source stringclasses 1
value | problem stringlengths 120 401 | solution stringlengths 991 3.56k | messages listlengths 2 2 | prompt_for_new_question stringlengths 1.34k 2.25k | new_question stringlengths 123 418 | __original_row_idx int64 0 19 |
|---|---|---|---|---|---|---|
olympiads | Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy for all \( x, y \in \mathbb{R} \):
\[
f(y-f(x)) = f(x) - 2x + f(f(y))
\] |
Nous voulons trouver toutes les fonctions \( f: \mathbb{R} \rightarrow \mathbb{R} \) qui satisfont l'équation fonctionnelle suivante pour tous \( x, y \in \mathbb{R} \):
\[
f(y - f(x)) = f(x) - 2x + f(f(y))
\]
1. **Injectivité de \( f \)**:
Pour prouver que \( f \) est injective, supposons que \( f(a) = f(b... | [
{
"content": "Find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfy for all \\( x, y \\in \\mathbb{R} \\):\n\n\\[ \nf(y-f(x)) = f(x) - 2x + f(f(y)) \n\\]",
"role": "user"
},
{
"content": "\nNous voulons trouver toutes les fonctions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 12:
Consider the following figure:
 that satisfy for all \( x, y \in \mathbb{R} \):
\[
g(y - g(x)) = g(x) - 3x + g(g(y))
\] | 0 |
olympiads | Find all real numbers \( x, y, z \) such that
\[ x + y + z = 3, \quad x^2 + y^2 + z^2 = 3, \quad x^3 + y^3 + z^3 = 3 \] |
1. **Identify and Define the Given Conditions:**
We are given the following equations involving the real numbers $x, y,$ and $z$:
\[
x + y + z = 3, \quad x^{2} + y^{2} + z^{2} = 3, \quad x^{3} + y^{3} + z^{3} = 3
\]
2. **Apply Newton's Sums:**
Use the relationships provided by Newton's sums on ... | [
{
"content": "Find all real numbers \\( x, y, z \\) such that\n\n\\[ x + y + z = 3, \\quad x^2 + y^2 + z^2 = 3, \\quad x^3 + y^3 + z^3 = 3 \\]",
"role": "user"
},
{
"content": "\n1. **Identify and Define the Given Conditions:**\n\n We are given the following equations involving the real numbers $... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 3:
In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triang... | Find all real numbers \( a, b, c \) such that
\[ a + b + c = 6, \quad a^2 + b^2 + c^2 = 14, \quad a^3 + b^3 + c^3 = 36. \] | 15 |
olympiads | How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight. |
Given the problem, we are to find how many plums (sliv) balance with one pear (grusha). We have the following information:
- 3 apples (yabloka) and 1 pear (grusha) together balance 10 plums (sliv).
- 1 apple (yabloko) and 6 plums (slivy) balance 1 pear (grusha).
We begin by setting up equations based on the balance ... | [
{
"content": "How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight.",
"role": "user"
},
{
"content": "\nGiven the problem, we are to find how many plums (... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 7:
Given that \(1 \leq x, y, z \leq 6\), how many cases are there in which the product of natural numbers \(x, y, z\)... | How many oranges will balance two bananas, given that 4 apples and two bananas weigh as much as 12 oranges, and one apple and 8 oranges balance one banana? Assume that fruits of the same kind have the same weight. | 13 |
olympiads | In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triangles $ABC$ and $EFC$ share the same incenter. Prove that $DE \cdot DF = DA^2$. | 1. **Identify Key Elements**: Consider \( \triangle ABC \) where \( CA = CB \) and \( D \) is the midpoint of \( AB \). Line \( \mathrm{EF} \) passes through \( D \) such that \( \triangle ABC \) and \( \triangle \mathrm{EFC} \) share the same incircle (inscribed circle).
2. **Given Common Incircle**: Let the shared i... | [
{
"content": "In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triangles $ABC$ and $EFC$ share the same incenter. Prove that $DE \\cdot DF = DA^2$.",
"role": "user"
},
{
"content": "1. **Identify Key Elements**: Consider \\( \\triangle ABC \... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 12:
Consider the following figure:
, let \( CA = CB \), and let \( D \) be the midpoint of \( AB \). A line \( EF \) is drawn through point \( D \) such that triangles \( ABC \) and \( EFC \) share the same circumcenter. Prove that \( DE \cdot DF = DA^2 \). | 6 |
olympiads | A set consists of five different odd positive integers, each greater than 2. When these five integers are multiplied together, their product is a five-digit integer of the form $AB0AB$, where $A$ and $B$ are digits with $A \neq 0$ and $A \neq B$. (The hundreds digit of the product is zero.) In total, how many different... |
1. **Observe the Structure of \( N \)**:
Let \( N = AB0AB \) and let \( t \) be the two-digit integer \( AB \).
We recognize that \( N = 1001 \cdot t \), where \( 1001 = 11 \cdot 91 = 11 \cdot 7 \cdot 13 \).
Thus,
\[
N = t \cdot 7 \cdot 11 \cdot 13
\]
2. **Formulate the Problem**:
We n... | [
{
"content": "A set consists of five different odd positive integers, each greater than 2. When these five integers are multiplied together, their product is a five-digit integer of the form $AB0AB$, where $A$ and $B$ are digits with $A \\neq 0$ and $A \\neq B$. (The hundreds digit of the product is zero.) In t... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 4:
Let \( p = 2^{3009}, q = 3^{2006}, \) and \( r = 5^{1003} \). Which of the following statements is true?
(A) \( p ... | A set consists of six different even positive integers, each greater than 4. When these six integers are multiplied together, their product is a six-digit integer of the form \( ABC0AB \), where \( A \), \( B \), and \( C \) are digits with \( A \neq 0 \) and \( A \neq B \). (The thousands digit of the product is zero.... | 5 |
olympiads | In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triangles $ABC$ and $EFC$ share the same incenter. Prove that $DE \cdot DF = DA^2$. | 1. **Identify Key Elements**: Consider \( \triangle ABC \) where \( CA = CB \) and \( D \) is the midpoint of \( AB \). Line \( \mathrm{EF} \) passes through \( D \) such that \( \triangle ABC \) and \( \triangle \mathrm{EFC} \) share the same incircle (inscribed circle).
2. **Given Common Incircle**: Let the shared i... | [
{
"content": "In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triangles $ABC$ and $EFC$ share the same incenter. Prove that $DE \\cdot DF = DA^2$.",
"role": "user"
},
{
"content": "1. **Identify Key Elements**: Consider \\( \\triangle ABC \... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 1:
A set consists of five different odd positive integers, each greater than 2. When these five integers are multipli... | In triangle \( XYZ \), \( XY = XZ \), and \( M \) is the midpoint of \( YZ \). Line \( NP \) passes through point \( M \) such that triangles \( XYZ \) and \( NPM \) share the same circumcenter. Prove that \( MN \cdot MP = MY^2 \). | 9 |
olympiads | Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of triangle $ABC$. It turns out that the perimeter of $\triangle AMC$ is equal to the perimeter of $\triangle CNA$, and the perimeter of $\triangle ANB$ is equal to the perimeter of $\triangle CMB$. Prove that $\triangle ABC$ is isosceles. |
1. Denote the perimeter with the letter $P$. According to the conditions of the problem, we know the following equalities:
\[
P(\triangle AMC) = P(\triangle CAN)
\]
and
\[
P(\triangle ANB) = P(\triangle CMB)
\]
2. This implies that:
\[
P(\triangle AMC) + P(\triangle CMB) = P(\triangle C... | [
{
"content": "Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of triangle $ABC$. It turns out that the perimeter of $\\triangle AMC$ is equal to the perimeter of $\\triangle CNA$, and the perimeter of $\\triangle ANB$ is equal to the perimeter of $\\triangle CMB$. Prove that $\\triangle ABC... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 0:
Let \( a, b, c \) be positive real numbers. Prove that
$$
\frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq... | Let \( P, Q, R \) be points on the sides \( AB, BC, \) and \( CA \) respectively of triangle \( ABC \). It is given that the perimeter of triangle \( APB \) is equal to the perimeter of triangle \( QRC \), and the perimeter of triangle \( AQC \) is equal to the perimeter of triangle \( PRB \). Prove that triangle \( AB... | 11 |
olympiads | Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of triangle $ABC$. It turns out that the perimeter of $\triangle AMC$ is equal to the perimeter of $\triangle CNA$, and the perimeter of $\triangle ANB$ is equal to the perimeter of $\triangle CMB$. Prove that $\triangle ABC$ is isosceles. |
1. Denote the perimeter with the letter $P$. According to the conditions of the problem, we know the following equalities:
\[
P(\triangle AMC) = P(\triangle CAN)
\]
and
\[
P(\triangle ANB) = P(\triangle CMB)
\]
2. This implies that:
\[
P(\triangle AMC) + P(\triangle CMB) = P(\triangle C... | [
{
"content": "Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of triangle $ABC$. It turns out that the perimeter of $\\triangle AMC$ is equal to the perimeter of $\\triangle CNA$, and the perimeter of $\\triangle ANB$ is equal to the perimeter of $\\triangle CMB$. Prove that $\\triangle ABC... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 3:
In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triang... | Points $P$ and $Q$ are taken on the sides $AC$ and $BC$ respectively of triangle $ABC$. It is given that the perimeter of triangle $APB$ is equal to the perimeter of triangle $CQA$, and the perimeter of triangle $BQC$ is equal to the perimeter of triangle $APC$. Prove that triangle $ABC$ is isosceles. | 1 |
olympiads | Let \( a, b, c \) be positive real numbers. Prove that
$$
\frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{3}{1+abc},
$$
and that equality occurs if and only if \( a = b = c = 1 \). | 1. Consider the given inequality:
\[
\frac{1}{a(1+b)}+ \frac{1}{b(1+c)}+ \frac{1}{c(1+a)} \geq \frac{3}{1 + abc}
\]
To simplify, we add \( \frac{3}{1 + abc} \) to both sides. The new inequality becomes:
\[
\frac{1}{a(1+b)} + \frac{1}{b(1+c)} + \frac{1}{c(1+a)} + \frac{3}{1 + abc} \geq \frac{6}{1 + abc}
\]
2. Let's ... | [
{
"content": "Let \\( a, b, c \\) be positive real numbers. Prove that\n\n$$\n\\frac{1}{a(1+b)}+\\frac{1}{b(1+c)}+\\frac{1}{c(1+a)} \\geq \\frac{3}{1+abc},\n$$\n\nand that equality occurs if and only if \\( a = b = c = 1 \\).",
"role": "user"
},
{
"content": "1. Consider the given inequality:\n\n\\[... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 1:
A set consists of five different odd positive integers, each greater than 2. When these five integers are multipli... | Let \( x, y, z \) be positive real numbers. Prove that
$$
\frac{1}{x(1+y)}+\frac{1}{y(1+z)}+\frac{1}{z(1+x)} \geq \frac{3}{1+xyz},
$$
and that equality occurs if and only if \( x = y = z = 1 \). | 7 |
olympiads | Write the first $n$ natural numbers in decimal form on a (very long) strip of paper, then cut the strip so that each piece contains exactly one digit. Place these pieces in a box, mix them up, and draw one at random. Let $p_{n}$ denote the probability that the drawn piece of paper has the digit 0 on it. Determine the l... |
1. **Define Variables and Notation:**
Let's analyze the problem by denoting the number of digits in the number \(n\) by \(k\). Each of the numbers less than \(n\) is padded with leading zeros to construct \(k\)-digit numbers. Suppose we consider all numbers from \(0\) to \(n\) with exactly \(k\) digits. The number ... | [
{
"content": "Write the first $n$ natural numbers in decimal form on a (very long) strip of paper, then cut the strip so that each piece contains exactly one digit. Place these pieces in a box, mix them up, and draw one at random. Let $p_{n}$ denote the probability that the drawn piece of paper has the digit 0 ... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 4:
Let \( p = 2^{3009}, q = 3^{2006}, \) and \( r = 5^{1003} \). Which of the following statements is true?
(A) \( p ... | Write the first \( n \) natural numbers in binary form on a (very long) strip of paper, then cut the strip so that each piece contains exactly one digit. Place these pieces in a box, mix them up, and draw one at random. Let \( q_n \) denote the probability that the drawn piece of paper has the digit 1 on it. Determine ... | 16 |
olympiads | Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of triangle $ABC$. It turns out that the perimeter of $\triangle AMC$ is equal to the perimeter of $\triangle CNA$, and the perimeter of $\triangle ANB$ is equal to the perimeter of $\triangle CMB$. Prove that $\triangle ABC$ is isosceles. |
1. Denote the perimeter with the letter $P$. According to the conditions of the problem, we know the following equalities:
\[
P(\triangle AMC) = P(\triangle CAN)
\]
and
\[
P(\triangle ANB) = P(\triangle CMB)
\]
2. This implies that:
\[
P(\triangle AMC) + P(\triangle CMB) = P(\triangle C... | [
{
"content": "Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of triangle $ABC$. It turns out that the perimeter of $\\triangle AMC$ is equal to the perimeter of $\\triangle CNA$, and the perimeter of $\\triangle ANB$ is equal to the perimeter of $\\triangle CMB$. Prove that $\\triangle ABC... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
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Example 19:
Let \(a \star b = ab - 2\). Compute the remainder when \((((579 \star 569) \star 559) \star \cdots \star 19) \sta... | Points \( P \) and \( Q \) are taken on the sides \( AC \) and \( BC \) respectively of triangle \( ABC \). It turns out that the perimeter of \( \triangle ABP \) is equal to the perimeter of \( \triangle BCQ \), and the perimeter of \( \triangle APQ \) is equal to the perimeter of \( \triangle BCP \). Prove that \( \t... | 17 |
olympiads | When \( a < -1 \), the nature of the roots for the equation
$$
\left(a^{3}+1\right) x^{2}+\left(a^{2}+1\right) x-(a+1)=0
$$
is:
(A) Two negative roots.
(B) One positive root and one negative root, with the absolute value of the negative root being larger.
(C) One positive root and one negative root, with the absolute v... |
Given the quadratic equation:
\[
(a^3 + 1) x^2 + (a^2 + 1) x - (a + 1) = 0
\]
where \( a < -1 \).
1. **Identify the coefficients and analyze their signs:**
- \( a^3 + 1 \)
- \( a^2 + 1 \)
- \( -(a+1) \)
2. **Signs of the coefficients under the condition \( a < -1 \):**
- Since \( a < -1 \), we ha... | [
{
"content": "When \\( a < -1 \\), the nature of the roots for the equation\n$$\n\\left(a^{3}+1\\right) x^{2}+\\left(a^{2}+1\\right) x-(a+1)=0\n$$\nis:\n(A) Two negative roots.\n(B) One positive root and one negative root, with the absolute value of the negative root being larger.\n(C) One positive root and one... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
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Example 11:
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy for all \( x, y \in \mathbb{R} \):
\[... | When \( b > 2 \), the nature of the roots for the equation
$$
\left(b^{2}-4\right) x^{2} + \left(3b-5\right) x + (b-3) = 0
$$
is:
(A) Two positive roots.
(B) One positive root and one negative root, with the absolute value of the negative root being larger.
(C) One positive root and one negative root, with the absolute... | 4 |
olympiads | Let \(a \star b = ab - 2\). Compute the remainder when \((((579 \star 569) \star 559) \star \cdots \star 19) \star 9\) is divided by 100. |
Given the operation \(a \star b = ab - 2\), we need to compute the remainder when \(\left( ( ( (579 \star 569) \star 559) \star \cdots \star 19) \star 9 \right)\) is divided by 100.
We start by examining the operation \(\star\):
\[ (10a + 9) \star (10b + 9) = (10a + 9)(10b + 9) - 2 \]
\[ = 100ab + 90a + 90b + 81 - 2 ... | [
{
"content": "Let \\(a \\star b = ab - 2\\). Compute the remainder when \\((((579 \\star 569) \\star 559) \\star \\cdots \\star 19) \\star 9\\) is divided by 100.",
"role": "user"
},
{
"content": "\nGiven the operation \\(a \\star b = ab - 2\\), we need to compute the remainder when \\(\\left( ( ( (... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 7:
Given that \(1 \leq x, y, z \leq 6\), how many cases are there in which the product of natural numbers \(x, y, z\)... | Let \(a \star b = a^2 + b - 3\). Compute the remainder when \((((5 \star 4) \star 3) \star \cdots \star 1) \star 2\) is divided by 50. | 12 |
olympiads | Let \( p = 2^{3009}, q = 3^{2006}, \) and \( r = 5^{1003} \). Which of the following statements is true?
(A) \( p < q < r \)
(B) \( p < r < q \)
(C) \( q < p < r \)
(D) \( r < p < q \)
(E) \( q < r < p \) | Given the values:
\[ p = 2^{3009}, \quad q = 3^{2006}, \quad r = 5^{1003} \]
1. Express \( p \) and \( q \) in terms of powers of the same base:
\[ p = 2^{3009} = 2^{3 \times 1003} = (2^3)^{1003} = 8^{1003} \]
\[ q = 3^{2006} = 3^{2 \times 1003} = (3^2)^{1003} = 9^{1003} \]
Note: \( r = 5^{1003} \) is already... | [
{
"content": "Let \\( p = 2^{3009}, q = 3^{2006}, \\) and \\( r = 5^{1003} \\). Which of the following statements is true?\n(A) \\( p < q < r \\)\n(B) \\( p < r < q \\)\n(C) \\( q < p < r \\)\n(D) \\( r < p < q \\)\n(E) \\( q < r < p \\)",
"role": "user"
},
{
"content": "Given the values:\n\\[ p = 2... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
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Example 17:
Participants in a chess tournament played one game with each other. For each participant \( A \), the number of p... | Let \( x = 4^{2010}, y = 2^{3015}, \) and \( z = 6^{1505} \). Which of the following statements is true?
(A) \( x < y < z \)
(B) \( x < z < y \)
(C) \( y < x < z \)
(D) \( z < x < y \)
(E) \( y < z < x \) | 14 |
olympiads | Given real numbers \( a, b, c \) and a positive number \( \lambda \) such that the polynomial \( f(x) = x^3 + a x^2 + b x + c \) has three real roots \( x_1, x_2, x_3 \), and the conditions \( x_2 - x_1 = \lambda \) and \( x_3 > \frac{1}{2}(x_1 + x_2) \) are satisfied, find the maximum value of \( \frac{2 a^3 + 27 c - ... |
We begin by analyzing the function \( f(x) = x^3 + a x^2 + b x + c \), which has three real roots \( x_1, x_2, x_3 \). We are given the following conditions:
1. \( x_2 - x_1 = \lambda \)
2. \( x_3 > \frac{1}{2} (x_1 + x_2) \)
We aim to find the maximum value of \( \frac{2a^3 + 27c - 9ab}{\lambda^3} \).
1. **Transfor... | [
{
"content": "Given real numbers \\( a, b, c \\) and a positive number \\( \\lambda \\) such that the polynomial \\( f(x) = x^3 + a x^2 + b x + c \\) has three real roots \\( x_1, x_2, x_3 \\), and the conditions \\( x_2 - x_1 = \\lambda \\) and \\( x_3 > \\frac{1}{2}(x_1 + x_2) \\) are satisfied, find the maxi... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
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Example 16:
Write the first $n$ natural numbers in decimal form on a (very long) strip of paper, then cut the strip so that e... | Given real numbers \( p, q, r \) and a positive number \( \mu \) such that the polynomial \( g(x) = x^3 + p x^2 + q x + r \) has three real roots \( y_1, y_2, y_3 \), and the conditions \( y_2 - y_1 = \mu \) and \( y_3 > \frac{1}{2}(y_1 + y_2) \) are satisfied, find the minimum value of \( \frac{3 p^3 + 18 r - 6 p q}{\... | 3 |
olympiads | Let \( a, b, c \) be positive real numbers. Prove that
$$
\frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{3}{1+abc},
$$
and that equality occurs if and only if \( a = b = c = 1 \). | 1. Consider the given inequality:
\[
\frac{1}{a(1+b)}+ \frac{1}{b(1+c)}+ \frac{1}{c(1+a)} \geq \frac{3}{1 + abc}
\]
To simplify, we add \( \frac{3}{1 + abc} \) to both sides. The new inequality becomes:
\[
\frac{1}{a(1+b)} + \frac{1}{b(1+c)} + \frac{1}{c(1+a)} + \frac{3}{1 + abc} \geq \frac{6}{1 + abc}
\]
2. Let's ... | [
{
"content": "Let \\( a, b, c \\) be positive real numbers. Prove that\n\n$$\n\\frac{1}{a(1+b)}+\\frac{1}{b(1+c)}+\\frac{1}{c(1+a)} \\geq \\frac{3}{1+abc},\n$$\n\nand that equality occurs if and only if \\( a = b = c = 1 \\).",
"role": "user"
},
{
"content": "1. Consider the given inequality:\n\n\\[... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 5:
When \( a < -1 \), the nature of the roots for the equation
$$
\left(a^{3}+1\right) x^{2}+\left(a^{2}+1\right) x-(... | Let \( x, y, z \) be positive real numbers. Prove that
$$
\frac{1}{x(1+y)}+\frac{1}{y(1+z)}+\frac{1}{z(1+x)} \geq \frac{3}{1+xyz},
$$
and that equality occurs if and only if \( x = y = z = 1 \). | 8 |
olympiads | When \( a < -1 \), the nature of the roots for the equation
$$
\left(a^{3}+1\right) x^{2}+\left(a^{2}+1\right) x-(a+1)=0
$$
is:
(A) Two negative roots.
(B) One positive root and one negative root, with the absolute value of the negative root being larger.
(C) One positive root and one negative root, with the absolute v... |
Given the quadratic equation:
\[
(a^3 + 1) x^2 + (a^2 + 1) x - (a + 1) = 0
\]
where \( a < -1 \).
1. **Identify the coefficients and analyze their signs:**
- \( a^3 + 1 \)
- \( a^2 + 1 \)
- \( -(a+1) \)
2. **Signs of the coefficients under the condition \( a < -1 \):**
- Since \( a < -1 \), we ha... | [
{
"content": "When \\( a < -1 \\), the nature of the roots for the equation\n$$\n\\left(a^{3}+1\\right) x^{2}+\\left(a^{2}+1\\right) x-(a+1)=0\n$$\nis:\n(A) Two negative roots.\n(B) One positive root and one negative root, with the absolute value of the negative root being larger.\n(C) One positive root and one... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 17:
Participants in a chess tournament played one game with each other. For each participant \( A \), the number of p... | When \( b < -2 \), the nature of the roots for the equation
$$
\left(b^{4}+1\right) x^{2}+\left(b^{3}+1\right) x-(b+2)=0
$$
is:
(A) Two negative roots.
(B) One positive root and one negative root, with the absolute value of the negative root being larger.
(C) One positive root and one negative root, with the absolute v... | 19 |
olympiads | Prove that for each \( n \geq 2 \), there is a set \( S \) of \( n \) integers such that \((a - b)^{2}\) divides \( ab \) for every distinct \( a, b \in S \). |
Given the problem:
> Prove that for each \( n \geq 2 \), there is a set \( S \) of \( n \) integers such that \( (a-b)^2 \) divides \( ab \) for every distinct \( a, b \in S \).
We aim to show that such a set \( S \) exists for all \( n \geq 2 \). To do this, we rely on the direct corollary of a more general result p... | [
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"content": "Prove that for each \\( n \\geq 2 \\), there is a set \\( S \\) of \\( n \\) integers such that \\((a - b)^{2}\\) divides \\( ab \\) for every distinct \\( a, b \\in S \\).",
"role": "user"
},
{
"content": "\nGiven the problem:\n> Prove that for each \\( n \\geq 2 \\), there is a set \... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 15:
Let \( a, b, c \) be given complex numbers, with \( |a+b| = m \) and \( |a-b| = n \). Given that \( m \cdot n \ne... | Prove that for each \( n \geq 3 \), there exists a set \( T \) of \( n \) integers such that \( (a + b)^{2} \) divides \( ab \) for every distinct \( a, b \in T \). | 18 |
olympiads | Let \( a, b, c \) be given complex numbers, with \( |a+b| = m \) and \( |a-b| = n \). Given that \( m \cdot n \neq 0 \), prove that:
$$
\max \{|ac+b|, |a+bc|\} \geqslant \frac{mn}{\sqrt{m^2 + n^2}}
$$ | Given that \( a, b, c \) are complex numbers with \( |a+b|=m \) and \( |a-b|=n \), and \( mn \neq 0 \), we need to prove:
\[
\max \{|ac+b|, |a+bc|\} \geq \frac{mn}{\sqrt{m^2+n^2}}
\]
#### Proof Method 1:
1. Considering the inequality:
\[
\max \{|ac+b|, |a+bc|\} \geq \frac{|b| \cdot |ac+b| + |a| \cdot |a+bc|}{|... | [
{
"content": "Let \\( a, b, c \\) be given complex numbers, with \\( |a+b| = m \\) and \\( |a-b| = n \\). Given that \\( m \\cdot n \\neq 0 \\), prove that:\n$$\n\\max \\{|ac+b|, |a+bc|\\} \\geqslant \\frac{mn}{\\sqrt{m^2 + n^2}}\n$$",
"role": "user"
},
{
"content": "Given that \\( a, b, c \\) are c... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 18:
How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple a... | Let \( x, y, z \) be given complex numbers, with \( |x+y| = p \) and \( |x-y| = q \). Given that \( p \cdot q \neq 0 \), prove that:
$$
\max \{|xz+y|, |x+yz|\} \geqslant \frac{pq}{\sqrt{p^2 + q^2}}.
$$ | 10 |
olympiads | Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy for all \( x, y \in \mathbb{R} \):
\[
f(y-f(x)) = f(x) - 2x + f(f(y))
\] |
Nous voulons trouver toutes les fonctions \( f: \mathbb{R} \rightarrow \mathbb{R} \) qui satisfont l'équation fonctionnelle suivante pour tous \( x, y \in \mathbb{R} \):
\[
f(y - f(x)) = f(x) - 2x + f(f(y))
\]
1. **Injectivité de \( f \)**:
Pour prouver que \( f \) est injective, supposons que \( f(a) = f(b... | [
{
"content": "Find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfy for all \\( x, y \\in \\mathbb{R} \\):\n\n\\[ \nf(y-f(x)) = f(x) - 2x + f(f(y)) \n\\]",
"role": "user"
},
{
"content": "\nNous voulons trouver toutes les fonctions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R... | Help the user to create a new math problem similar to a given one.
Make the new problem reasonable and solvable.
Here are some examples of how to complete this task:
Example 9:
Prove that for each \( n \geq 2 \), there is a set \( S \) of \( n \) integers such that \((a - b)^{2}\) divides \... | Find all functions \( g: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy for all \( x, y \in \mathbb{R} \):
\[
g(x + g(y)) = g(y) + 3x - g(g(x))
\] | 2 |
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