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value | problem stringlengths 72 3.36k | solution stringlengths 312 10.4k | messages listlengths 2 2 |
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aops_forum | Let be a natural number $ n\ge 2 $ and $ n $ positive real numbers $ a_1,a_2,\ldots ,a_n $ whose product is $ 1. $ Prove that the function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R} ,\quad f(x)=\prod_{i=1}^n \left( 1+a_i^x \right) , $ is nondecreasing. | 1. **Define the function and its domain:**
We are given a function \( f:\mathbb{R}_{>0} \to \mathbb{R} \) defined by
\[
f(x) = \prod_{i=1}^n (1 + a_i^x)
\]
where \( a_1, a_2, \ldots, a_n \) are positive real numbers such that their product is 1, i.e., \( \prod_{i=1}^n a_i = 1 \).
2. **Differentiate the ... | [
{
"content": "Let be a natural number $ n\\ge 2 $ and $ n $ positive real numbers $ a_1,a_2,\\ldots ,a_n $ whose product is $ 1. $ Prove that the function $ f:\\mathbb{R}_{>0}\\longrightarrow\\mathbb{R} ,\\quad f(x)=\\prod_{i=1}^n \\left( 1+a_i^x \\right) , $ is nondecreasing.",
"role": "user"
},... |
aops_forum | Let $ABCD$ be a quadrilateral inscribed in a circle $k$ . Let the lines $AC\cap BD=O$ , $AD\cap BC=P$ , and $AB\cap CD=Q$ . Line $QO$ intersects $k$ in points $M$ and $N$ . Prove that $PM$ and $PN$ are tangent to $k$ . | 1. **Define the problem and setup:**
- Let \(ABCD\) be a quadrilateral inscribed in a circle \(k\).
- Let the lines \(AC\) and \(BD\) intersect at point \(O\).
- Let the lines \(AD\) and \(BC\) intersect at point \(P\).
- Let the lines \(AB\) and \(CD\) intersect at point \(Q\).
- The line \(QO\) interse... | [
{
"content": "Let $ABCD$ be a quadrilateral inscribed in a circle $k$ . Let the lines $AC\\cap BD=O$ , $AD\\cap BC=P$ , and $AB\\cap CD=Q$ . Line $QO$ intersects $k$ in points $M$ and $N$ . Prove that $PM$ and $PN$ are tangent to $k$ .",
"role": "user"
},
{
"content": "1. **Define ... |
aops_forum | Let $a, b, c$ be positive integers whose greatest common divisor is $1$ . Determine whether there always exists a positive integer $n$ such that, for every positive integer $k$ , the number $2^n$ is not a divisor of $a^k+b^k+c^k$ . | 1. **Given**: \(a, b, c\) are positive integers with \(\gcd(a, b, c) = 1\). We need to determine if there always exists a positive integer \(n\) such that for every positive integer \(k\), \(2^n\) is not a divisor of \(a^k + b^k + c^k\).
2. **Objective**: Prove that there exists such an \(n\).
3. **Approach**: We wil... | [
{
"content": "Let $a, b, c$ be positive integers whose greatest common divisor is $1$ . Determine whether there always exists a positive integer $n$ such that, for every positive integer $k$ , the number $2^n$ is not a divisor of $a^k+b^k+c^k$ .",
"role": "user"
},
{
"content": "1. **Given*... |
aops_forum | Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that
(1) For all integer $m$ , $f(m) \ne 0$ .
(2) For all positive odd integer $n$ , there exist positive integer $k$ such that $f(k)$ is divided by $n$ . | We need to prove that there exists a monic polynomial \( f(x) \) of degree 6 with integer coefficients such that:
1. For all integers \( m \), \( f(m) \neq 0 \).
2. For all positive odd integers \( n \), there exists a positive integer \( k \) such that \( f(k) \) is divisible by \( n \).
We propose the polynomial \( ... | [
{
"content": "Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that\n(1) For all integer $m$ , $f(m) \\ne 0$ .\n(2) For all positive odd integer $n$ , there exist positive integer $k$ such that $f(k)$ is divided by $n$ .",
"role": "user"
},
... |
aops_forum | Let $ABCD$ be a tetrahedron and let $S$ be its center of gravity. A line through $S$ intersects the surface of $ABCD$ in the points $K$ and $L$ . Prove that \[\frac{1}{3}\leq \frac{KS}{LS}\leq 3\] | 1. **Identify the center of gravity \( S \) of the tetrahedron \( ABCD \):**
The center of gravity \( S \) of a tetrahedron is the point where the medians intersect. Each median of a tetrahedron is a line segment connecting a vertex to the centroid of the opposite face. The centroid of a face of a tetrahedron is the... | [
{
"content": "Let $ABCD$ be a tetrahedron and let $S$ be its center of gravity. A line through $S$ intersects the surface of $ABCD$ in the points $K$ and $L$ . Prove that \\[\\frac{1}{3}\\leq \\frac{KS}{LS}\\leq 3\\]",
"role": "user"
},
{
"content": "1. **Identify the center of gravity \\... |
aops_forum | Let $S = \{x_0, x_1,\dots , x_n\}$ be a finite set of numbers in the interval $[0, 1]$ with $x_0 = 0$ and $x_1 = 1$ . We consider pairwise distances between numbers in $S$ . If every distance that appears, except the distance $1$ , occurs at least twice, prove that all the $x_i$ are rational. | 1. **Define the set and distances:**
Let \( S = \{x_0, x_1, \dots, x_n\} \) be a finite set of numbers in the interval \([0, 1]\) with \( x_0 = 0 \) and \( x_1 = 1 \). We consider the pairwise distances between the numbers in \( S \).
2. **Condition on distances:**
Every distance that appears, except the distan... | [
{
"content": "Let $S = \\{x_0, x_1,\\dots , x_n\\}$ be a finite set of numbers in the interval $[0, 1]$ with $x_0 = 0$ and $x_1 = 1$ . We consider pairwise distances between numbers in $S$ . If every distance that appears, except the distance $1$ , occurs at least twice, prove that all the $x_i$ are ... |
aops_forum | The sequence $a_1 = 1$ , $a_2, a_3, \cdots$ is defined as follows: if $a_n - 2$ is a natural number not already occurring on the board, then $a_{n+1} = a_n-2$ ; otherwise, $a_{n+1} = a_n + 3$ . Prove that every nonzero perfect square occurs in the sequence as the previous term increased by $3$ . | 1. **Define the sequence and the problem statement:**
The sequence \( \{a_n\} \) is defined as follows:
- \( a_1 = 1 \)
- For \( n \geq 1 \), if \( a_n - 2 \) is a natural number not already in the sequence, then \( a_{n+1} = a_n - 2 \); otherwise, \( a_{n+1} = a_n + 3 \).
We need to prove that every nonze... | [
{
"content": "The sequence $a_1 = 1$ , $a_2, a_3, \\cdots$ is defined as follows: if $a_n - 2$ is a natural number not already occurring on the board, then $a_{n+1} = a_n-2$ ; otherwise, $a_{n+1} = a_n + 3$ . Prove that every nonzero perfect square occurs in the sequence as the previous term increased ... |
aops_forum | The square $ABCD$ is inscribed in a circle with center $O$ . Let $E$ be the midpoint of $AD$ . The line $CE$ meets the circle again at $F$ . The lines $FB$ and $AD$ meet at $H$ . Prove $HD = 2AH$ | 1. **Define the problem geometrically:**
- Let \(ABCD\) be a square inscribed in a circle with center \(O\).
- \(E\) is the midpoint of \(AD\).
- The line \(CE\) intersects the circle again at \(F\).
- The lines \(FB\) and \(AD\) intersect at \(H\).
2. **Establish key properties and relationships:**
- S... | [
{
"content": "The square $ABCD$ is inscribed in a circle with center $O$ . Let $E$ be the midpoint of $AD$ . The line $CE$ meets the circle again at $F$ . The lines $FB$ and $AD$ meet at $H$ . Prove $HD = 2AH$ ",
"role": "user"
},
{
"content": "1. **Define the problem geometrically:*... |
aops_forum | A group of $4046$ friends will play a videogame tournament. For that, $2023$ of them will go to one room which the computers are labeled with $a_1,a_2,\dots,a_{2023}$ and the other $2023$ friends go to another room which the computers are labeled with $b_1,b_2,\dots,b_{2023}$ . The player of computer $a_i$ *... | 1. Define \( i_a \) and \( j_b \) as the \( i \)-th player from room \( a \) and the \( j \)-th player from room \( b \) respectively.
2. Assume that player \( 1_a \) does not switch computers. We need to show that no other player switches computers.
3. Since player \( 1_a \) does not switch, they remain at computer \(... | [
{
"content": "A group of $4046$ friends will play a videogame tournament. For that, $2023$ of them will go to one room which the computers are labeled with $a_1,a_2,\\dots,a_{2023}$ and the other $2023$ friends go to another room which the computers are labeled with $b_1,b_2,\\dots,b_{2023}$ . The play... |
aops_forum | Problem 3.
Let $f:\left[ 0,\frac{\pi }{2} \right]\to \left[ 0,\infty \right)$ an increasing function .Prove that:
(a) $\int_{0}^{\frac{\pi }{2}}{\left( f\left( x \right)-f\left( \frac{\pi }{4} \right) \right)}\left( \sin x-\cos x \right)dx\ge 0.$ (b) Exist $a\in \left[ \frac{\pi }{4},\frac{\pi }{2} \right]$ suc... | ### Part (a)
We need to prove that:
\[
\int_{0}^{\frac{\pi}{2}} \left( f(x) - f\left( \frac{\pi}{4} \right) \right) \left( \sin x - \cos x \right) dx \geq 0
\]
1. **Substitute \( u = \frac{\pi}{2} - x \):**
Let \( u = \frac{\pi}{2} - x \). Then \( du = -dx \) and the limits of integration change as follows:
- ... | [
{
"content": "Problem 3.\nLet $f:\\left[ 0,\\frac{\\pi }{2} \\right]\\to \\left[ 0,\\infty \\right)$ an increasing function .Prove that:\n(a) $\\int_{0}^{\\frac{\\pi }{2}}{\\left( f\\left( x \\right)-f\\left( \\frac{\\pi }{4} \\right) \\right)}\\left( \\sin x-\\cos x \\right)dx\\ge 0.$ (b) Exist $a\\in \\... |
aops_forum | **Q.** Show that for any given positive integers $k, l$ , there exists infinitely many positive integers $m$ , such that $i) m \geqslant k$ $ii) \text{gcd}\left(\binom{m}{k}, l\right)=1$ *Suggested by pigeon_in_a_hole* | 1. **Define the sequence \( m_n \):**
Consider the arithmetic sequence \( m_n = k + nl^a \) where \( n \in \mathbb{N} \) and \( a \) is sufficiently large such that \( l^a > k \). This ensures that \( m_n > k \) for all \( n \in \mathbb{N} \).
2. **Expression for the binomial coefficient:**
The binomial coeffici... | [
{
"content": "**Q.** Show that for any given positive integers $k, l$ , there exists infinitely many positive integers $m$ , such that $i) m \\geqslant k$ $ii) \\text{gcd}\\left(\\binom{m}{k}, l\\right)=1$ *Suggested by pigeon_in_a_hole*",
"role": "user"
},
{
"content": "1. **Define the sequence ... |
aops_forum | Let $L$ be a subset in the coordinate plane defined by $L = \{(41x + 2y, 59x + 15y) | x, y \in \mathbb Z \}$ , where $\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$ , there exist at least two points of $L$ located in it. | 1. **Define the set \( L \) and the problem context:**
\[
L = \{(41x + 2y, 59x + 15y) \mid x, y \in \mathbb{Z}\}
\]
We need to prove that for any parallelogram centered at the origin with area 1990, there exist at least two points of \( L \) located in it.
2. **Consider a specific parallelogram \( P \) wit... | [
{
"content": "Let $L$ be a subset in the coordinate plane defined by $L = \\{(41x + 2y, 59x + 15y) | x, y \\in \\mathbb Z \\}$ , where $\\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$ , there exist at least two points of $L$ locate... |
aops_forum | Let $ABCD$ be a trapezoid, with $AB \parallel CD$ (the vertices are listed in cyclic order). The diagonals of this trapezoid are perpendicular to one another and intersect at $O$ . The base angles $\angle DAB$ and $\angle CBA$ are both acute. A point $M$ on the line sgement $OA$ is such that $\angle BMD =... | 1. **Given Information and Setup:**
- Trapezoid \(ABCD\) with \(AB \parallel CD\).
- Diagonals \(AC\) and \(BD\) are perpendicular and intersect at \(O\).
- Base angles \(\angle DAB\) and \(\angle CBA\) are acute.
- Point \(M\) on \(OA\) such that \(\angle BMD = 90^\circ\).
- Point \(N\) on \(OB\) such t... | [
{
"content": "Let $ABCD$ be a trapezoid, with $AB \\parallel CD$ (the vertices are listed in cyclic order). The diagonals of this trapezoid are perpendicular to one another and intersect at $O$ . The base angles $\\angle DAB$ and $\\angle CBA$ are both acute. A point $M$ on the line sgement $OA$ is... |
aops_forum | We have three circles $w_1$ , $w_2$ and $\Gamma$ at the same side of line $l$ such that $w_1$ and $w_2$ are tangent to $l$ at $K$ and $L$ and to $\Gamma$ at $M$ and $N$ , respectively. We know that $w_1$ and $w_2$ do not intersect and they are not in the same size. A circle passing through $K... | 1. **Define the centers and radii:**
Let \( O_1 \), \( O_2 \), and \( O \) be the centers of circles \( w_1 \), \( w_2 \), and \( \Gamma \) respectively. Let the radii of \( w_1 \), \( w_2 \), and \( \Gamma \) be \( r_1 \), \( r_2 \), and \( R \) respectively.
2. **Angles involving tangents:**
Since \( w_1 \) an... | [
{
"content": "We have three circles $w_1$ , $w_2$ and $\\Gamma$ at the same side of line $l$ such that $w_1$ and $w_2$ are tangent to $l$ at $K$ and $L$ and to $\\Gamma$ at $M$ and $N$ , respectively. We know that $w_1$ and $w_2$ do not intersect and they are not in the same size. A ci... |
aops_forum | Let $ABC$ be a triangle. The ex-circles touch sides $BC, CA$ and $AB$ at points $U, V$ and $W$ , respectively. Be $r_u$ a straight line that passes through $U$ and is perpendicular to $BC$ , $r_v$ the straight line that passes through $V$ and is perpendicular to $AC$ and $r_w$ the straight line th... | 1. **Lemma:** Let \(ABC\) be a triangle with the orthic triangle \(DEF\). Let \(A', B', C'\) be the feet of perpendiculars from the points \(A, B, C\) to \(FE, FD, DE\) respectively. Then the lines \(AA', BB',\) and \(CC'\) are concurrent.
**Proof of Lemma:**
- The triangles \(ABC\) and \(AEF\) are similar, so ... | [
{
"content": "Let $ABC$ be a triangle. The ex-circles touch sides $BC, CA$ and $AB$ at points $U, V$ and $W$ , respectively. Be $r_u$ a straight line that passes through $U$ and is perpendicular to $BC$ , $r_v$ the straight line that passes through $V$ and is perpendicular to $AC$ and $r_w$... |
aops_forum | Let $ n\geq 3 $ be an integer and let $ \mathcal{S} \subset \{1,2,\ldots, n^3\} $ be a set with $ 3n^2 $ elements. Prove that there exist nine distinct numbers $ a_1,a_2,\ldots,a_9 \in \mathcal{S} $ such that the following system has a solution in nonzero integers:
\begin{eqnarray*} a_1x + a_2y +a_3 z &=& 0 ... | 1. **Define the sets and function:**
Let \( \mathcal{S} \subset \{1, 2, \ldots, n^3\} \) be a set with \( 3n^2 \) elements. Suppose \( x_1 < x_2 < \ldots < x_{3n^2} \) are the elements of \( \mathcal{S} \). Define the subsets:
\[
X_1 = \{x_1, x_2, \ldots, x_{n^2}\}, \quad X_2 = \{x_{n^2+1}, x_{n^2+2}, \ldots, ... | [
{
"content": "Let $ n\\geq 3 $ be an integer and let $ \\mathcal{S} \\subset \\{1,2,\\ldots, n^3\\} $ be a set with $ 3n^2 $ elements. Prove that there exist nine distinct numbers $ a_1,a_2,\\ldots,a_9 \\in \\mathcal{S} $ such that the following system has a solution in nonzero integers:\r\n\\begin{eqn... |
aops_forum | The extension of the median $AM$ of the triangle $ABC$ intersects its circumcircle at $D$ . The circumcircle of triangle $CMD$ intersects the line $AC$ at $C$ and $E$ .The circumcircle of triangle $AME$ intersects the line $AB$ at $A$ and $F$ . Prove that $CF$ is the altitude of triangle $ABC$ . | 1. **Identify the given elements and relationships:**
- $AM$ is the median of $\triangle ABC$.
- $D$ is the point where the extension of $AM$ intersects the circumcircle of $\triangle ABC$.
- The circumcircle of $\triangle CMD$ intersects $AC$ at $C$ and $E$.
- The circumcircle of $\triangle AME$ intersects... | [
{
"content": "The extension of the median $AM$ of the triangle $ABC$ intersects its circumcircle at $D$ . The circumcircle of triangle $CMD$ intersects the line $AC$ at $C$ and $E$ .The circumcircle of triangle $AME$ intersects the line $AB$ at $A$ and $F$ . Prove that $CF$ is the altitude ... |
aops_forum | Let $p$ and $q$ be given prime numbers and $S$ be a subset of ${1,2,3,\dots ,p-2,p-1}$ . Prove that the number of elements in the set $A=\{ (x_1,x_2,…,x_q ):x_i\in S,\sum_{i=1}^q x_i \equiv 0(mod\: p)\}$ is multiple of $q$ .
| 1. **Base Case: \( |S| = 1 \)**
If \( S = \{a\} \) for some \( 1 \le a \le p-1 \), then the set \( A \) is defined as:
\[
A = \{ (x_1, x_2, \ldots, x_q) : x_i \in S, \sum_{i=1}^q x_i \equiv 0 \pmod{p} \}
\]
Since \( S = \{a\} \), every \( x_i = a \). Therefore, the sum \( \sum_{i=1}^q x_i = qa \). For \... | [
{
"content": "Let $p$ and $q$ be given prime numbers and $S$ be a subset of ${1,2,3,\\dots ,p-2,p-1}$ . Prove that the number of elements in the set $A=\\{ (x_1,x_2,…,x_q ):x_i\\in S,\\sum_{i=1}^q x_i \\equiv 0(mod\\: p)\\}$ is multiple of $q$ .\n",
"role": "user"
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"content": "1. **Base ... |
aops_forum | Let $l, l'$ be two lines in $3$ -space and let $A,B,C$ be three points taken on $l$ with $B$ as midpoint of the segment $AC$ . If $a, b, c$ are the distances of $A,B,C$ from $l'$ , respectively, show that $b \leq \sqrt{ \frac{a^2+c^2}{2}}$ , equality holding if $l, l'$ are parallel. | 1. Let $l$ and $l'$ be two lines in 3-space, and let $A, B, C$ be three points on $l$ such that $B$ is the midpoint of the segment $AC$. This implies that $B$ divides $AC$ into two equal segments, i.e., $AB = BC$.
2. Let $a, b, c$ be the perpendicular distances from points $A, B, C$ to the line $l'$, respectively. We n... | [
{
"content": "Let $l, l'$ be two lines in $3$ -space and let $A,B,C$ be three points taken on $l$ with $B$ as midpoint of the segment $AC$ . If $a, b, c$ are the distances of $A,B,C$ from $l'$ , respectively, show that $b \\leq \\sqrt{ \\frac{a^2+c^2}{2}}$ , equality holding if $l, l'$ are par... |
aops_forum | A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figu... | 1. **Identify the Midpoints of Diagonals:**
Consider the midpoints of the diagonals of length $\sqrt{2}$ drawn as the rectangle is divided into the described figures.
2. **Connect Midpoints with Red Segments:**
Connect the midpoints of the diagonals as follows:
- The midpoints of the two equal sides of lengt... | [
{
"content": "A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove t... |
aops_forum | The point $O$ is situated inside the parallelogram $ABCD$ such that $\angle AOB+\angle COD=180^{\circ}$ . Prove that $\angle OBC=\angle ODC$ . | 1. **Construct Point \( O' \)**:
Construct point \( O' \) such that \( DAO'O \) is a parallelogram. This means \( O' \) is the reflection of \( O \) across the line \( DA \).
2. **Properties of Parallelograms**:
Since \( AB \parallel CD \) and \( AD \parallel BC \) (properties of parallelogram \( ABCD \)), and ... | [
{
"content": "The point $O$ is situated inside the parallelogram $ABCD$ such that $\\angle AOB+\\angle COD=180^{\\circ}$ . Prove that $\\angle OBC=\\angle ODC$ .",
"role": "user"
},
{
"content": "1. **Construct Point \\( O' \\)**: \n Construct point \\( O' \\) such that \\( DAO'O \\) is a pa... |
aops_forum | A real valued function $f$ is defined on the interval $(-1,2)$ . A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$ . Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$ . Show that if $f'(1)>1$ , then $f$ has a fixed point in the interval $(0,1)$ . | To show that the function \( f \) has a fixed point in the interval \( (0,1) \), we will use the Intermediate Value Theorem (IVT) and the given conditions.
1. **Given Conditions:**
- \( f \) is differentiable on \( (-1,2) \).
- \( f(0) > 0 \).
- \( f(1) = 1 \).
- \( f'(1) > 1 \).
2. **Intermediate Value T... | [
{
"content": "A real valued function $f$ is defined on the interval $(-1,2)$ . A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$ . Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$ . Show that if $f'(1)>1$ , then $f$ has a fixed point in the interval $... |
aops_forum | Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with a rational area. | 1. **Base Case: \( n = 3 \)**
We need to find three points such that the distance between any two points is irrational and the area of the triangle formed by any three points is rational. Consider the points \((0,0)\), \((1,1)\), and \((2,4)\).
- Distance between \((0,0)\) and \((1,1)\):
\[
\sqrt{(1-0... | [
{
"content": "Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with a rational area.",
"role": "user"
},
{
"content... |
aops_forum | Let $n>k \geq 1$ be integers and let $p$ be a prime dividing $\tbinom{n}{k}$ . Prove that the $k$ -element subsets of $\{1,\ldots,n\}$ can be split into $p$ classes of equal size, such that any two subsets with the same sum of elements belong to the same class.
*Ankan Bhattacharya* | 1. **Generating Function Setup**:
Consider the generating function:
\[
F(x,y) := \prod_{j=1}^n (1 + x^j y).
\]
The coefficient of \(x^s y^k\) in this expansion counts the number of subsets of \([n] := \{1, \ldots, n\}\) with \(k\) elements and sum \(s\).
2. **Lucas' Theorem Application**:
By Lucas' t... | [
{
"content": "Let $n>k \\geq 1$ be integers and let $p$ be a prime dividing $\\tbinom{n}{k}$ . Prove that the $k$ -element subsets of $\\{1,\\ldots,n\\}$ can be split into $p$ classes of equal size, such that any two subsets with the same sum of elements belong to the same class.\n\n*Ankan Bhattachary... |
aops_forum | Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$ . The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\hat{BAC}<\frac{\pi}{8}$ then ... | 1. **Define the given elements and their relationships:**
- Let \( A \) and \( B \) be points on a circle with center \( O \) and radius \( R \).
- Let \( M \) be the midpoint of the arc \( AB \) that does not contain the other arc \( AB \).
- Let \( l \) be the tangent to the circle at \( A \).
- Let \( C ... | [
{
"content": "Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$ . The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\\hat{BAC... |
aops_forum | The sequence ${a_1, a_2, ..., a_{2019}}$ satisfies the following condition. $a_1=1, a_{n+1}=2019a_{n}+1$ Now let $x_1, x_2, ..., x_{2019}$ real numbers such that $x_1=a_{2019}, x_{2019}=a_1$ (The others are arbitary.)
Prove that $\sum_{k=1}^{2018} (x_{k+1}-2019x_k-1)^2 \ge \sum_{k=1}^{2018} (a_{2019-k}-2019a_{20... | 1. **Define the sequence \(a_n\):**
Given the sequence \( \{a_n\} \) defined by:
\[
a_1 = 1, \quad a_{n+1} = 2019a_n + 1
\]
We need to find a general form for \(a_n\).
2. **Find the general form of \(a_n\):**
Let's compute the first few terms to identify a pattern:
\[
a_1 = 1
\]
\[
a_2... | [
{
"content": "The sequence ${a_1, a_2, ..., a_{2019}}$ satisfies the following condition. $a_1=1, a_{n+1}=2019a_{n}+1$ Now let $x_1, x_2, ..., x_{2019}$ real numbers such that $x_1=a_{2019}, x_{2019}=a_1$ (The others are arbitary.)\nProve that $\\sum_{k=1}^{2018} (x_{k+1}-2019x_k-1)^2 \\ge \\sum_{k=1}^{2... |
aops_forum | Three real numbers are given. Fractional part of the product of every two of them is $ 1\over 2$ . Prove that these numbers are irrational.
*Proposed by A. Golovanov* | 1. Let \( a, b, c \) be the three given real numbers. According to the problem, the fractional part of the product of every two of them is \( \frac{1}{2} \). This can be written as:
\[
\{ab\} = \frac{1}{2}, \quad \{bc\} = \frac{1}{2}, \quad \{ca\} = \frac{1}{2}
\]
where \(\{x\}\) denotes the fractional part... | [
{
"content": "Three real numbers are given. Fractional part of the product of every two of them is $ 1\\over 2$ . Prove that these numbers are irrational.\r\n\r\n*Proposed by A. Golovanov*",
"role": "user"
},
{
"content": "1. Let \\( a, b, c \\) be the three given real numbers. According to the pro... |
aops_forum | Let $ x,y,z$ be real numbers greater than or equal to $ 1.$ Prove that
\[ \prod(x^{2} \minus{} 2x \plus{} 2)\le (xyz)^{2} \minus{} 2xyz \plus{} 2.\] | 1. Let \( x = a + 1 \), \( y = b + 1 \), and \( z = c + 1 \) where \( a, b, c \geq 0 \). Then the inequality transforms into:
\[
\prod (x^2 - 2x + 2) \leq (xyz)^2 - 2xyz + 2
\]
Substituting \( x = a + 1 \), \( y = b + 1 \), and \( z = c + 1 \), we get:
\[
(a^2 + 1)(b^2 + 1)(c^2 + 1) \leq ((a + 1)(b + ... | [
{
"content": "Let $ x,y,z$ be real numbers greater than or equal to $ 1.$ Prove that\r\n\\[ \\prod(x^{2} \\minus{} 2x \\plus{} 2)\\le (xyz)^{2} \\minus{} 2xyz \\plus{} 2.\\]",
"role": "user"
},
{
"content": "1. Let \\( x = a + 1 \\), \\( y = b + 1 \\), and \\( z = c + 1 \\) where \\( a, b, c \\g... |
aops_forum | Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\pi]$ and $f''(x) \geq 0$ for all $x$ in $[0,2\pi]$ . Show that
\[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\] | 1. **Given Conditions and Definitions:**
- Let \( f(x) \) be a continuous function on \([0, 2\pi]\).
- The first and second derivatives of \( f(x) \) are continuous on \([0, 2\pi]\).
- \( f''(x) \geq 0 \) for all \( x \) in \([0, 2\pi]\), which implies that \( f(x) \) is convex on \([0, 2\pi]\).
2. **Lemma 1:... | [
{
"content": "Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\\pi]$ and $f''(x) \\geq 0$ for all $x$ in $[0,2\\pi]$ . Show that\n\\[\\int_{0}^{2\\pi} f(x)\\cos x dx \\geq 0\\]",
"role": "user"
},
{
"content": "1. **Given Conditions and Definit... |
aops_forum | Let $p(n)\geq 0$ for all positive integers $n$ . Furthermore, $x(0)=0, v(0)=1$ , and \[x(n)=x(n-1)+v(n-1), \qquad v(n)=v(n-1)-p(n)x(n) \qquad (n=1,2,\dots).\]
Assume that $v(n)\to 0$ in a decreasing manner as $n \to \infty$ . Prove that the sequence $x(n)$ is bounded if and only if $\sum_{n=1}^{\infty}n\cdot ... | 1. **Notation and Assumptions**:
- Let \( p_n, v_n, x_n \) denote \( p(n), v(n), x(n) \) respectively.
- Assume \( v_n \) is decreasing and positive.
- Let \( S_n \) denote the sequence of partial sums of \( \sum_{n=1}^{\infty} v_n \).
2. **Expressing \( x_n \) in terms of \( S_n \)**:
- Given \( x(n) = x(... | [
{
"content": "Let $p(n)\\geq 0$ for all positive integers $n$ . Furthermore, $x(0)=0, v(0)=1$ , and \\[x(n)=x(n-1)+v(n-1), \\qquad v(n)=v(n-1)-p(n)x(n) \\qquad (n=1,2,\\dots).\\]\nAssume that $v(n)\\to 0$ in a decreasing manner as $n \\to \\infty$ . Prove that the sequence $x(n)$ is bounded if and only... |
aops_forum | Let $a, b, c$ be naturals. The equation $ax^2-bx + c = 0$ has two roots at $[0, 1]$ . Prove that $a\ge 5$ and $b\ge 5$ . | To prove that \(a \geq 5\) and \(b \geq 5\) for the quadratic equation \(ax^2 - bx + c = 0\) with roots in the interval \([0, 1]\), we will use the properties of quadratic equations and the conditions given.
1. **Identify the roots and their properties:**
Let the roots of the quadratic equation \(ax^2 - bx + c = 0\... | [
{
"content": "Let $a, b, c$ be naturals. The equation $ax^2-bx + c = 0$ has two roots at $[0, 1]$ . Prove that $a\\ge 5$ and $b\\ge 5$ .",
"role": "user"
},
{
"content": "To prove that \\(a \\geq 5\\) and \\(b \\geq 5\\) for the quadratic equation \\(ax^2 - bx + c = 0\\) with roots in the in... |
aops_forum | $a,b$ are odd numbers. Prove, that exists natural $k$ that $b^k-a^2$ or $a^k-b^2$ is divided by $2^{2018}$ . | 1. **Base Case:**
- Given that \(a\) and \(b\) are odd numbers, we know that \(a^2\) and \(b^2\) are also odd.
- Let \(t = \nu_2(b^2 - 1)\), where \(\nu_2(x)\) denotes the highest power of 2 dividing \(x\).
- We need to show that there exists a positive integer \(k_t\) such that \(2^t \mid b^{k_t} - a^2\).
... | [
{
"content": " $a,b$ are odd numbers. Prove, that exists natural $k$ that $b^k-a^2$ or $a^k-b^2$ is divided by $2^{2018}$ .",
"role": "user"
},
{
"content": "1. **Base Case:**\n - Given that \\(a\\) and \\(b\\) are odd numbers, we know that \\(a^2\\) and \\(b^2\\) are also odd.\n - Let \... |
aops_forum | Let $a, b, c, d$ be positive real numbers. Prove the following inequality and determine all cases in which the equality holds : $$ \frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a}+\frac{d - a}{a + b} \ge 0. $$ | 1. **Introduce the inequality and the variables:**
Let \( a, b, c, d \) be positive real numbers. We need to prove the inequality:
\[
\frac{a - b}{b + c} + \frac{b - c}{c + d} + \frac{c - d}{d + a} + \frac{d - a}{a + b} \geq 0.
\]
2. **Consider the cyclic nature of the terms:**
Notice that the terms in ... | [
{
"content": "Let $a, b, c, d$ be positive real numbers. Prove the following inequality and determine all cases in which the equality holds : $$ \\frac{a - b}{b + c}+\\frac{b - c}{c + d}+\\frac{c - d}{d + a}+\\frac{d - a}{a + b} \\ge 0. $$ ",
"role": "user"
},
{
"content": "1. **Introduce the i... |
aops_forum | Let be given a circle with center $O$ and a point $P$ outside the circle. A line $l$ passes through $P$ and cuts the circle at $A$ and $B$ . Let $C$ be the point symmetric to $A$ with respect to $OP$ , and let $m$ be the line $BC$ . Prove that all lines $m$ have a common point as $l$ varies. | 1. **Define the problem and setup:**
- Let \( O \) be the center of the circle.
- Let \( P \) be a point outside the circle.
- Let \( l \) be a line passing through \( P \) and intersecting the circle at points \( A \) and \( B \).
- Let \( C \) be the point symmetric to \( A \) with respect to \( OP \).
... | [
{
"content": "Let be given a circle with center $O$ and a point $P$ outside the circle. A line $l$ passes through $P$ and cuts the circle at $A$ and $B$ . Let $C$ be the point symmetric to $A$ with respect to $OP$ , and let $m$ be the line $BC$ . Prove that all lines $m$ have a common point... |
aops_forum | A finite set of unit circles is given in a plane such that the area of their union $U$ is $S$ . Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that $\frac{2S}{9}.$ | 1. **Constructing the Collection Greedily**:
- We start by constructing a collection \( T \) of mutually disjoint circles. We do this by iteratively selecting the largest circle that does not intersect any of the circles already chosen in \( T \). This ensures that all circles in \( T \) are disjoint.
2. **Forming ... | [
{
"content": "A finite set of unit circles is given in a plane such that the area of their union $U$ is $S$ . Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that $\\frac{2S}{9}.$ ",
"role": "user"
},
{
"content": "1. **Constructing the ... |
aops_forum | A palindrome is a number that remains the same when its digits are reversed. Let $n$ be a product of distinct primes not divisible by $10$ . Prove that infinitely many multiples of $n$ are palindromes. | 1. **Define the problem and the given conditions:**
- A palindrome is a number that reads the same forwards and backwards.
- Let \( n \) be a product of distinct primes not divisible by 10.
- We need to prove that there are infinitely many multiples of \( n \) that are palindromes.
2. **Express \( n \) as a p... | [
{
"content": "A palindrome is a number that remains the same when its digits are reversed. Let $n$ be a product of distinct primes not divisible by $10$ . Prove that infinitely many multiples of $n$ are palindromes.",
"role": "user"
},
{
"content": "1. **Define the problem and the given conditi... |
aops_forum | In three piles there are $51, 49$ , and $5$ stones, respectively. You can combine any two piles into one pile or divide a pile consisting of an even number of stones into two equal piles. Is it possible to get $105$ piles with one stone in each? | 1. **Understanding the operations**:
- You can combine any two piles into one pile.
- You can divide a pile consisting of an even number of stones into two equal piles.
2. **Analyzing the initial piles**:
- The initial piles have 51, 49, and 5 stones respectively.
3. **Divisibility by 2**:
- Notice that o... | [
{
"content": "In three piles there are $51, 49$ , and $5$ stones, respectively. You can combine any two piles into one pile or divide a pile consisting of an even number of stones into two equal piles. Is it possible to get $105$ piles with one stone in each?",
"role": "user"
},
{
"content": "1... |
aops_forum | Peter has a deck of $1001$ cards, and with a blue pen he has written the numbers $1,2,\ldots,1001$ on the cards (one number on each card). He replaced cards in a circle so that blue numbers were on the bottom side of the card. Then, for each card $C$ , he took $500$ consecutive cards following $C$ (clockwise o... | 1. **Identify the Problem**: We need to prove that Basil can determine the blue number on each card by only looking at the red numbers.
2. **Understanding the Function \( f(C) \)**: For each card \( C \), \( f(C) \) is the number of blue numbers on the next 500 cards that are greater than the blue number on \( C \).
... | [
{
"content": "Peter has a deck of $1001$ cards, and with a blue pen he has written the numbers $1,2,\\ldots,1001$ on the cards (one number on each card). He replaced cards in a circle so that blue numbers were on the bottom side of the card. Then, for each card $C$ , he took $500$ consecutive cards follo... |
aops_forum | Let $a, \ b \in \mathbb{Z_{+}}$ . Denote $f(a, b)$ the number sequences $s_1, \ s_2, \ ..., \ s_a$ , $s_i \in \mathbb{Z}$ such that $|s_1|+|s_2|+...+|s_a| \le b$ . Show that $f(a, b)=f(b, a)$ . | To show that \( f(a, b) = f(b, a) \), we need to establish a bijection between the sequences counted by \( f(a, b) \) and those counted by \( f(b, a) \).
1. **Define the sequences and partial sums:**
Let \( s_1, s_2, \ldots, s_a \) be a sequence of integers such that \( |s_1| + |s_2| + \cdots + |s_a| \leq b \). De... | [
{
"content": "Let $a, \\ b \\in \\mathbb{Z_{+}}$ . Denote $f(a, b)$ the number sequences $s_1, \\ s_2, \\ ..., \\ s_a$ , $s_i \\in \\mathbb{Z}$ such that $|s_1|+|s_2|+...+|s_a| \\le b$ . Show that $f(a, b)=f(b, a)$ .",
"role": "user"
},
{
"content": "To show that \\( f(a, b) = f(b, a) \\), w... |
aops_forum | may be challenge for beginner section, but anyone is able to solve it if you really try.
show that for every natural $n > 1$ we have: $(n-1)^2|\ n^{n-1}-1$ | 1. We start with the expression \( n^{n-1} - 1 \). We can factor this expression as follows:
\[
n^{n-1} - 1 = (n-1) \left( \sum_{k=0}^{n-2} n^k \right)
\]
This factorization is based on the formula for the sum of a geometric series.
2. Next, we need to show that \( (n-1) \) divides the sum \( \sum_{k=0}^{n... | [
{
"content": "may be challenge for beginner section, but anyone is able to solve it if you really try.\r\n\r\nshow that for every natural $n > 1$ we have: $(n-1)^2|\\ n^{n-1}-1$ ",
"role": "user"
},
{
"content": "1. We start with the expression \\( n^{n-1} - 1 \\). We can factor this expression a... |
aops_forum | The set of numbers $M=\{4k-3 | k\in N\}$ is considered. A number of of this set is called “simple” if it is impossible to put in the form of a product of numbers from $M$ other than $1$ . Show that in this set, the decomposition of numbers in the product of "simple" factors is ambiguous. | 1. First, we need to understand the set \( M = \{4k-3 \mid k \in \mathbb{N}\} \). This set consists of numbers of the form \( 4k-3 \) where \( k \) is a natural number. For example, when \( k = 1, 2, 3, \ldots \), the corresponding elements of \( M \) are \( 1, 5, 9, 13, 17, \ldots \).
2. A number in this set is calle... | [
{
"content": "The set of numbers $M=\\{4k-3 | k\\in N\\}$ is considered. A number of of this set is called “simple” if it is impossible to put in the form of a product of numbers from $M$ other than $1$ . Show that in this set, the decomposition of numbers in the product of \"simple\" factors is ambiguous.... |
aops_forum | We call a function $g$ *special* if $g(x)=a^{f(x)}$ (for all $x$ ) where $a$ is a positive integer and $f$ is polynomial with integer coefficients such that $f(n)>0$ for all positive integers $n$ .
A function is called an *exponential polynomial* if it is obtained from the product or sum of special functi... | 1. **Assume the existence of the functions:**
Suppose there exists a non-zero exponential polynomial \( f(x) \) and a non-constant polynomial \( P(x) \) with integer coefficients such that \( P(n) \mid f(n) \) for all positive integers \( n \).
2. **Properties of \( P(x) \):**
Since \( P(x) \) is a non-constant ... | [
{
"content": "We call a function $g$ *special* if $g(x)=a^{f(x)}$ (for all $x$ ) where $a$ is a positive integer and $f$ is polynomial with integer coefficients such that $f(n)>0$ for all positive integers $n$ .\n\nA function is called an *exponential polynomial* if it is obtained from the product ... |
aops_forum | Real numbers $x$ and $y$ satisfy the system of equalities $\sin x + \cos y = 1$ $\cos x + \sin y = -1$ .
Prove that $\cos 2x = \cos 2y$ . | 1. Given the system of equations:
\[
\sin x + \cos y = 1
\]
\[
\cos x + \sin y = -1
\]
2. Square both equations:
\[
(\sin x + \cos y)^2 = 1^2
\]
\[
(\cos x + \sin y)^2 = (-1)^2
\]
3. Expand both squared equations:
\[
\sin^2 x + 2 \sin x \cos y + \cos^2 y = 1
\]
\[
\c... | [
{
"content": "Real numbers $x$ and $y$ satisfy the system of equalities $\\sin x + \\cos y = 1$ $\\cos x + \\sin y = -1$ .\nProve that $\\cos 2x = \\cos 2y$ .",
"role": "user"
},
{
"content": "1. Given the system of equations:\n \\[\n \\sin x + \\cos y = 1\n \\]\n \\[\n \\cos x + \\s... |
aops_forum | Let there be given three circles $K_1,K_2,K_3$ with centers $O_1,O_2,O_3$ respectively, which meet at a common point $P$ . Also, let $K_1 \cap K_2 = \{P,A\}, K_2 \cap K_3 = \{P,B\}, K_3 \cap K_1 = \{P,C\}$ . Given an arbitrary point $X$ on $K_1$ , join $X$ to $A$ to meet $K_2$ again in $Y$ , and join ... | ### Part (a)
1. **Given**: The circles \( K_1, K_2, K_3 \) intersect at a common point \( P \). The intersections are \( K_1 \cap K_2 = \{P, A\} \), \( K_2 \cap K_3 = \{P, B\} \), and \( K_3 \cap K_1 = \{P, C\} \). An arbitrary point \( X \) on \( K_1 \) is joined to \( A \) to meet \( K_2 \) again at \( Y \), and \( X... | [
{
"content": "Let there be given three circles $K_1,K_2,K_3$ with centers $O_1,O_2,O_3$ respectively, which meet at a common point $P$ . Also, let $K_1 \\cap K_2 = \\{P,A\\}, K_2 \\cap K_3 = \\{P,B\\}, K_3 \\cap K_1 = \\{P,C\\}$ . Given an arbitrary point $X$ on $K_1$ , join $X$ to $A$ to meet $K_... |
aops_forum | Suppose that both $x^{3}-x$ and $x^{4}-x$ are integers for some real number $x$ . Show that $x$ is an integer. | 1. **Define the problem and given conditions:**
We are given that both \(x^3 - x\) and \(x^4 - x\) are integers for some real number \(x\). We need to show that \(x\) is an integer.
2. **Establish that \(x\) is an algebraic integer:**
Since \(x^3 - x\) and \(x^4 - x\) are integers, \(x\) must be a root of the po... | [
{
"content": "Suppose that both $x^{3}-x$ and $x^{4}-x$ are integers for some real number $x$ . Show that $x$ is an integer.",
"role": "user"
},
{
"content": "1. **Define the problem and given conditions:**\n We are given that both \\(x^3 - x\\) and \\(x^4 - x\\) are integers for some real ... |
aops_forum | Let $n$ be a positive integer. Andrés has $n+1$ cards and each of them has a positive integer written, in such a way that the sum of the $n+1$ numbers is $3n$ . Show that Andrés can place one or more cards in a red box and one or more cards in a blue box in such a way that the sum of the numbers of the cards in ... | We will prove the statement by induction on \( n \).
1. **Base Case:**
For \( n = 1 \):
- Andrés has \( n+1 = 2 \) cards.
- The sum of the numbers on the cards is \( 3n = 3 \).
- The only possible pairs of positive integers that sum to 3 are (1, 2) and (2, 1).
- We can place the card with 1 in the blue ... | [
{
"content": "Let $n$ be a positive integer. Andrés has $n+1$ cards and each of them has a positive integer written, in such a way that the sum of the $n+1$ numbers is $3n$ . Show that Andrés can place one or more cards in a red box and one or more cards in a blue box in such a way that the sum of the nu... |
aops_forum | Given that $x$ and $y$ are positive integers such that $xy^{10}$ is perfect 33rd power of a positive integer, prove that $x^{10}y$ is also a perfect 33rd power! | 1. **Given Information:**
- \( x \) and \( y \) are positive integers.
- \( xy^{10} \) is a perfect 33rd power of a positive integer.
2. **Express \( xy^{10} \) as a 33rd power:**
- Let \( xy^{10} = k^{33} \) for some positive integer \( k \).
3. **Prime Factorization:**
- Let \( x = \prod_{i=1}^n p_i^{e_... | [
{
"content": "Given that $x$ and $y$ are positive integers such that $xy^{10}$ is perfect 33rd power of a positive integer, prove that $x^{10}y$ is also a perfect 33rd power!",
"role": "user"
},
{
"content": "1. **Given Information:**\n - \\( x \\) and \\( y \\) are positive integers.\n ... |
aops_forum | Let $\displaystyle \mathcal K$ be a finite field. Prove that the following statements are equivalent:
(a) $\displaystyle 1+1=0$ ;
(b) for all $\displaystyle f \in \mathcal K \left[ X \right]$ with $\displaystyle \textrm{deg} \, f \geq 1$ , $\displaystyle f \left( X^2 \right)$ is reducible. | To prove that the statements (a) and (b) are equivalent, we need to show both directions: (a) implies (b) and (b) implies (a).
1. **(a) implies (b):**
Assume that $1 + 1 = 0$ in the finite field $\mathcal{K}$. This implies that the characteristic of $\mathcal{K}$ is 2.
Let $f \in \mathcal{K}[X]$ be a polynomi... | [
{
"content": "Let $\\displaystyle \\mathcal K$ be a finite field. Prove that the following statements are equivalent:\r\n\r\n(a) $\\displaystyle 1+1=0$ ;\r\n\r\n(b) for all $\\displaystyle f \\in \\mathcal K \\left[ X \\right]$ with $\\displaystyle \\textrm{deg} \\, f \\geq 1$ , $\\displaystyle f \\left(... |
aops_forum | The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$ .
**a.)** Prove that $0\le b_n<2$ .
**b.)** Given $c$ satisfying $0\le c<2$ , prove that we can find $a_n$ so that $b_n>c$ for all suffi... | ### Part (a)
We need to prove that \(0 \leq b_n < 2\).
1. **Define \(u_n\):**
Let \(u_n = \sqrt{a_n}\) for all \(n \geq 0\). This simplifies our expressions involving square roots.
2. **Rewrite \(b_n\):**
\[
b_n = \sum_{k=1}^n \left(1 - \frac{a_{k-1}}{a_k}\right) \frac{1}{\sqrt{a_k}} = \sum_{k=1}^n \left(1 ... | [
{
"content": "The real numbers $a_0,a_1,a_2,\\ldots$ satisfy $1=a_0\\le a_1\\le a_2\\le\\ldots. b_1,b_2,b_3,\\ldots$ are defined by $b_n=\\sum_{k=1}^n{1-{a_{k-1}\\over a_k}\\over\\sqrt a_k}$ .\r\n\r**a.)** Prove that $0\\le b_n<2$ .\r\n\r**b.)** Given $c$ satisfying $0\\le c<2$ , prove that we can find... |
aops_forum | Are there natural $a, b >1000$ , such that for any $c$ that is a perfect square, the three numbers $a, b$ and $c$ are not the lengths of the sides of a triangle? | 1. **Define the values of \(a\) and \(b\):**
Let \(a = 10^{100} + 1002\) and \(b = 1001\). Note that both \(a\) and \(b\) are greater than 1000, satisfying the given condition.
2. **Triangle inequality conditions:**
For \(a\), \(b\), and \(c\) to be the sides of a triangle, the following inequalities must hold:
... | [
{
"content": "Are there natural $a, b >1000$ , such that for any $c$ that is a perfect square, the three numbers $a, b$ and $c$ are not the lengths of the sides of a triangle?",
"role": "user"
},
{
"content": "1. **Define the values of \\(a\\) and \\(b\\):**\n Let \\(a = 10^{100} + 1002\\)... |
aops_forum | It is given sequence wih length of $2017$ which consists of first $2017$ positive integers in arbitrary order (every number occus exactly once). Let us consider a first term from sequence, let it be $k$ . From given sequence we form a new sequence of length 2017, such that first $k$ elements of new sequence are ... | 1. **Base Case:**
- Consider a sequence of length 1. The sequence is simply \([1]\). The first element is already 1, so the statement is trivially true for \(n = 1\).
2. **Inductive Hypothesis:**
- Assume that for a sequence of length \(n\), the statement holds true. That is, if we continue transforming the sequ... | [
{
"content": "It is given sequence wih length of $2017$ which consists of first $2017$ positive integers in arbitrary order (every number occus exactly once). Let us consider a first term from sequence, let it be $k$ . From given sequence we form a new sequence of length 2017, such that first $k$ element... |
aops_forum | Real numbers $x_1, x_2, x_3, x_4$ are roots of the fourth degree polynomial $W (x)$ with integer coefficients.
Prove that if $x_3 + x_4$ is a rational number and $x_3x_4$ is a irrational number, then $x_1 + x_2 = x_3 + x_4$ . | 1. Let \( W(x) \) be a fourth-degree polynomial with integer coefficients, and let \( x_1, x_2, x_3, x_4 \) be its roots. By Vieta's formulas, we know:
\[
x_1 + x_2 + x_3 + x_4 \in \mathbb{Q}
\]
\[
x_1x_2 + x_1x_3 + x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4 \in \mathbb{Q}
\]
\[
x_1x_2x_3 + x_1x_2x_4 + x... | [
{
"content": "Real numbers $x_1, x_2, x_3, x_4$ are roots of the fourth degree polynomial $W (x)$ with integer coefficients. \nProve that if $x_3 + x_4$ is a rational number and $x_3x_4$ is a irrational number, then $x_1 + x_2 = x_3 + x_4$ .",
"role": "user"
},
{
"content": "1. Let \\( W(x)... |
aops_forum | Let $ABCD$ be a concyclic quadrilateral such that $AB:AD=CD:CB.$ The line $AD$ intersects the line $BC$ at $X$ , and the line $AB$ intersects the line $CD$ at $Y$ . Let $E,\ F,\ G$ and $H$ are the midpoints of the edges $AB,\ BC,\ CD$ and $DA$ respectively. The bisector of angle $AXB$ intersects... | 1. Given that $ABCD$ is a concyclic quadrilateral, we know that the opposite angles sum to $180^\circ$. This property will be useful in our proof.
2. We are given the ratio $AB:AD = CD:CB$. This implies that $\triangle XBA \sim \triangle XDC$ by the Angle-Angle (AA) similarity criterion.
3. Since $\triangle XBA \sim \t... | [
{
"content": "Let $ABCD$ be a concyclic quadrilateral such that $AB:AD=CD:CB.$ The line $AD$ intersects the line $BC$ at $X$ , and the line $AB$ intersects the line $CD$ at $Y$ . Let $E,\\ F,\\ G$ and $H$ are the midpoints of the edges $AB,\\ BC,\\ CD$ and $DA$ respectively. The bisector o... |
aops_forum | Each cell of a $100$ x $100$ board is painted in one of $100$ different colors so that there are exactly $100$ cells of each color. Prove that there is a row or column in which there are at least $10$ cells of different colors. | 1. **Define Sets and Projections:**
Let \( S_i \) be the set of cells painted in color \( i \) for \( 1 \leq i \leq 100 \). Each \( S_i \) contains exactly 100 cells. Let \( P_i \) be the projection of \( S_i \) on the horizontal and vertical sides of the board. We aim to show that \( \# P_i \geq 20 \).
2. **Boundi... | [
{
"content": "Each cell of a $100$ x $100$ board is painted in one of $100$ different colors so that there are exactly $100$ cells of each color. Prove that there is a row or column in which there are at least $10$ cells of different colors.",
"role": "user"
},
{
"content": "1. **Define S... |
aops_forum | Let $ABCD$ be a rectangle with $AB\ge BC$ Point $M$ is located on the side $(AD)$ , and the perpendicular bisector of $[MC]$ intersects the line $BC$ at the point $N$ . Let ${Q} =MN\cup AB$ . Knowing that $\angle MQA= 2\cdot \angle BCQ $ , show that the quadrilateral $ABCD$ is a square. | 1. **Define the points and angles:**
- Let $ABCD$ be a rectangle with $AB \ge BC$.
- Point $M$ is on side $AD$.
- The perpendicular bisector of $MC$ intersects $BC$ at point $N$.
- Let $Q$ be the intersection of $MN$ and $AB$.
- Given that $\angle MQA = 2 \cdot \angle BCQ$.
2. **Introduce auxiliary poin... | [
{
"content": "Let $ABCD$ be a rectangle with $AB\\ge BC$ Point $M$ is located on the side $(AD)$ , and the perpendicular bisector of $[MC]$ intersects the line $BC$ at the point $N$ . Let ${Q} =MN\\cup AB$ . Knowing that $\\angle MQA= 2\\cdot \\angle BCQ $ , show that the quadrilateral $ABCD... |
aops_forum | Let $a_1$ , $a_2$ ,..., $a_{99}$ be a sequence of digits from the set ${0,...,9}$ such that if for some $n$ ∈ $N$ , $a_n = 1$ , then $a_{n+1} \ne 2$ , and if $a_n = 3$ then $a_{n+1} \ne 4$ . Prove that there exist indices $k,l$ ∈ ${1,...,98}$ such that $a_k = a_l$ and $a_{k+1} = a_{l+1}$ . | 1. **Define the problem and assumptions:**
We are given a sequence of digits \(a_1, a_2, \ldots, a_{99}\) from the set \(\{0, 1, \ldots, 9\}\) with the conditions:
- If \(a_n = 1\), then \(a_{n+1} \neq 2\).
- If \(a_n = 3\), then \(a_{n+1} \neq 4\).
We need to prove that there exist indices \(k, l \in \{1,... | [
{
"content": "Let $a_1$ , $a_2$ ,..., $a_{99}$ be a sequence of digits from the set ${0,...,9}$ such that if for some $n$ ∈ $N$ , $a_n = 1$ , then $a_{n+1} \\ne 2$ , and if $a_n = 3$ then $a_{n+1} \\ne 4$ . Prove that there exist indices $k,l$ ∈ ${1,...,98}$ such that $a_k = a_l$ and $a_{k... |
aops_forum | Let $ABCDE$ be a pentagon inscribed in a circle $\Omega$ . A line parallel to the segment $BC$ intersects $AB$ and $AC$ at points $S$ and $T$ , respectively. Let $X$ be the intersection of the line $BE$ and $DS$ , and $Y$ be the intersection of the line $CE$ and $DT$ .
Prove that, if the line... | 1. **Identify Key Points and Lines:**
- Let $ABCDE$ be a pentagon inscribed in a circle $\Omega$.
- A line parallel to $BC$ intersects $AB$ and $AC$ at points $S$ and $T$, respectively.
- Let $X$ be the intersection of the line $BE$ and $DS$, and $Y$ be the intersection of the line $CE$ and $DT$.
2. **Apply P... | [
{
"content": " Let $ABCDE$ be a pentagon inscribed in a circle $\\Omega$ . A line parallel to the segment $BC$ intersects $AB$ and $AC$ at points $S$ and $T$ , respectively. Let $X$ be the intersection of the line $BE$ and $DS$ , and $Y$ be the intersection of the line $CE$ and $DT$ .\n\n... |
aops_forum | Given a set $X=\{x_1,\ldots,x_n\}$ of natural numbers in which for all $1< i \leq n$ we have $1\leq x_i-x_{i-1}\leq 2$ , call a real number $a$ **good** if there exists $1\leq j \leq n$ such that $2|x_j-a|\leq 1$ . Also a subset of $X$ is called **compact** if the average of its elements is a good number.
... | 1. **Define the problem and notation:**
- Given a set \( X = \{x_1, x_2, \ldots, x_n\} \) of natural numbers where for all \( 1 < i \leq n \), \( 1 \leq x_i - x_{i-1} \leq 2 \).
- A real number \( a \) is called **good** if there exists \( 1 \leq j \leq n \) such that \( 2|x_j - a| \leq 1 \).
- A subset of \( ... | [
{
"content": "Given a set $X=\\{x_1,\\ldots,x_n\\}$ of natural numbers in which for all $1< i \\leq n$ we have $1\\leq x_i-x_{i-1}\\leq 2$ , call a real number $a$ **good** if there exists $1\\leq j \\leq n$ such that $2|x_j-a|\\leq 1$ . Also a subset of $X$ is called **compact** if the average of i... |
aops_forum | On a $55\times 55$ square grid, $500$ unit squares were cut out as well as $400$ L-shaped pieces consisting of 3 unit squares (each piece can be oriented in any way) [refer to the figure]. Prove that at least two of the cut out pieces bordered each other before they were cut out.
[asy]size(2.013cm);
draw ((0,0)--... | 1. **Define the problem and the grid:**
We have a \(55 \times 55\) square grid, which contains \(55^2 = 3025\) unit squares. We are cutting out 500 unit squares and 400 L-shaped pieces, each consisting of 3 unit squares.
2. **Calculate the total number of unit squares removed:**
Each L-shaped piece consists of 3... | [
{
"content": "On a $55\\times 55$ square grid, $500$ unit squares were cut out as well as $400$ L-shaped pieces consisting of 3 unit squares (each piece can be oriented in any way) [refer to the figure]. Prove that at least two of the cut out pieces bordered each other before they were cut out.\n[asy]size... |
aops_forum | Given $n$ points in a circle, Arnaldo write 0 or 1 in all the points. Bernado can do a operation, he can chosse some point and change its number and the numbers of the points on the right and left side of it. Arnaldo wins if Bernado can´t change all the numbers in the circle to 0, and Bernado wins if he can
a) Show ... | ### Part (a): Show that Bernado can win if \( n = 101 \)
1. **Initial Setup**: Consider the circle with 101 points. Each point is either 0 or 1. Bernado's goal is to change all the numbers to 0.
2. **Operation Description**: Bernado can choose any point and change its number and the numbers of the points on the right... | [
{
"content": "Given $n$ points in a circle, Arnaldo write 0 or 1 in all the points. Bernado can do a operation, he can chosse some point and change its number and the numbers of the points on the right and left side of it. Arnaldo wins if Bernado can´t change all the numbers in the circle to 0, and Bernado wi... |
aops_forum | Let $ABC$ be an equilateral triangle and $D$ the midpoint of $BC$ . Let $E$ and $F$ be points on $AC$ and $AB$ respectively such that $AF=CE$ . $P=BE$ $\cap$ $CF$ .
Show that $\angle $ $APF=$ $\angle$ $ BPD$ | 1. **Define the Rotation and Images:**
Let \( F_1 \) and \( P_1 \) be the images of \( F \) and \( P \) under the rotation about \( C \) that takes \( A \) to \( B \). This means that \( F_1 \) is the point on \( BC \) such that \( CF_1 = AF \) and \( P_1 \) is the point on \( BE \) such that \( CP_1 = AP \).
2. **... | [
{
"content": "Let $ABC$ be an equilateral triangle and $D$ the midpoint of $BC$ . Let $E$ and $F$ be points on $AC$ and $AB$ respectively such that $AF=CE$ . $P=BE$ $\\cap$ $CF$ . \nShow that $\\angle $ $APF=$ $\\angle$ $ BPD$ ",
"role": "user"
},
{
"content": "1. **Define th... |
aops_forum | Given $101$ distinct non-negative integers less than $5050$ show that one can choose four $a, b, c, d$ such that $a + b - c - d$ is a multiple of $5050$ | To solve this problem, we will use the Pigeonhole Principle and properties of modular arithmetic. The goal is to show that among 101 distinct non-negative integers less than 5050, we can find four distinct integers \(a, b, c, d\) such that \(a + b - c - d\) is a multiple of 5050.
1. **Consider the differences modulo 5... | [
{
"content": "Given $101$ distinct non-negative integers less than $5050$ show that one can choose four $a, b, c, d$ such that $a + b - c - d$ is a multiple of $5050$ ",
"role": "user"
},
{
"content": "To solve this problem, we will use the Pigeonhole Principle and properties of modular ari... |
aops_forum | Let $ABC$ be a triangle and $D, E, F$ be the foots of altitudes drawn from $A,B,C$ respectively. Let $H$ be the orthocenter of $ABC$ . Lines $EF$ and $AD$ intersect at $G$ . Let $K$ the point on circumcircle of $ABC$ such that $AK$ is a diameter of this circle. $AK$ cuts $BC$ in $M$ . Prove th... | 1. **Define the complex numbers for the vertices of the triangle:**
Let \( a, b, c \) be complex numbers lying on the unit circle such that they are the vertices of the triangle \( \triangle ABC \).
2. **Determine the orthocenter \( H \):**
The orthocenter \( H \) of \( \triangle ABC \) is given by the sum of t... | [
{
"content": "Let $ABC$ be a triangle and $D, E, F$ be the foots of altitudes drawn from $A,B,C$ respectively. Let $H$ be the orthocenter of $ABC$ . Lines $EF$ and $AD$ intersect at $G$ . Let $K$ the point on circumcircle of $ABC$ such that $AK$ is a diameter of this circle. $AK$ cuts $B... |
aops_forum | Let $a_1=1,$ $a_2=2,$ $a_n=2a_{n-1}+a_{n-2},$ $n=3,4,\cdots.$ Prove that for any integer $n\geq5,$ $a_n$ has at least one prime factor $p,$ such that $p\equiv 1\pmod{4}.$ | 1. **Define the sequence and characteristic polynomial:**
Given the sequence \(a_1 = 1\), \(a_2 = 2\), and \(a_n = 2a_{n-1} + a_{n-2}\) for \(n \geq 3\), we can find the characteristic polynomial of the recurrence relation:
\[
x^2 - 2x - 1 = 0
\]
Solving this quadratic equation, we get the roots:
\[
... | [
{
"content": "Let $a_1=1,$ $a_2=2,$ $a_n=2a_{n-1}+a_{n-2},$ $n=3,4,\\cdots.$ Prove that for any integer $n\\geq5,$ $a_n$ has at least one prime factor $p,$ such that $p\\equiv 1\\pmod{4}.$ ",
"role": "user"
},
{
"content": "1. **Define the sequence and characteristic polynomial:**\n ... |
aops_forum | All integer numbers are colored in 3 colors in arbitrary way. Prove that there are two distinct numbers whose difference is a perfect square and the numbers are colored in the same color. | 1. **Assume the contrary**: Suppose that there is a way to color all integers using three colors such that no two distinct numbers whose difference is a perfect square are colored the same.
2. **Consider the set of numbers**: Let's examine the numbers \( n, n+9, n+16, n+25 \). These numbers are chosen because the diff... | [
{
"content": "All integer numbers are colored in 3 colors in arbitrary way. Prove that there are two distinct numbers whose difference is a perfect square and the numbers are colored in the same color.",
"role": "user"
},
{
"content": "1. **Assume the contrary**: Suppose that there is a way to color... |
aops_forum | We call a natural number *almost a square* if it can be represented as a product of two numbers that differ by no more than one percent of the larger of them. Prove that there are infinitely many consecutive quadruples of almost squares. | 1. **Definition and Initial Setup**:
We need to prove that there are infinitely many consecutive quadruples of almost squares. A natural number \( n \) is called *almost a square* if it can be represented as a product of two numbers that differ by no more than one percent of the larger of them.
2. **Identifying Alm... | [
{
"content": "We call a natural number *almost a square* if it can be represented as a product of two numbers that differ by no more than one percent of the larger of them. Prove that there are infinitely many consecutive quadruples of almost squares.",
"role": "user"
},
{
"content": "1. **Definitio... |
aops_forum | Let $a,b, c$ be $3$ distinct numbers from $\{1, 2,3, 4, 5, 6\}$ Show that $7$ divides $abc + (7 - a)(7 - b)(7 - c)$ | 1. We start by noting that the numbers \(a, b, c\) are distinct elements from the set \(\{1, 2, 3, 4, 5, 6\}\). We need to show that \(7\) divides \(abc + (7 - a)(7 - b)(7 - c)\).
2. First, observe that for any integer \(x\), we have:
\[
x \equiv x \pmod{7}
\]
This is a basic property of congruences.
3. N... | [
{
"content": "Let $a,b, c$ be $3$ distinct numbers from $\\{1, 2,3, 4, 5, 6\\}$ Show that $7$ divides $abc + (7 - a)(7 - b)(7 - c)$ ",
"role": "user"
},
{
"content": "1. We start by noting that the numbers \\(a, b, c\\) are distinct elements from the set \\(\\{1, 2, 3, 4, 5, 6\\}\\). We need... |
aops_forum | Let the diagonals of cyclic quadrilateral $ABCD$ meet at point $P$ . The line passing through $P$ and perpendicular to $PD$ meets $AD$ at point $D_1$ , a point $A_1$ is defined similarly. Prove that the tangent at $P$ to the circumcircle of triangle $D_1PA_1$ is parallel to $BC$ . | 1. **Inversion at Point \( P \)**:
Consider a negative inversion centered at point \( P \) that swaps points \( A \) and \( C \), and points \( B \) and \( D \). Under this inversion, the line \( AD \) inverts to the circumcircle of \( \triangle BCP \), denoted as \( \omega \).
2. **Mapping of Points \( A_1 \) and ... | [
{
"content": "Let the diagonals of cyclic quadrilateral $ABCD$ meet at point $P$ . The line passing through $P$ and perpendicular to $PD$ meets $AD$ at point $D_1$ , a point $A_1$ is defined similarly. Prove that the tangent at $P$ to the circumcircle of triangle $D_1PA_1$ is parallel to $BC$ .... |
aops_forum | A point $O$ lies inside of the square $ABCD$ . Prove that the difference between the sum of angles $OAB, OBC, OCD , ODA$ and $180^{\circ}$ does not exceed $45^{\circ}$ . | 1. **Place the vertices of the square and the point \( O \):**
- Let the vertices of the square \( ABCD \) be \( A(1,1) \), \( B(-1,1) \), \( C(-1,-1) \), and \( D(1,-1) \).
- Let the point \( O \) be at coordinates \( (x, y) \), where \( -1 < x < 1 \) and \( -1 < y < 1 \).
2. **Express the tangent of each angle... | [
{
"content": "A point $O$ lies inside of the square $ABCD$ . Prove that the difference between the sum of angles $OAB, OBC, OCD , ODA$ and $180^{\\circ}$ does not exceed $45^{\\circ}$ .",
"role": "user"
},
{
"content": "1. **Place the vertices of the square and the point \\( O \\):**\n - L... |
aops_forum | Let $f(x)=x^3+17$ . Prove that for every integer $n\ge 2$ there exists a natural number $x$ for which $f(x)$ is divisible by $3^n$ but not by $3^{n+1}$ . | To prove that for every integer \( n \ge 2 \), there exists a natural number \( x \) for which \( f(x) = x^3 + 17 \) is divisible by \( 3^n \) but not by \( 3^{n+1} \), we will use induction.
1. **Base Case: \( n = 2 \)**
We need to find \( x \) such that \( f(x) \) is divisible by \( 3^2 = 9 \) but not by \( 3^3 ... | [
{
"content": "Let $f(x)=x^3+17$ . Prove that for every integer $n\\ge 2$ there exists a natural number $x$ for which $f(x)$ is divisible by $3^n$ but not by $3^{n+1}$ .",
"role": "user"
},
{
"content": "To prove that for every integer \\( n \\ge 2 \\), there exists a natural number \\( x \... |
aops_forum | Let $ABC$ be an acute triangle. Let $P$ be a point such that $PB$ and $PC$ are tangent to circumcircle of $ABC$ . Let $X$ and $Y$ be variable points on $AB$ and $AC$ , respectively, such that $\angle XPY = 2\angle BAC$ and $P$ lies in the interior of triangle $AXY$ . Let $Z$ be the reflection of ... | 1. **Lemma:** Given a quadrilateral \(ABCD\) and a point \(P\) in the plane of the quadrilateral, there exists a point \(Q\), the isogonal conjugate of \(P\) with respect to \(ABCD\) if and only if \(\measuredangle APB = \measuredangle DPC\).
**Proof of Lemma:** See [this](https://artofproblemsolving.com/community/... | [
{
"content": "Let $ABC$ be an acute triangle. Let $P$ be a point such that $PB$ and $PC$ are tangent to circumcircle of $ABC$ . Let $X$ and $Y$ be variable points on $AB$ and $AC$ , respectively, such that $\\angle XPY = 2\\angle BAC$ and $P$ lies in the interior of triangle $AXY$ . Let $Z... |
aops_forum | Let $P(x)$ be a polynomial with real non-negative coefficients. Let $k$ be a positive integer and $x_1, x_2, \dots, x_k$ positive real numbers such that $x_1x_2\cdots x_k=1$ . Prove that $$ P(x_1)+P(x_2)+\cdots+P(x_k)\geq kP(1). $$ | 1. **Apply the AM-GM Inequality:**
The Arithmetic Mean-Geometric Mean (AM-GM) Inequality states that for any non-negative real numbers \(a_1, a_2, \ldots, a_k\),
\[
\frac{a_1 + a_2 + \cdots + a_k}{k} \geq \sqrt[k]{a_1 a_2 \cdots a_k}.
\]
Given \(x_1, x_2, \ldots, x_k\) are positive real numbers such that... | [
{
"content": "Let $P(x)$ be a polynomial with real non-negative coefficients. Let $k$ be a positive integer and $x_1, x_2, \\dots, x_k$ positive real numbers such that $x_1x_2\\cdots x_k=1$ . Prove that $$ P(x_1)+P(x_2)+\\cdots+P(x_k)\\geq kP(1). $$ ",
"role": "user"
},
{
"content": "1. **... |
aops_forum | The point $M$ is the midpoint of the side $BC$ of triangle $ABC$ . A circle is passing through $B$ , is tangent to the line $AM$ at $M$ , and intersects the segment $AB$ secondary at the point $P$ .
Prove that the circle, passing through $A$ , $P$ , and the midpoint of the segment $AM$ , is tangent to th... | 1. **Define Points and Circle Properties:**
- Let \( M \) be the midpoint of side \( BC \) in triangle \( ABC \).
- A circle passes through \( B \), is tangent to line \( AM \) at \( M \), and intersects segment \( AB \) at point \( P \).
2. **Introduce Parallelogram and Midpoint:**
- Let \( D \) be a point s... | [
{
"content": "The point $M$ is the midpoint of the side $BC$ of triangle $ABC$ . A circle is passing through $B$ , is tangent to the line $AM$ at $M$ , and intersects the segment $AB$ secondary at the point $P$ .\nProve that the circle, passing through $A$ , $P$ , and the midpoint of the segment ... |
aops_forum | Let $a,b$ be two different positive integers. Suppose that $a,b$ are relatively prime. Prove that $\dfrac{2a(a^2+b^2)}{a^2-b^2}$ is not an integer. | To prove that $\dfrac{2a(a^2+b^2)}{a^2-b^2}$ is not an integer, we will proceed by contradiction. Assume that $\dfrac{2a(a^2+b^2)}{a^2-b^2}$ is an integer. This implies that $a^2 - b^2$ divides $2a(a^2 + b^2)$.
1. **Assume the contrary**: Suppose $\dfrac{2a(a^2+b^2)}{a^2-b^2}$ is an integer. This implies:
\[
a^2... | [
{
"content": "Let $a,b$ be two different positive integers. Suppose that $a,b$ are relatively prime. Prove that $\\dfrac{2a(a^2+b^2)}{a^2-b^2}$ is not an integer.",
"role": "user"
},
{
"content": "To prove that $\\dfrac{2a(a^2+b^2)}{a^2-b^2}$ is not an integer, we will proceed by contradiction... |
aops_forum | Five points $A$ , $B$ , $C$ , $D$ and $E$ lie on a circle $\tau$ clockwise in that order such that $AB \parallel CE$ and $\angle ABC > 90^{\circ}$ . Let $k$ be a circle tangent to $AD$ , $CE$ and $\tau$ such that $k$ and $\tau$ touch on the arc $\widehat{DE}$ not containing $A$ , $B$ and $C... | 1. **Restate the problem in terms of cyclic quadrilateral and mixtilinear circles:**
- Given five points \( A, B, C, D, E \) on a circle \(\tau\) such that \( AB \parallel CE \) and \( \angle ABC > 90^\circ \).
- Circle \( k \) is tangent to \( AD \), \( CE \), and \(\tau\), touching \(\tau\) on the arc \(\wideha... | [
{
"content": "Five points $A$ , $B$ , $C$ , $D$ and $E$ lie on a circle $\\tau$ clockwise in that order such that $AB \\parallel CE$ and $\\angle ABC > 90^{\\circ}$ . Let $k$ be a circle tangent to $AD$ , $CE$ and $\\tau$ such that $k$ and $\\tau$ touch on the arc $\\widehat{DE}$ not co... |
aops_forum | Prove that the polynomial $z^{2n} + z^n + 1\ (n \in \mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$ . | 1. **Identify the roots of the polynomial \( z^2 + z + 1 \):**
The roots of the polynomial \( z^2 + z + 1 \) are the non-real cube roots of unity, denoted as \( \omega \) and \( \omega^2 \), where \( \omega = e^{2\pi i / 3} \) and \( \omega^2 = e^{-2\pi i / 3} \). These roots satisfy the equation:
\[
\omega^2 ... | [
{
"content": "Prove that the polynomial $z^{2n} + z^n + 1\\ (n \\in \\mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$ .",
"role": "user"
},
{
"content": "1. **Identify the roots of the polynomial \\( z^2 + z + 1 \\):**\n The roots of the ... |
aops_forum | Consider a regular $2^k$ -gon with center $O$ and label its sides clockwise by $l_1,l_2,...,l_{2^k}$ . Reflect $O$ with respect to $l_1$ , then reflect the resulting point with respect to $l_2$ and do this process until the last side. Prove that the distance between the final point and $O$ is less than the p... | 1. **Initial Setup and Reflection Formula:**
Consider a regular $2^k$-gon centered at $O$ with vertices labeled as $a_1, a_2, \ldots, a_{2^k}$. Let $z_1 = O = 0$. Reflect $O$ with respect to $l_1$, then reflect the resulting point with respect to $l_2$, and continue this process until the last side.
Suppose $z_2... | [
{
"content": "Consider a regular $2^k$ -gon with center $O$ and label its sides clockwise by $l_1,l_2,...,l_{2^k}$ . Reflect $O$ with respect to $l_1$ , then reflect the resulting point with respect to $l_2$ and do this process until the last side. Prove that the distance between the final point and $... |
aops_forum | Let $k$ be a positive integer and $m$ be an odd integer. Prove that there exists a positive integer $n$ such that $n^n-m$ is divisible by $2^k$ . | 1. **Base Case:**
- For \( k = 1 \):
- If \( m \equiv 1 \pmod{4} \), choose \( n \equiv 1 \pmod{4} \).
- If \( m \equiv 3 \pmod{4} \), choose \( n \equiv 3 \pmod{4} \).
- For \( k = 2 \):
- If \( m \equiv 1 \pmod{4} \), choose \( n \equiv 1 \pmod{4} \).
- If \( m \equiv 3 \pmod{4} \), choose \... | [
{
"content": "Let $k$ be a positive integer and $m$ be an odd integer. Prove that there exists a positive integer $n$ such that $n^n-m$ is divisible by $2^k$ .",
"role": "user"
},
{
"content": "1. **Base Case:**\n - For \\( k = 1 \\):\n - If \\( m \\equiv 1 \\pmod{4} \\), choose \\( n... |
aops_forum | Given positive integers $n$ and $k$ , say that $n$ is $k$ -*solvable* if there are positive integers $a_1$ , $a_2$ , ..., $a_k$ (not necessarily distinct) such that
\[
\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k} = 1
\]
and
\[
a_1 + a_2 + \cdots + a_k = n.
\]
Prove that if $n$ is $k$ -solvabl... | 1. **Assume \( n \) is \( k \)-solvable**:
Given that \( n \) is \( k \)-solvable, there exist positive integers \( a_1, a_2, \ldots, a_k \) such that:
\[
\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k} = 1
\]
and
\[
a_1 + a_2 + \cdots + a_k = n.
\]
2. **Construct a new tuple for \( 42n... | [
{
"content": "Given positive integers $n$ and $k$ , say that $n$ is $k$ -*solvable* if there are positive integers $a_1$ , $a_2$ , ..., $a_k$ (not necessarily distinct) such that\n\\[\n \\frac{1}{a_1} + \\frac{1}{a_2} + \\cdots + \\frac{1}{a_k} = 1\n\\]\nand\n\\[\n a_1 + a_2 + \\cdots + a_k = n.\n\\... |
aops_forum | Prove that for $ |x| < 1 $ , $ |z| > 1 $ , \[ 1 + \displaystyle\sum_{j=1}^{\infty} \left( 1 + x^j \right) P_j = 0, \]where $P_j$ is \[ \dfrac {(1-z)(1-zx)(1-zx^2) \cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3)\cdots(z-x^j)}. \] | 1. **Define the problem and the given series:**
We need to prove that for \( |x| < 1 \) and \( |z| > 1 \),
\[
1 + \sum_{j=1}^{\infty} \left( 1 + x^j \right) P_j = 0,
\]
where \( P_j \) is given by
\[
P_j = \frac{(1-z)(1-zx)(1-zx^2) \cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3) \cdots (z-x^j)}.
\]
2.... | [
{
"content": "Prove that for $ |x| < 1 $ , $ |z| > 1 $ , \\[ 1 + \\displaystyle\\sum_{j=1}^{\\infty} \\left( 1 + x^j \\right) P_j = 0, \\]where $P_j$ is \\[ \\dfrac {(1-z)(1-zx)(1-zx^2) \\cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3)\\cdots(z-x^j)}. \\]",
"role": "user"
},
{
"content": "1. **Define th... |
aops_forum | Let $ p$ be a prime number with $ p>5$ . Consider the set $ X \equal{} \left\{p \minus{} n^2 \mid n\in \mathbb{N} ,\ n^2 < p\right\}$ .
Prove that the set $ X$ has two distinct elements $ x$ and $ y$ such that $ x\neq 1$ and $ x\mid y$ .
*Albania* | 1. Let \( p \) be a prime number with \( p > 5 \). Consider the set \( X = \{ p - n^2 \mid n \in \mathbb{N},\ n^2 < p \} \).
2. We need to prove that there exist two distinct elements \( x \) and \( y \) in \( X \) such that \( x \neq 1 \) and \( x \mid y \).
First, let's analyze the structure of the set \( X \). For ... | [
{
"content": "Let $ p$ be a prime number with $ p>5$ . Consider the set $ X \\equal{} \\left\\{p \\minus{} n^2 \\mid n\\in \\mathbb{N} ,\\ n^2 < p\\right\\}$ .\r\nProve that the set $ X$ has two distinct elements $ x$ and $ y$ such that $ x\\neq 1$ and $ x\\mid y$ .\r\n\r\n*Albania*",
"role": "... |
aops_forum | On a circle $n \geq 1$ real numbers are written, their sum is $n-1$ . Prove that one can denote these numbers as $x_1, x_2, ..., x_n$ consecutively, starting from a number and moving clockwise, such that for any $k$ ( $1\leq k \leq n$ ) $ x_1 + x_2+...+x_k \geq k-1$ . | 1. **Transform the problem**:
By subtracting 1 from each number, the problem becomes equivalent to:
On a circle, \( n \geq 1 \) real numbers \( a_1, a_2, \ldots, a_n \) are written clockwise around the circle where \( a_1 + a_2 + \ldots + a_n = -1 \). Prove that one can choose \( j \in \{1, 2, \ldots, n\} \) suc... | [
{
"content": "On a circle $n \\geq 1$ real numbers are written, their sum is $n-1$ . Prove that one can denote these numbers as $x_1, x_2, ..., x_n$ consecutively, starting from a number and moving clockwise, such that for any $k$ ( $1\\leq k \\leq n$ ) $ x_1 + x_2+...+x_k \\geq k-1$ .",
"role": "u... |
aops_forum | Let $XY$ be a chord of a circle $\Omega$ , with center $O$ , which is not a diameter. Let $P, Q$ be two distinct points inside the segment $XY$ , where $Q$ lies between $P$ and $X$ . Let $\ell$ the perpendicular line drawn from $P$ to the diameter which passes through $Q$ . Let $M$ be the intersectio... | 1. Let $XY$ be a chord of a circle $\Omega$ with center $O$, which is not a diameter. Let $P$ and $Q$ be two distinct points inside the segment $XY$, where $Q$ lies between $P$ and $X$. Let $\ell$ be the perpendicular line drawn from $P$ to the diameter which passes through $Q$. Let $M$ be the intersection point of $\e... | [
{
"content": "Let $XY$ be a chord of a circle $\\Omega$ , with center $O$ , which is not a diameter. Let $P, Q$ be two distinct points inside the segment $XY$ , where $Q$ lies between $P$ and $X$ . Let $\\ell$ the perpendicular line drawn from $P$ to the diameter which passes through $Q$ . Let ... |
aops_forum | The sequences $(a_n),(b_n)$ are defined by $a_0=1,b_0=4$ and for $n\ge 0$ \[a_{n+1}=a_n^{2001}+b_n,\ \ b_{n+1}=b_n^{2001}+a_n\]
Show that $2003$ is not divisor of any of the terms in these two sequences. | 1. **Initial Setup and Assumptions:**
We are given the sequences \((a_n)\) and \((b_n)\) defined by \(a_0 = 1\) and \(b_0 = 4\), with the recurrence relations:
\[
a_{n+1} = a_n^{2001} + b_n, \quad b_{n+1} = b_n^{2001} + a_n
\]
We need to show that \(2003\) is not a divisor of any term in these sequences.... | [
{
"content": "The sequences $(a_n),(b_n)$ are defined by $a_0=1,b_0=4$ and for $n\\ge 0$ \\[a_{n+1}=a_n^{2001}+b_n,\\ \\ b_{n+1}=b_n^{2001}+a_n\\]\nShow that $2003$ is not divisor of any of the terms in these two sequences.",
"role": "user"
},
{
"content": "1. **Initial Setup and Assumptions... |
aops_forum | Let $M=\{A_{1},A_{2},\ldots,A_{5}\}$ be a set of five points in the plane such that the area of each triangle $A_{i}A_{j}A_{k}$ , is greater than 3. Prove that there exists a triangle with vertices in $M$ and having the area greater than 4.
*Laurentiu Panaitopol* | 1. Consider the set \( M = \{A_1, A_2, A_3, A_4, A_5\} \) of five points in the plane. We are given that the area of each triangle formed by any three points from \( M \) is greater than 3. We need to prove that there exists at least one triangle with vertices in \( M \) having an area greater than 4.
2. Let's denote ... | [
{
"content": "Let $M=\\{A_{1},A_{2},\\ldots,A_{5}\\}$ be a set of five points in the plane such that the area of each triangle $A_{i}A_{j}A_{k}$ , is greater than 3. Prove that there exists a triangle with vertices in $M$ and having the area greater than 4.\r\n\r\n*Laurentiu Panaitopol*",
"role": "user... |
aops_forum | Let $n$ be a positive integer. Show that there exist three distinct integers
between $n^{2}$ and $n^{2}+n+3\sqrt{n}$ , such that one of them divides the product of the other two. | 1. **Define the integers:**
Let \( a = (n+k)(n-k+1) \), \( b = (n+k+1)(n-k+1) \), and \( c = (n+k)(n-k+2) \) where \( k \) is a positive integer.
2. **Check the divisibility condition:**
We need to show that \( a \) divides the product \( bc \). First, let's expand \( a \), \( b \), and \( c \):
\[
a = (n+... | [
{
"content": "Let $n$ be a positive integer. Show that there exist three distinct integers \r\nbetween $n^{2}$ and $n^{2}+n+3\\sqrt{n}$ , such that one of them divides the product of the other two.",
"role": "user"
},
{
"content": "1. **Define the integers:**\n Let \\( a = (n+k)(n-k+1) \\), \... |
aops_forum | Let $a,b,c,d$ be positive integers such that $ab\equal{}cd$ . Prove that $a\plus{}b\plus{}c\plus{}d$ is a composite number. | To prove that \(a + b + c + d\) is a composite number given that \(ab = cd\), we will use two different approaches.
### Approach 1:
1. **Lemma**: If \(m\) divides \(xy\) and \(0 < |x|, |y| < m\), then \(m\) is composite.
2. Given \(ab = cd\), we can rewrite it as:
\[
a(a + b + c + d) = (a + c)(a + d)
\]
3. Th... | [
{
"content": "Let $a,b,c,d$ be positive integers such that $ab\\equal{}cd$ . Prove that $a\\plus{}b\\plus{}c\\plus{}d$ is a composite number.",
"role": "user"
},
{
"content": "To prove that \\(a + b + c + d\\) is a composite number given that \\(ab = cd\\), we will use two different approaches.... |
aops_forum | Prove that for any natural number $n < 2310 $ , $n(2310-n)$ is not divisible by $2310$ . | To prove that for any natural number \( n < 2310 \), \( n(2310 - n) \) is not divisible by \( 2310 \), we can proceed as follows:
1. **Prime Factorization of 2310**:
\[
2310 = 2 \times 3 \times 5 \times 7 \times 11
\]
This means that any number divisible by \( 2310 \) must be divisible by all these prime f... | [
{
"content": "Prove that for any natural number $n < 2310 $ , $n(2310-n)$ is not divisible by $2310$ .",
"role": "user"
},
{
"content": "To prove that for any natural number \\( n < 2310 \\), \\( n(2310 - n) \\) is not divisible by \\( 2310 \\), we can proceed as follows:\n\n1. **Prime Factoriz... |
aops_forum | Given a positive real number $a_1$ , we recursively define $a_{n+1} = 1+a_1 a_2 \cdots \cdot a_n.$ Furthermore, let $$ b_n = \frac{1}{a_1 } + \frac{1}{a_2 } +\cdots + \frac{1}{a_n }. $$ Prove that $b_n < \frac{2}{a_1}$ for all positive integers $n$ and that this is the smallest possible bound. | 1. We start with the given recursive definition \( a_{n+1} = 1 + a_1 a_2 \cdots a_n \) and the sequence \( b_n = \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n} \).
2. To prove the inequality \( b_n < \frac{2}{a_1} \), we first express \( \frac{1}{a_{n+1}} \) in terms of the previous terms:
\[
\frac{1}{a_... | [
{
"content": "Given a positive real number $a_1$ , we recursively define $a_{n+1} = 1+a_1 a_2 \\cdots \\cdot a_n.$ Furthermore, let $$ b_n = \\frac{1}{a_1 } + \\frac{1}{a_2 } +\\cdots + \\frac{1}{a_n }. $$ Prove that $b_n < \\frac{2}{a_1}$ for all positive integers $n$ and that this is the smallest pos... |
aops_forum | Let $S$ be a set of $n\ge2$ positive integers. Prove that there exist at least $n^2$ integers that can be written in the form $x+yz$ with $x,y,z\in S$ .
*Proposed by Gerhard Woeginger, Austria* | 1. Let \( S = \{a_1, a_2, \ldots, a_n\} \) with \( a_1 < a_2 < \cdots < a_n \). We need to show that there are at least \( n^2 \) integers that can be written in the form \( x + yz \) with \( x, y, z \in S \).
2. Consider the function \( \phi: S \times S \to \mathbb{Z}^+ \) defined by \( \phi(x, y) = x + y \cdot a_n \... | [
{
"content": "Let $S$ be a set of $n\\ge2$ positive integers. Prove that there exist at least $n^2$ integers that can be written in the form $x+yz$ with $x,y,z\\in S$ .\n\n*Proposed by Gerhard Woeginger, Austria*",
"role": "user"
},
{
"content": "1. Let \\( S = \\{a_1, a_2, \\ldots, a_n\\} ... |
aops_forum | Show that it is possible to express $1$ as a sum of $6$ , and as a sum of $9$ reciprocals of odd positive integers. Generalize the problem. | 1. **Expressing 1 as a sum of 6 reciprocals of odd positive integers:**
We need to find six odd positive integers \( n_1, n_2, n_3, n_4, n_5, n_6 \) such that:
\[
1 = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} + \frac{1}{n_4} + \frac{1}{n_5} + \frac{1}{n_6}
\]
Let's consider the following integers: \... | [
{
"content": "Show that it is possible to express $1$ as a sum of $6$ , and as a sum of $9$ reciprocals of odd positive integers. Generalize the problem.",
"role": "user"
},
{
"content": "1. **Expressing 1 as a sum of 6 reciprocals of odd positive integers:**\n\n We need to find six odd posit... |
aops_forum | Kevin and Yang are playing a game. Yang has $2017 + \tbinom{2017}{2}$ cards with their front sides face down on the table. The cards are constructed as follows:
- For each $1 \le n \le 2017$ , there is a blue card with $n$ written on the back, and a fraction $\tfrac{a_n}{b_n}$ written on the front, where $\gc... | 1. **Understanding the Problem:**
- We have \(2017 + \binom{2017}{2}\) cards.
- There are 2017 blue cards, each with a fraction \(\frac{a_n}{b_n}\) on the front.
- There are \(\binom{2017}{2}\) red cards, each with a fraction \(\frac{a_i + a_j}{b_i + b_j}\) on the front.
- No two cards have the same fractio... | [
{
"content": "Kevin and Yang are playing a game. Yang has $2017 + \\tbinom{2017}{2}$ cards with their front sides face down on the table. The cards are constructed as follows: \n- For each $1 \\le n \\le 2017$ , there is a blue card with $n$ written on the back, and a fraction $\\tfrac{a_n}{b_n}$ writte... |
aops_forum | Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$ , the number $P(n)$ has a proper divisor $d_n$ , i.e., $1<d_n<P(n)$ , such that the sequence $d_1,d_2,d_3,\ldots$ is increasing. Prove that either $P(n)$ is the product of two linear polynomials with integer coeffic... | To prove the given statement, we will consider two cases: when the quadratic polynomial \( P(n) \) is irreducible over the integers and when it is reducible. We will show that if \( P(n) \) is irreducible, then all values of \( P(n) \) for positive integers \( n \) are divisible by the same integer \( m > 1 \). If \( P... | [
{
"content": "Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$ , the number $P(n)$ has a proper divisor $d_n$ , i.e., $1<d_n<P(n)$ , such that the sequence $d_1,d_2,d_3,\\ldots$ is increasing. Prove that either $P(n)$ is the product of two linear polynomia... |
aops_forum | Let a, b, c be the sidelengths and S the area of a triangle ABC. Denote $x=a+\frac{b}{2}$ , $y=b+\frac{c}{2}$ and $z=c+\frac{a}{2}$ . Prove that there exists a triangle with sidelengths x, y, z, and the area of this triangle is $\geq\frac94 S$ . | 1. **Verify that \(x, y, z\) form the sides of a triangle:**
To prove that \(x, y, z\) form the sides of a triangle, we need to show that they satisfy the triangle inequality:
\[
x + y > z, \quad y + z > x, \quad z + x > y
\]
Given:
\[
x = a + \frac{b}{2}, \quad y = b + \frac{c}{2}, \quad z = c + ... | [
{
"content": "Let a, b, c be the sidelengths and S the area of a triangle ABC. Denote $x=a+\\frac{b}{2}$ , $y=b+\\frac{c}{2}$ and $z=c+\\frac{a}{2}$ . Prove that there exists a triangle with sidelengths x, y, z, and the area of this triangle is $\\geq\\frac94 S$ .",
"role": "user"
},
{
"content... |
aops_forum | Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies
\[f(x + y) \leq yf(x) + f(f(x))\]
for all real numbers $x$ and $y$ . Prove that $f(x) = 0$ for all $x \leq 0$ .
*Proposed by Igor Voronovich, Belarus* | 1. Let \( P(x, y) \) denote the given assertion:
\[
P(x, y) : f(x + y) \leq y f(x) + f(f(x))
\]
for all real numbers \( x \) and \( y \).
2. First, consider \( P(x, 0) \):
\[
P(x, 0) : f(x + 0) \leq 0 \cdot f(x) + f(f(x)) \implies f(x) \leq f(f(x))
\]
This implies:
\[
f(f(x)) \geq f(x)
... | [
{
"content": "Let $f : \\mathbb R \\to \\mathbb R$ be a real-valued function defined on the set of real numbers that satisfies\n\\[f(x + y) \\leq yf(x) + f(f(x))\\]\nfor all real numbers $x$ and $y$ . Prove that $f(x) = 0$ for all $x \\leq 0$ .\n\n*Proposed by Igor Voronovich, Belarus*",
"role": "us... |
aops_forum | Define the sequence of positive integers $f_n$ by $f_0 = 1, f_1 = 1, f_{n+2} = f_{n+1} + f_n$ . Show that $f_n =\frac{ (a^{n+1} - b^{n+1})}{\sqrt5}$ , where $a, b$ are real numbers such that $a + b = 1, ab = -1$ and $a > b$ . | 1. **Define the sequence and initial conditions:**
The sequence \( f_n \) is defined by:
\[
f_0 = 1, \quad f_1 = 1, \quad f_{n+2} = f_{n+1} + f_n
\]
We need to show that:
\[
f_n = \frac{a^{n+1} - b^{n+1}}{\sqrt{5}}
\]
where \( a \) and \( b \) are real numbers such that \( a + b = 1 \), \( ab... | [
{
"content": "Define the sequence of positive integers $f_n$ by $f_0 = 1, f_1 = 1, f_{n+2} = f_{n+1} + f_n$ . Show that $f_n =\\frac{ (a^{n+1} - b^{n+1})}{\\sqrt5}$ , where $a, b$ are real numbers such that $a + b = 1, ab = -1$ and $a > b$ .",
"role": "user"
},
{
"content": "1. **Define the... |
aops_forum | Show that the equation $ y^{37}\equiv x^3\plus{}11 \pmod p$ is solvable for every prime $ p$ , where $ p\leq100$ . | 1. **Understanding the Problem:**
We need to show that the equation \( y^{37} \equiv x^3 + 11 \pmod{p} \) is solvable for every prime \( p \leq 100 \). This means that for each prime \( p \leq 100 \), there exist integers \( x \) and \( y \) such that the congruence holds.
2. **Analyzing the Congruence:**
The eq... | [
{
"content": "Show that the equation $ y^{37}\\equiv x^3\\plus{}11 \\pmod p$ is solvable for every prime $ p$ , where $ p\\leq100$ .",
"role": "user"
},
{
"content": "1. **Understanding the Problem:**\n We need to show that the equation \\( y^{37} \\equiv x^3 + 11 \\pmod{p} \\) is solvable for... |
aops_forum | Let $p$ be a prime number greater than $5$ . Prove that there exists a positive integer $n$ such that $p$ divides $20^n+ 15^n-12^n$ . | 1. **Identify the problem and the claim:**
We need to prove that for any prime number \( p > 5 \), there exists a positive integer \( n \) such that \( p \) divides \( 20^n + 15^n - 12^n \). The claim is that \( n = p-3 \) works for every \( p \).
2. **Apply Fermat's Little Theorem:**
Fermat's Little Theorem sta... | [
{
"content": "Let $p$ be a prime number greater than $5$ . Prove that there exists a positive integer $n$ such that $p$ divides $20^n+ 15^n-12^n$ .",
"role": "user"
},
{
"content": "1. **Identify the problem and the claim:**\n We need to prove that for any prime number \\( p > 5 \\), ther... |
aops_forum | Let $O$ be an interior point in the equilateral triangle $ABC$ , of side length $a$ . The lines $AO, BO$ , and $CO$ intersect the sides of the triangle in the points $A_1, B_1$ , and $C_1$ . Show that $OA_1 + OB_1 + OC_1 < a$ . | 1. **General Result Application**:
We start by considering the more general result mentioned in the solution. Let \( M \) be an interior point in an arbitrary triangle \( \triangle ABC \) with sides \( a, b, \) and \( c \). The lines \( AM, BM, \) and \( CM \) intersect the sides of the triangle at points \( A_1, B_... | [
{
"content": "Let $O$ be an interior point in the equilateral triangle $ABC$ , of side length $a$ . The lines $AO, BO$ , and $CO$ intersect the sides of the triangle in the points $A_1, B_1$ , and $C_1$ . Show that $OA_1 + OB_1 + OC_1 < a$ .",
"role": "user"
},
{
"content": "1. **General ... |
aops_forum | $n$ positive numbers are given. Is it always possible to find a convex polygon with $n+3$ edges and a triangulation of it so that the length of the diameters used in the triangulation are the given $n$ numbers?
*Proposed by Morteza Saghafian* | 1. **Initial Setup and Assumptions**:
- We are given \( n \) positive numbers \( a_1, a_2, \ldots, a_n \).
- We need to find a convex polygon with \( n+3 \) edges and a triangulation such that the lengths of the diagonals used in the triangulation are the given \( n \) numbers.
2. **Base Case**:
- For \( n \l... | [
{
"content": " $n$ positive numbers are given. Is it always possible to find a convex polygon with $n+3$ edges and a triangulation of it so that the length of the diameters used in the triangulation are the given $n$ numbers?\n\n*Proposed by Morteza Saghafian*",
"role": "user"
},
{
"content": "... |
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