 194,879 | [
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"Algebra"
] | On a research vessel, three containers of a special liquid hold fractions $u$, $v$, and $w$ of the total supply, where $u$, $v$, $w>0$ and $u+v+w=1$. Prove that
\[
\frac{u v}{1+w}+\frac{v w}{1+u}+\frac{w u}{1+v}\le\frac14\,.
\] |
aops_1160690 | We can first get that Rowena can paint $\frac{1}{14}$ every hour and Ruby can paint $\frac{1}{6}$ of the room every hour. We can set up the following equations: $\frac{1}{14}x+\frac{1}{6}y=\frac{1}{2}$ and $\frac{1}{14}y+\frac{1}{6}x=1$. Solving for $x$ and $y$ gives us $\boxed{(x,y)=(\frac{231}{40},\frac{21}{40})}... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Rowena can paint a room in $14$ hours, while Ruby can paint it in $6$ hours. If Rowena paints for $x$ hours and Ruby paints for $y$ hours, they will finish half of the painting, while if Rowena paints for ... | Rowena can paint a room in $14$ hours, while Ruby can paint it in $6$ hours. If Rowena paints for $x$ hours and Ruby paints for $y$ hours, they will finish half of the painting, while if Rowena paints for $y$ hours and Ruby paints for $x$ hours they will paint the whole room. Find the ordered pair $(x,y)$. | [
"/Mathematics/Algebra/AlgebraicEquations/LinearEquation",
"/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations",
"/Mathematics/Algebra/RateProblems"
] | Express each painter's contribution as rate × time and set up two linear equations for half and the whole work, then solve the system. | 146,093 | [
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"Algebra"
] | Kai can mow a lawn in $14$ hours, while Sam can mow the same lawn in $6$ hours. If Kai mows for $a$ hours and Sam mows for $b$ hours, together they will have mowed exactly half of the lawn. If Kai mows for $b$ hours and Sam mows for $a$ hours, together they will mow the entire lawn. Find the ordered pair $(a,b)$. |
aops_625060 | [quote="achilles04"]If $3x+4y+z=5$ where $x,y,z $ real numbers, then find the minimum value of $26(x^2+y^2+z^2)$ .[/quote]
Use cauchy-swarz inequality to find the minimum of x^2+y^2+z^2. Then the problem is done. :) | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "If $3x+4y+z=5$ where $x,y,z $ real numbers, then find the minimum value of $26(x^2+y^2+z^2)$ .",
"content_html": "If <img src=\"//latex.artofproblemsolving.com/d/6/6/d660975885da0d1c850f87c63dae48808... | If \(3x+4y+z=5\) where \(x,y,z\) are real numbers, find the minimum value of \(26(x^2+y^2+z^2)\). | [
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"/Mathematics/Algebra/LinearAlgebra",
"/Mathematics/Algebra/VectorAlgebra/DotProduct",
"/Mathematics/Algebra/VectorAlgebra/RealVector",
"/Mathematics/Algebra/VectorAlgebra/ScalarProduct",
"/Mathematics/Algebra/VectorAlgebra/VectorNorm"
] | Apply the Cauchy–Schwarz inequality to bound the sum of squares using the given linear constraint. | 277,015 | [
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0.052642822265625,
0.043487548828125,
0,
0.208374023437... | [
"Algebra"
] | During a satellite‑calibration routine, three real adjustment parameters \(\alpha\), \(\beta\) and \(\gamma\) are required to obey the linear condition \(4\alpha+3\beta+\gamma=5\). Determine the minimum possible value of \(26\big(\alpha^{2}+\beta^{2}+\gamma^{2}\big)\). |
numina_10100184 | $$
\begin{array}{l}
\sum\left(\frac{a^{2}}{c}+\frac{b^{2}}{c}\right)+7 \sum a \\
\geqslant \frac{\left(\sum a\right)^{3}}{\sum a b}+\frac{2\left(\sum a b\right)^{2}}{a b c} \\
\Leftrightarrow \sum \frac{(a-b)^{2}}{c} \\
\geqslant \frac{\left(\sum a\right)^{3}-3\left(\sum a\right)\left(\sum a b\right)}{\sum a b}+ \\
\fr... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | High $\mathbf{5 2 2}$ Let $a, b, c \in \mathbf{R}_{+}$. Prove:
$$
\begin{aligned}
\sum & \left(\frac{a^{2}}{c}+\frac{c^{2}}{a}\right)+7 \sum a \\
& \geqslant \frac{\left(\sum a\right)^{3}}{\sum a b}+\frac{2\left(\sum a b\right)^{2}}{a b c},
\end{aligned}
$$
where, “ $\sum$ ” denotes the cyclic sum. | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Sums/Sum"
] | Rewrite the inequality as a weighted sum of squares ∑(1/c-(∑a)/(2∑ab))(a-b)^2≥0 and verify the weights are non‑negative. | 126,010 | [
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"Algebra"
] | **Quantum Crystals Challenge**.
Three autonomous mining drones—Alpha, Beta and Gamma—have extracted positive amounts $u$, $v$, $w$ of quantum crystals, respectively. Prove that
\[
\sum_{\text{cyc}}\Bigl(\frac{u^{2}}{w}+\frac{w^{2}}{u}\Bigr)+7\sum_{\text{cyc}}u
\;\ge\;
\frac{\bigl(\sum_{\text{cyc}}u\bigr)^{3}}{\sum... |
aops_1589788 | It is simple, just start from $sin^{2}x+cos^{2}x=$1, thus sin$^{4}x+cos^{4}x=1-\frac{sin^{2}2x}{2}$, and $sin^{8}x-cos^{8}x=(sin^{2}x-cos^{2}x)(sin^{4}x+cos^{4}x),$ so we got the equation $cos2xsin^{2}2x=0$ (we are used $cos2x=cos^{2}x-sin^{2}x=1-2sin^{2}x)$ | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Solve the equation: $sin^8x-cos^8x-2sin^2x+1=0$",
"content_html": "Solve the equation: <img src=\"//latex.artofproblemsolving.com/8/a/9/8a9c2c7afe60ce29e5f042c0bad0b781465f07cc.png\" class=\"latex\" ... | Solve the equation:
\[
\sin^8 x - \cos^8 x - 2\sin^2 x + 1 = 0.
\] | [
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation",
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicExpression",
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/AlgebraicIdentities/PolynomialIdentity"
] | Factor sin⁸x−cos⁸x as (sin²x−cos²x)(sin⁴x+cos⁴x) and rewrite using sin²x+cos²x=1 and double‑angle identities to get a product equation. | 168,680 | [
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"Algebra"
] | During the calibration of a lunar rover's navigation system, the steering angle \(\alpha\) (measured in radians) must satisfy
\[
\sin^{8}\alpha - \cos^{8}\alpha - 2\sin^{2}\alpha + 1 = 0.
\]
Determine all real values of \(\alpha\) that fulfill this condition. |
numina_10128063 | 【Solution】Solution: Assume all are soccer balls,
$$
96 \div 6=16 \text { (units), } 4 \times 6=24 \text { (people), }
$$
Basketballs: $24 \div(6-3)$,
$$
\begin{array}{l}
=24 \div 3, \\
=8 \text { (units); }
\end{array}
$$
Soccer balls: $20-8=12$ (units);
Answer: There are 12 soccer balls.
Therefore, the answer is: 12... | 12 | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 10. (3 points) The school has a total of 20 soccer balls and basketballs, which can accommodate 96 students playing at the same time. Each soccer ball is shared by 6 students, and each basketball is shared by 3 students. Among them, there are $\qquad$ soccer balls. | [
"/Mathematics/Algebra/AlgebraicEquations/LinearEquation",
"/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations",
"/Mathematics/Algebra/RateProblems"
] | Set up a pair of linear equations for total balls and total student capacity and solve for the number of soccer balls. | 35,520 | [
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"Algebra"
] | A factory operates a total of 20 machines, which together can produce 96 widgets per minute. Each type‑A machine can produce 6 widgets per minute, whereas each type‑B machine can produce 3 widgets per minute. How many type‑A machines are there among the 20 machines? \(\underline{\qquad}\) |
numina_10162807 | IV/2. If we substitute $y=-x$ into the equation, we get $f(f(x)-x)=f(0)+f(2012)$. Since the right side is a constant and $f$ is an injective function, $f(x)-x$ must also be a constant. Therefore, $f(x)=x+c$ for some real number $c$. If we substitute this into the original equation, we get $x+y+2c=x+y+2c+2012$, which gi... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 2. Prove that there does not exist an injective function $f: \mathbb{R} \rightarrow \mathbb{R}$ for which
$$
f(f(x)+y)=f(x+y)+f(2012) \quad \text { for all } x, y \in \mathbb{R}
$$ | [
"/Mathematics/Algebra/GeneralAlgebra/AffineEquation",
"/Mathematics/Algebra/GeneralAlgebra/Algebra"
] | Use injectivity to force f(x)-x to be constant by swapping variables or setting y = -x, leading to a contradiction. | 133,280 | [
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"Algebra"
] | Let \(T:\mathbb{R}\rightarrow\mathbb{R}\) be an injective (one‑to‑one) function. Prove that there is no such function for which
\[
T\bigl(T(u)+v\bigr)=T(u+v)+T\bigl(\pi\bigr)\qquad\text{holds for all real numbers }u,v.
\] |
numina_10067757 | Substitute the values $x=0$ and $x=-1$ into the given equation. We get: $\left\{\begin{array}{c}-f(-1)=f(0), \\ -2 f(0)=-1+f(-1)\end{array}\right.$. Therefore, $2 f(-1)=-1+f(-1)$, which means $f(-1)=-1$.
## Answer
-1. | -1 | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | Folklore
The function $f(x)$ is defined for all $x$, except 1, and satisfies the equation: $(x-1) f\left(\frac{x+1}{x-1}\right)=x+f(x)$. Find $f(-1)$. | [
"/Mathematics/Algebra/AlgebraicEquations/LinearEquation",
"/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations"
] | Plug in x=0 and x=-1 to generate two simple equations linking f(-1) and f(0) and solve them | 22,545 | [
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"Algebra"
] | In a futuristic laboratory, a researcher defines a response function $R(t)$ for every real number $t$ except $t=1$. The function obeys the rule
\[
(t-1)\,R\!\Bigl(\frac{t+1}{\,t-1\,}\Bigr)=t+R(t).
\]
Determine the value of $R(-1)$. |
aops_1492517 | $$\tan \left(\frac{\pi}{4}+\frac y2\right)=\tan^3 \left(\frac{\pi}{4}+\frac x2\right)$$
Applying the tangent addition formula, we have:
$$\frac{1+\tan\frac y2}{1-\tan\frac y2}=\left(\frac{1+\tan\frac x2}{1-\tan\frac x2}\right) ^3$$
$$\frac{1+\frac {\sin \frac y2}{\cos \frac y2}}{1-\frac {\sin \frac y2}{\cos \frac y2}}=... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "If tan(π/4+y/2) =tan[sup]3[/sup]( π/4+x/2)\nProve that:\nSiny/sinx = 3+sin[sup]2[/sup]x/1+3sin[sup]2[/sup]x\n",
"content_html": "If tan(π/4+y/2) =tan<sup>3</sup>( π/4+x/2)<br>\nProve that:<br>\nSiny/... | If \(\tan\!\left(\frac{\pi}{4}+\frac{y}{2}\right)=\tan^3\!\left(\frac{\pi}{4}+\frac{x}{2}\right)\), prove that
\[
\frac{\sin y}{\sin x}=\frac{3+\sin^2 x}{1+3\sin^2 x}.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/AlgebraicIdentities/PolynomialIdentity"
] | Convert the tan(π/4+θ) expression to (cosθ+sinθ)/(cosθ−sinθ) and square to obtain a rational relation in sinθ. | 163,712 | [
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"Algebra"
] | In a deep‑space exploration scenario, two spacecraft, **Voyager α** and **Voyager β**, have their antennae inclined at angles \(\alpha\) and \(\beta\) (in radians) with respect to a fixed reference direction. Assume that these inclination angles satisfy
\[
\tan\!\Bigl(\frac{\pi}{4}+\frac{\beta}{2}\Bigr)=\tan^{3}\!\B... |
aops_254900 | [quote="Brut3Forc3"]If $ \frac {xy}{x \plus{} y} \equal{} a, \frac {xz}{x \plus{} z} \equal{} b, \frac {yz}{y \plus{} z} \equal{} c$, where $ a,b,c$ are other than zero, then $ x$ equals:
$ \text{(A)}\, \frac {abc}{ab \plus{} ac \plus{} bc} \qquad\text{(B)}\, \frac {2abc}{ab \plus{} bc \plus{} ac} \qquad\text{(C)}\, \f... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "If $ \\frac {xy}{x \\plus{} y} \\equal{} a, \\frac {xz}{x \\plus{} z} \\equal{} b, \\frac {yz}{y \\plus{} z} \\equal{} c$, where $ a,b,c$ are other than zero, then $ x$ equals:\r\n\r\n$ \\textbf{(A)}\\ \\f... | If \(\dfrac{xy}{x+y}=a,\ \dfrac{xz}{x+z}=b,\ \dfrac{yz}{y+z}=c\), where \(a,b,c\) are nonzero, then \(x\) equals:
(A) \(\dfrac{abc}{ab+ac+bc}\)
(B) \(\dfrac{2abc}{ab+bc+ac}\)
(C) \(\dfrac{2abc}{ab+ac-bc}\)
(D) \(\dfrac{2abc}{ab+bc-ac}\)
(E) \(\dfrac{2abc}{ac+bc-ab}\) | [
"/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations",
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/AlgebraicOperations/GeneralAlgebraicOperations"
] | Rewrite each ½xy/(x+y)=a as 1/x+1/y=1/a to obtain linear equations in the reciprocals of x, y, z. | 206,749 | [
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"Algebra"
] | Three capacitors with unknown capacitances \(C_{1},C_{2},C_{3}\) are each paired in a separate series circuit.
The equivalent capacitance of the series pair \((C_{1},C_{2})\) is \(\alpha\), the equivalent capacitance of \((C_{1},C_{3})\) is \(\beta\), and the equivalent capacitance of \((C_{2},C_{3})\) is \(\gamma\);... |
aops_115303 | [quote="Phelpedo"]If $a+b+c=1$, find (with proof) the minimum and maximum possible value of $ab+bc+ca$.[/quote]
I think a minimum requires positive reals. :huh:
[hide="Because"] We can have the numbers be $-n$, $0$, $n+1$, $n\to\infty$. Thus, $ab+bc+ca=-n^{2}-n$
$\lim_{n\to\infty}-n^{2}-n=-\infty$ [/hide] | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "If $a+b+c=1$, find (with proof) the minimum and maximum possible value of $ab+bc+ca$.",
"content_html": "If <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/4/6/0463a9... | If \(a+b+c=1\), find (with proof) the minimum and maximum possible values of \(ab+bc+ca\). | [
"/Mathematics/Algebra"
] | Use (a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca) to bound ab+bc+ca above, and make one variable large positive and another large negative to drive it to -∞. | 145,510 | [
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"Algebra"
] | Let $p$, $q$, and $r$ be real numbers representing the shares of a single galactic credit allocated to three space stations, and assume that $p+q+r=1$. Determine (with proof) the minimum and maximum possible values of the expression $pq+qr+rp$. |
numina_10152868 | $$
\begin{array}{l}
x+y=(\sqrt{x}+\sqrt{y})^{2}-2 \sqrt{x y} \\
\Rightarrow(\sqrt{x}+\sqrt{y})^{2}=18+2 \sqrt{x y} \leq 18+2\left(\frac{x+y}{2}\right)=36 \quad(\mathrm{GM} \leq \mathrm{AM})
\end{array}
$$
$\sqrt{x}+\sqrt{y} \leq 6=d$ (It is easy to get the answer by letting $x=y$ in $x+y=18$ )
Remark The original quest... | 6 | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | G3.4 Let $x \geq 0$ and $y \geq 0$. Given that $x+y=18$. If the maximum value of $\sqrt{x}+\sqrt{y}$ is $d$, find the value of $d$. (Reference: 1999 FGS.2) | [
"/Mathematics/Algebra/Sums/Sum"
] | Rewrite (√x+√y)^2 = x+y+2√{xy} and bound √{xy} using AM–GM under the fixed sum x+y=18 | 40,388 | [
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"Algebra"
] | A spacecraft’s two cargo holds contain non‑negative quantities of a rare mineral, measured in tons. Let the amounts be $p$ and $q$, with $p\ge0$, $q\ge0$ and $p+q=18$. If $d$ denotes the greatest possible value of $\sqrt{p}+\sqrt{q}$, find $d$. |
numina_10048013 | ## Solution.
$$
\begin{aligned}
& 4 \cos \alpha \cos \varphi \cos (\alpha-\varphi)-2 \cos ^{2}(\alpha-\varphi)-\cos 2 \varphi= \\
& =2 \cos (\alpha-\varphi)(2 \cos \alpha \cos \varphi-\cos (\alpha-\varphi))-\cos 2 \varphi=
\end{aligned}
$$
$$
\begin{aligned}
& =2 \cos (\alpha-\varphi)(\cos \alpha \cos \varphi-\sin \a... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 3.223. $4 \cos \alpha \cos \varphi \cos (\alpha-\varphi)-2 \cos ^{2}(\alpha-\varphi)-\cos 2 \varphi=\cos 2 \alpha$. | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity"
] | Apply the product-to-sum identity to rewrite the cosine products and cancel terms. | 119,274 | [
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"Algebra"
] | In a signal‑processing experiment, two sinusoidal waves have phase angles $\theta$ and $\psi$ (measured in radians) relative to a common reference. Prove that, for any real $\theta$ and $\psi$, the following trigonometric identity holds:
\[
4\cos\theta\cos\psi\cos(\theta-\psi)\;-\;2\cos^{2}(\theta-\psi)\;-\;\cos 2\psi... |
aops_1486733 | [quote=AOPS12142015][hide=Expereince with Problem 11]
Wait I actually think I got this problem except I couldn't get my answer into the cotangent form. I just used a lot of double angle and some other trig properties.
[/hide][/quote]
[hide=Hint]
You can write $\tan(\alpha)$ as $\cot(\alpha)-2\cot(2\alpha)$. | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "The sum $\\sum\\limits_{n=1}^{102} (2^{n-1} \\tan 2^{2n-1}A)$ can be written in the form $x \\cot A + y \\cot z A$ where $x$ and $y$ and $z$ are integers. \n\nCompute the last three digits of $x+y+z$.\n\n[... | The sum
\[
\sum_{n=1}^{102} \bigl(2^{n-1}\tan(2^{2n-1}A)\bigr)
\]
can be written in the form \(x\cot A + y\cot(zA)\) where \(x\), \(y\), and \(z\) are integers. Compute the last three digits of \(x+y+z\). | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Sums/Sum",
"/Mathematics/Algebra/Sums/TelescopingSeries",
"/Mathematics/Algebra/Sums/TelescopingSum"
] | Rewrite each tan term using tan α = cot α − 2 cot 2α to turn the sum into a telescoping series of cotangents. | 163,449 | [
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"Algebra"
] | A spacecraft’s navigation system logs a series of angle measurements denoted by \(\theta\). The accumulated value
\[
S=\sum_{k=1}^{102}\Bigl(2^{\,k-1}\,\tan\!\bigl(2^{\,2k-1}\theta\bigr)\Bigr)
\]
can be expressed in the form \(X\cot\theta+Y\cot(Z\theta)\), where \(X\), \(Y\), and \(Z\) are integers. Find the last thr... |
numina_10050537 | Solution.
$$
\begin{aligned}
& \frac{\left(\sin ^{2} \alpha+\tan ^{2} \alpha+1\right)\left(\cos ^{2} \alpha-\cot ^{2} \alpha+1\right)}{\left(\cos ^{2} \alpha+\cot ^{2} \alpha+1\right)\left(\sin ^{2} \alpha+\tan ^{2} \alpha-1\right)}= \\
& =\frac{\left(\sin ^{2} \alpha+\frac{\sin ^{2} \alpha}{\cos ^{2} \alpha}+1\right)... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 3.048. $\frac{\left(\sin ^{2} \alpha+\tan^{2} \alpha+1\right)\left(\cos ^{2} \alpha-\cot^{2} \alpha+1\right)}{\left(\cos ^{2} \alpha+\cot^{2} \alpha+1\right)\left(\sin ^{2} \alpha+\tan^{2} \alpha-1\right)}=1$. | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity"
] | Rewrite tan²α and cot²α as sin²α/ cos²α and cos²α/ sin²α, then simplify using common denominators to cancel terms. | 119,357 | [
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"Algebra"
] | A quantum navigation system records an angle $\theta$ (assume $\sin\theta\neq0$ and $\cos\theta\neq0$). Confirm that its calibration relation
\[
\frac{\bigl(\sin^{2}\theta+\tan^{2}\theta+1\bigr)\,\bigl(\cos^{2}\theta-\cot^{2}\theta+1\bigr)}
{\bigl(\cos^{2}\theta+\cot^{2}\theta+1\bigr)\,\bigl(\sin^{2}\theta+\tan^{2}\th... |
aops_410907 | Since $\cos 3x=4\cos^3 x-3\cos x$, we have
$A=(4\cos^29^\circ-3)(4\cos^227^\circ-3)={\cos(3\cdot 9^\circ)\over\cos 9^\circ}\cdot{\cos(3\cdot 27^\circ)\over\cos 27^\circ}$
$A={\cos 81^\circ\over\cos 9^\circ}={\sin 9^\circ\over\cos 9^\circ}=\tan 9^\circ$ | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Prove that\n\\[(4\\cos^2 9^\\circ - 3)(4\\cos^2 27^\\circ - 3) = \\tan 9^\\circ\\]",
"content_html": "Prove that<br>\n<img src=\"//latex.artofproblemsolving.com/6/a/a/6aa604b2a086744ce39d38fc3aad5b8d... | Prove that
\[
(4\cos^2 9^\circ - 3)(4\cos^2 27^\circ - 3) = \tan 9^\circ.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity"
] | Rewrite each factor 4cos²θ−3 as cos(3θ)/cosθ using the cosine triple-angle identity | 250,513 | [
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"Algebra"
] | In the navigation system of the starship *Odyssey*, Lieutenant Aria sets the reference angle \(\theta\) to \(9^\circ\). Prove that
\[
\bigl(4\cos^{2}\theta-3\bigr)\,\bigl(4\cos^{2}(3\theta)-3\bigr)=\tan\theta.
\] |
aops_2456171 | [quote=Quantum_fluctuations]Let $a_1, a_2, a_3, \cdots , a_n , b_1, b_2, b_3, \cdots b_n, c_1, c_2, c_3, \cdots c_n$ be $3n$ nonnegative real numbers.
Find the largest real constant $k$ for which the inequality
$$\left(a_1^2+ a_2^2+ a_3^2+ \cdots + a_n^2 \right) \left( b_1^2+ b_2^2+ b_3^2+ \cdots + b_n^2\right)\lef... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let $a_1, a_2, a_3, \\cdots , a_n , b_1, b_2, b_3, \\cdots b_n, c_1, c_2, c_3, \\cdots c_n$ be $3n$ nonnegative real numbers. \n\nFind the largest real constant $k$ for which the inequality\n\n $$\\left(a_... | Let \(a_1,\dots,a_n,b_1,\dots,b_n,c_1,\dots,c_n\) be \(3n\) nonnegative real numbers. Find the largest real constant \(k\) for which the inequality
\[
\left(\sum_{i=1}^n a_i^2\right)\left(\sum_{i=1}^n b_i^2\right)\left(\sum_{i=1}^n c_i^2\right)
\ge k\left(\sum_{i=1}^n a_i b_i c_i\right)^2
\]
always holds. | [
"/Mathematics/Algebra/LinearAlgebra",
"/Mathematics/Algebra/Products/Product",
"/Mathematics/Algebra/Sums/PowerSum",
"/Mathematics/Algebra/Sums/Sum"
] | Apply Hölder (or Cauchy–Schwarz) twice to bound the triple sum by the product of the square‑sum roots, yielding the optimal constant k=1. | 203,732 | [
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"Algebra"
] | On a monitoring system three collections of $n$ non‑negative real readings are obtained: $p_1,p_2,\dots ,p_n$, $q_1,q_2,\dots ,q_n$, and $r_1,r_2,\dots ,r_n$.
Find the greatest real constant $C$ such that, for every possible choice of these $3n$ numbers, the inequality
\[
\Bigl(\sum_{i=1}^{n}p_i^{2}\Bigr)\,
\Bigl... |
aops_3320402 | [hide=Very very nice. Thank ytChen.][quote=ytChen][quote=sqing]Let $ a,b, c>0 $ and $ 8a^2+8b^2=c^2.$ Prove that$$ \frac{ab}{ c^2}+\frac{ c}{ a}+\frac{c}{b} \geq \frac{129}{16}$$[/quote]
[quote=sqing]Let $ a,b, c>0 $ and $ 3a^2+3b^2=c^2.$ Prove that
$$ \frac{ab}{ c^2}+\frac{ c}{ a}+\frac{c}{b} \geq \frac{1}{6}+2\sqr... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let $ a,b, c>0 $ and $ 8a^2+8b^2=c^2.$ Prove that$$ \\frac{ab}{ c^2}+\\frac{ c}{ a}+\\frac{c}{b} \\geq \\frac{129}{16}$$",
"content_html": "Let <img src=\"//latex.artofproblemsolving.com/e/1/d/e1d... | Let \(a,b,c>0\) and \(8a^2+8b^2=c^2\). Prove that
\[
\frac{ab}{c^2}+\frac{c}{a}+\frac{c}{b}\ge\frac{129}{16}.
\] | [
"/Mathematics/Algebra"
] | Normalize variables and apply a weighted AM‑GM using the constraint to bound xy+1/x+1/y | 234,074 | [
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"Algebra"
] | Let \(p,\;q,\) and \(r\) be three positive real numbers representing the calibrated outputs of a set of sensors, and assume they satisfy
\[
8p^{2}+8q^{2}=r^{2}.
\]
Show that
\[
\frac{pq}{r^{2}}+\frac{r}{p}+\frac{r}{q}\;\ge\;\frac{129}{16}\,.
\] |
numina_10050558 | ## Solution.
$$
\begin{aligned}
& \sin ^{2}\left(\frac{\alpha}{2}+2 \beta\right)-\sin ^{2}\left(\frac{\alpha}{2}-2 \beta\right)=\frac{1-\cos (\alpha+4 \beta)}{2}-\frac{1-\cos (\alpha-4 \beta)}{2}= \\
& =\frac{1}{2}-\frac{\cos (\alpha+4 \beta)}{2}-\frac{1}{2}+\frac{\cos (\alpha-4 \beta)}{2}=\frac{1}{2}(\cos (\alpha-4 \... | \sin\alpha\sin4\beta | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | 3.071. $\sin ^{2}\left(\frac{\alpha}{2}+2 \beta\right)-\sin ^{2}\left(\frac{\alpha}{2}-2 \beta\right)$. | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity"
] | Apply the power-reduction identity for sin² to convert to cosines, then use the cosine difference-to-product identity to obtain a product of sines. | 83,337 | [
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"Algebra"
] | During a satellite navigation calibration, the primary rotation angle is denoted by $\theta$ and a secondary correction angle by $\phi$. Compute the simplified form of
\[
\sin^{2}\!\bigg(\frac{\theta}{2}+2\phi\bigg)-\sin^{2}\!\bigg(\frac{\theta}{2}-2\phi\bigg).
\] |
aops_1139038 | Let $P(x,y,z)$ be the inequality above.
$P(0,0,0)\Rightarrow\left(f(0)-\frac{1}{2}\right)^2\leq0$. Hence $f(0)=\frac{1}{2}$.
$P(1,1,1)\Rightarrow\left(f(1)-\frac{1}{2}\right)^2\leq0$. Hence $f(1)=\frac{1}{2}$.
$P(x,0,0)\Rightarrow f(x)\leq\frac{1}{2}\forall x \in \mathbb{R}$
$P(1,y,0)\Rightarrow f(y)\geq\frac{1}{2}\for... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Find all $f :\\mathbb{R} \\Rightarrow \\mathbb{R}$ such that\n$\\frac{f(xy)}{2}+\\frac{f(xz)}{2}-f(x)f(yz) \\geq \\frac{1}{4}$ $\\forall x,y,z \\in \\mathbb{R}$",
"content_html": "Find all <img src=\... | Find all functions \(f:\mathbb{R}\to\mathbb{R}\) such that
\[
\frac{f(xy)}{2}+\frac{f(xz)}{2}-f(x)f(yz)\geq\frac{1}{4}\qquad\forall x,y,z\in\mathbb{R}.
\] | [
"/Mathematics/Algebra"
] | Substitute (0,0,0) and (1,1,1) to get perfect‑square constraints forcing f(0)=f(1)=1/2, then use (x,0,0) and (1,y,0) to bound f everywhere. | 144,412 | [
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"Algebra"
] | Let \(W:\mathbb{R}\to\mathbb{R}\) be a function. Assume that for every triple of real numbers \(s,t,u\) the inequality
\[
\frac{W(st)}{2}\;+\;\frac{W(su)}{2}\;-\;W(s)\,W(tu)\;\ge\;\frac14
\]
holds. Determine all possible functions \(W\). |
numina_10049636 | 4.44 Applying formulas (4.2), (4.3), (4.1), and (4.13) sequentially to the left side of the equation, we find
$$
A=\operatorname{tg} 2 \alpha+\operatorname{ctg} 2 \alpha+\operatorname{tg} 6 \alpha+\operatorname{ctg} 6 \alpha=\frac{\sin 2 \alpha}{\cos 2 \alpha}+\frac{\cos 2 \alpha}{\sin 2 \alpha}+
$$
$$
\begin{aligned... | \frac{8\cos^{2}4\alpha}{\sin12\alpha} | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | $4.44 \operatorname{tg} 2 \alpha+\operatorname{ctg} 2 \alpha+\operatorname{tg} 6 \alpha+\operatorname{ctg} 6 \alpha=\frac{8 \cos ^{2} 4 \alpha}{\sin 12 \alpha}$.
$4.44 \tan 2 \alpha+\cot 2 \alpha+\tan 6 \alpha+\cot 6 \alpha=\frac{8 \cos ^{2} 4 \alpha}{\sin 12 \alpha}$. | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Sums/Sum"
] | Replace each tan+cot pair by 1/(sin x cos x) and apply sum-to-product identities to combine the terms. | 83,094 | [
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"Algebra"
] | **Problem 7.3.**
During a navigation exercise a pilot measures an angle $\psi$. Verify that
\[
\tan(2\psi)+\cot(2\psi)+\tan(6\psi)+\cot(6\psi)=\frac{8\cos^{2}(4\psi)}{\sin(12\psi)}\,,
\]
provided all trigonometric expressions are defined. |
numina_10734432 | 122. Proof: Since
$$a^{2}+2 b^{2}+3=\left(a^{2}+b^{2}\right)+\left(b^{2}+1\right)+2 \geqslant \frac{2}{c}+2 b+2$$
Therefore,
$$\frac{1}{a^{2}+2 b^{2}+3} \leqslant \frac{c}{2(1+c+b c)}$$
Similarly, we have
$$\begin{array}{l}
\frac{1}{b^{2}+2 c^{2}+3} \leqslant \frac{a}{2(1+a+c a)} \\
\frac{1}{c^{2}+2 a^{2}+3} \leqslan... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "inequalities"
} | 122. ("Home of Math Olympiads" website, 2008. 04. 14, provided by polynasia) Let $a, b, c \in \mathbf{R}^{+}$, and $a b c=1$, then
$$\sum \frac{1}{a^{2}+2 b^{2}+3} \leqslant \frac{1}{2}$$ | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/GeneralAlgebra/Algebra"
] | Apply AM–GM to bound each denominator by 2/c+2b+2 and then use a=x/y, b=y/z, c=z/x to reduce the sum to 1. | 137,556 | [
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"Algebra"
] | 137. (Celestial Conundrums Journal, 2026‑07‑13, submitted by Orion)
Let $p,\;q,\;r$ be positive real numbers satisfying $pqr=1$. Prove that
\[
\frac{1}{p^{2}+2q^{2}+3}\;+\;\frac{1}{q^{2}+2r^{2}+3}\;+\;\frac{1}{r^{2}+2p^{2}+3}\;\le\;\frac12\,.
\] |
aops_80393 | Here's my Kalva version of a solution. :P (very brief)
[hide]Use arithmetic series sum formula (both versions). $\frac{n}{2}(29+2)=155\implies n=10$. Now, $\frac{10}{2}(2+29)=\frac{10}{2}(4+9d)\implies d=3$[/hide] | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "In a given arithmetic sequence the first term is $2$, the last term is $29$, and the sum of all the terms is $155$. The common difference is:\r\n\r\n$\\text{(A)} \\ 3 \\qquad \\text{(B)} \\ 2 \\qquad \\tex... | In an arithmetic sequence the first term is \(2\), the last term is \(29\), and the sum of all the terms is \(155\). The common difference is:
\(\text{(A)}\ 3\qquad \text{(B)}\ 2\qquad \text{(C)}\ \dfrac{27}{19}\qquad \text{(D)}\ \dfrac{13}{9}\qquad \text{(E)}\ \dfrac{23}{38}\) | [
"/Mathematics/Algebra/Sums/Sum"
] | Apply the arithmetic series sum formula to find n, then use the last term expression a+(n-1)d to solve for d. | 279,247 | [
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"Algebra"
] | In a steady linear progression where each successive term rises by the same amount, the initial term equals \(2\), the terminal term equals \(29\), and the total of all terms is \(155\). What is the uniform increment?
\(\text{(A)}\ 3\qquad \text{(B)}\ 2\qquad \text{(C)}\ \dfrac{27}{19}\qquad \text{(D)}\ \dfrac{13}{9}\... |
aops_54962 | [quote="manuel"]Find n such that $\dfrac{1}{1+ \sqrt3} + \dfrac{1}{ \sqrt3 + \sqrt5}+...+ \dfrac{1}{\sqrt{2n-1}+\sqrt{2n+1}}=100$[/quote]
here is my idea:
\[ \dfrac{1}{1+ \sqrt{3}} + \dfrac{1}{ \sqrt{3} + \sqrt{5}}+...+ \dfrac{1}{\sqrt{2n-1}+\sqrt{2n+1}}=\dfrac{\sqrt{3}-1}{2}+\dfrac{\sqrt{5}-\sqrt{3}}{2}+...+\dfrac{\... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Find n such that $\\dfrac{1}{1+ \\sqrt3} + \\dfrac{1}{ \\sqrt3 + \\sqrt5}+...+ \\dfrac{1}{\\sqrt{2n-1}+\\sqrt{2n+1}}=100$",
"content_html": "Find n such that <img src=\"//latex.artofproblemsolving.co... | Find \(n\) such that
\[
\frac{1}{1+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\cdots+\frac{1}{\sqrt{2n-1}+\sqrt{2n+1}}=100.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Sums/TelescopingSeries",
"/Mathematics/Algebra/Sums/TelescopingSum"
] | Rationalize each term to a difference of square roots, yielding a telescoping series. | 269,087 | [
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"Algebra"
] | A space explorer is cataloguing a series of signal pairs. For the $i$‑th pair the two signal frequencies are $\sqrt{2i-1}$ Hz and $\sqrt{2i+1}$ Hz, and the effort required to decode the pair is represented by the fraction
\[
\frac{1}{\sqrt{2i-1}+\sqrt{2i+1}}.
\]
Determine the positive integer $M$ for which the total d... |
aops_1127658 |
Given the system 1 $\ \left\{\begin{array}{ll} x^{2}y - x^{2} - 16y = 56 \\ x^{2}y - 4x^{2} + y^{2} - 5y = 8 \end{array}\right.\ $
$\left\{\begin{array}{ll} x^{2}(y - 1) = 16 y + 56 \\ x^{2}(y - 4) = 8 +5y - y^{2}\end{array}\right.\ $
$x = 0$ is not a solution.
Dividing: $\frac{y-1}{y-4}=\frac{16y+56}{8 +5y - y^{2}}$... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Okay, so here I have two systems of equations to solve. I put them in a single post because they are very similar, and the methods to solve them are similar with one key difference (this is a hint to figur... | System 1:
\[
\begin{cases}
x^2y - x^2 - 16y = 56,\\[4pt]
x^2y - 4x^2 + y^2 - 5y = 8.
\end{cases}
\]
System 2:
\[
\begin{cases}
xy + 3x + y^2 + 2y = 5,\\[4pt]
x^2y + 4x^2 + 2xy^2 + 8xy + y^3 + 3y^2 - 3y = -10.
\end{cases}
\] | [
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation",
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicExpression",
"/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations",
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Algebra/Polynomials/PolynomialEquation"
] | Solve each equation for x² and equate them, eliminating x to obtain a single polynomial equation in y. | 143,695 | [
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"Algebra"
] | A crew of space‑craft designers is analyzing two independent algebraic models that involve the variables $a$ (the thrust coefficient) and $b$ (the pressure factor).
**Model Alpha**
\[
\begin{cases}
a^{2}b - a^{2} - 16b = 56,\\[4pt]
a^{2}b - 4a^{2} + b^{2} - 5b = 8.
\end{cases}
\]
**Model Beta**
\[
\begin{cases}
ab + ... |
aops_290061 | I think this is a shorter way.
Without the current, he would have went 15 miles, but he only went 9. 15-9=6, and 6/3 for the mph of the current gives you the current is going at 2 mph.
Edit: Dope!!!! I forgot about that. GJ isabella. | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Jim paddles a canoe upstream at a rate of 5 mph but travels a distance of only 9 miles in 3 hours. What was the rate of the current in miles per hour?",
"content_html": "Jim paddles a canoe upstream ... | Jim paddles a canoe upstream at a rate of 5 mph but travels a distance of only 9 miles in 3 hours. What was the rate of the current in miles per hour? | [
"/Mathematics/Algebra/RateProblems"
] | Set upstream speed equal to paddling speed minus current speed, then use distance = speed × time to find the current. | 219,470 | [
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"Algebra"
] | Zara pilots a small drone that can fly at 5 mph relative to still air. While flying straight into a steady headwind, she covers only 9 miles in 3 hours. What is the speed of the headwind in miles per hour? |
aops_2508673 | @Above I don't think that knowledge is necessary to solve the problem, unless I am missing smth.
[hide = My attempt]
Let $\cos A + \cos B + \cos C = x$ and $\sin A + \sin B + \sin C = y$.
The cosine difference identity and some factoring on the left side of the given equation yields
\begin{align*} \cos A \cos B+ \s... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "If $A, B$ and $C$ are real angles such that\n$$\\cos (B-C)+\\cos (C-A)+\\cos (A-B)=-3/2,$$\nfind\n$$\\cos (A)+\\cos (B)+\\cos (C)$$",
"content_html": "If <img src=\"//latex.artofproblemsolving.com/b/... | If \(A,B,C\) are real angles such that
\[
\cos(B-C)+\cos(C-A)+\cos(A-B)=-\tfrac{3}{2},
\]
find
\[
\cos A+\cos B+\cos C.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Sums/Sum"
] | Rewrite the sum of cosine differences as ((∑cos)^2 + (∑sin)^2 - 3)/2 using the cosine difference identity. | 205,390 | [
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0.0869750... | [
"Algebra"
] | In a navigation experiment three sensors are positioned at real angular headings \(\phi\), \(\psi\) and \(\omega\) (measured in radians). Their pairwise heading differences satisfy
\[
\cos(\psi-\omega)+\cos(\omega-\phi)+\cos(\phi-\psi)=-\frac{3}{2}.
\]
Find the value of \(\cos\phi+\cos\psi+\cos\omega\). |
aops_195017 | [quote="zaya_yc"]$ a,b,c\geq 0$
$ \frac {1 \plus{} a}{1 \plus{} b} \plus{} \frac {1 \plus{} b}{1 \plus{} c} \plus{} \frac {1 \plus{} c}{1 \plus{} a}\leq 3 \plus{} a^{2} \plus{} b^{2} \plus{} c^{2}$[/quote]
Let $ c \equal{} \max\{a,b,c\}.$ Then
$ \frac {1 \plus{} a}{1 \plus{} b} \plus{} \frac {1 \plus{} b}{1 \plus{... | null | {
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "$ a,b,c\\geq 0$ \r\n\r\n$ \\frac{1\\plus{}a}{1\\plus{}b}\\plus{}\\frac{1\\plus{}b}{1\\plus{}c}\\plus{}\\frac{1\\plus{}c}{1\\plus{}a}\\leq 3\\plus{}a^{2}\\plus{}b^{2}\\plus{}c^{2}$",
"content_html": ... | Let \(a,b,c\ge 0\). Prove
\[
\frac{1+a}{1+b}+\frac{1+b}{1+c}+\frac{1+c}{1+a}\le 3+a^{2}+b^{2}+c^{2}.
\] | [
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity"
] | Assume the largest variable is c, rewrite the inequality so each term reduces to a simple bound using a^2 ≥ (a−b)^2 and c^2 ≥ (c−a)(c−b). | 187,552 | [
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"Algebra"
] | Captain Aurora recorded three non‑negative calibration coefficients $p$, $q$, and $r$ for her spacecraft’s guidance array. Prove that
\[
\frac{1+p}{1+q}+\frac{1+q}{1+r}+\frac{1+r}{1+p}\le 3+p^{2}+q^{2}+r^{2}.
\] |
numina_10098833 | Prove that for $y_{i}=\frac{1}{1+x_{i}}, i=1,2, \cdots, n+1$, then $\sum_{i=1}^{n+1} y_{i}=1$, let $s_{i}=\sum_{1 \leqslant j \leqslant n+1, j \neq i} y_{j}, p_{i}=$ $\prod_{1 \leqslant i \leqslant n+1, j \neq i} y_{j}$
By the arithmetic-geometric mean inequality, we have $\frac{1-y_{i}}{y_{i}}=\frac{s_{i}}{y_{i}} \geq... | proof | {
"competition": "Numina-1.5",
"dataset": "NuminaMath-1.5",
"posts": null,
"source": "olympiads"
} | Example 14 (1998 Canadian Olympiad Problem) Let $x_{1}, x_{2}, \cdots, x_{n+1}$ be positive real numbers satisfying the condition $\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\cdots+\frac{1}{1+x_{n+1}}=1$. Prove that: $x_{1} x_{2} \cdots x_{n+1} \geqslant n^{n-1}$. | [
"/Mathematics/Algebra/GeneralAlgebra/Algebra",
"/Mathematics/Algebra/Products/Product",
"/Mathematics/Algebra/Sums/Sum"
] | Introduce y_i = 1/(1+x_i) to obtain ∑y_i = 1, then apply AM‑GM to each (sum of the other y's)/y_i to bound the product. | 125,847 | [
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"Algebra"
] | A sorcerer has prepared $n+1$ distinct elixirs, assigning to the $i$‑th elixir a positive real potency $c_i$ ($i=1,2,\dots ,n+1$). Suppose that the sum of the dilution fractions of all the elixirs equals one, i.e.
\[
\frac{1}{1+c_1}+\frac{1}{1+c_2}+ \cdots + \frac{1}{1+c_{n+1}} = 1.
\]
Show that the product of the po... |
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