problem stringlengths 59 5.02k | reasoning stringlengths 1.51k 124k | deepseek_solution stringlengths 0 5.06k | ground_truth_solution stringlengths 97 9.16k | correct bool 2
classes | judge_reasoning stringlengths 31 1.55k | source stringclasses 1
value | solution stringlengths 97 9.16k | messages stringlengths 325 10.4k | 4o_response stringlengths 286 16.7k | __original_row_idx int64 0 17.3k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|---|---|---|---|---|
Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits. | Okay, so I need to find the smallest three-digit number that equals the sum of its digits plus twice the product of its digits. Let me break this down. Let's denote the three-digit number as ABC, where A, B, and C are its digits. Since it's a three-digit number, A can't be zero; it has to be at least 1. The number itse... | The least three-digit number that equals the sum of its digits plus twice the product of its digits is **397**.
### Step-by-Step Explanation:
1. **Define the Problem**: Let the three-digit number be \(100A + 10B + C\), where \(A, B, C\) are its digits. The equation derived from the problem is:
\[
100A + 10B + C ... | 1. Let the three-digit number be represented as \(100a + 10b + c\), where \(a, b, c\) are the digits of the number, and \(a \neq 0\) since it is a three-digit number.
2. According to the problem, the number is equal to the sum of its digits plus twice the product of its digits. Therefore, we have the equation:
\[
... | true | The provided solution correctly identifies the three-digit number that satisfies the equation derived from the problem statement. The step-by-step breakdown validates the arithmetic and logical deductions made throughout, demonstrating that 397 is indeed the solution that meets the criteria set forth. | aops_forum | 1. Let the three-digit number be represented as \(100a + 10b + c\), where \(a, b, c\) are the digits of the number, and \(a \neq 0\) since it is a three-digit number.
2. According to the problem, the number is equal to the sum of its digits plus twice the product of its digits. Therefore, we have the equation:
\[
... | [{'content': 'Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits.', 'role': 'user'}
{'content': '1. Let the three-digit number be represented as \\(100a + 10b + c\\), where \\(a, b, c\\) are the digits of the number, and \\(a \\neq 0\\) since it is a three-dig... | Let the three-digit number be represented as \( xyz \), where \( x \), \( y \), and \( z \) are its digits. The numerical value of the number can be expressed as:
\[
100x + 10y + z
\]
We are looking for a number such that:
\[
100x + 10y + z = x + y + z + 2xyz
\]
This can be rearranged to:
\[
100x + 10y + z - (x + ... | 9,969 | [
{
"from": "human",
"value": "Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits."
},
{
"from": "gpt",
"value": "Let the three-digit number be represented as \\( xyz \\), where \\( x \\), \\( y \\), and \\( z \\) are its digits. The numer... |
find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have $$ \{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\} $$ . | Okay, let's try to tackle this problem. So, the question is asking for all distinct integers a₁, a₂, ..., a_k such that there exists an injective function f from the real numbers to themselves. The condition is that for each positive integer n, the set {fⁿ(x) - x | x ∈ ℝ} equals {a₁ + n, a₂ + n, ..., a_k + n}.
First,... | To solve the problem, we need to find all distinct integers \(a_1, a_2, \ldots, a_k\) such that there exists an injective function \(f: \mathbb{R} \to \mathbb{R}\) with the property that for each positive integer \(n\), the set \(\{f^n(x) - x \mid x \in \mathbb{R}\}\) is equal to \(\{a_1 + n, a_2 + n, \ldots, a_k + n\}... | 1. **Claim 1: \(a_1, a_2, \ldots, a_k \geq 0\)**
**Proof:** Suppose for the sake of contradiction that \(a_1\) is the minimum and \(a_1 < 0\). Let \(a = -a_1\). Then, for some \(x\), we have:
\[
f^a(x) = x + a + a_1 = x \implies f^a(x) = x
\]
This implies that \(f\) has a periodic point with period \(a\... | false | The provided solution concludes that the only values for the integers must be zero, while the ground truth solution asserts a minimum condition for non-negativity of the integers. The ground truth solution ultimately leads to the conclusion that no positive integers can exist, making the distinct integers as empty, thu... | aops_forum | 1. **Claim 1: \(a_1, a_2, \ldots, a_k \geq 0\)**
**Proof:** Suppose for the sake of contradiction that \(a_1\) is the minimum and \(a_1 < 0\). Let \(a = -a_1\). Then, for some \(x\), we have:
\[
f^a(x) = x + a + a_1 = x \implies f^a(x) = x
\]
This implies that \(f\) has a periodic point with period \(a\... | [{'content': 'find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have $$ \\{f^n(x)-x| x \\in \\mathbb{R} \\}=\\{a_1+n,a_2+n,...,a_k+n\\} $$ .', 'role': 'user'}
{'content': '1. **Claim 1: \\(a_1,... | To solve for the integer values \( k \) such that there exist \( k \) distinct integers \( a_1, a_2, \ldots, a_k \) meeting the criteria laid out in the problem, we start by analyzing the implications of the condition on the sequence defined by \( f \).
The statement requires that for each positive integer \( n \), th... | 9,970 | [
{
"from": "human",
"value": "find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have $$ \\{f^n(x)-x| x \\in \\mathbb{R} \\}=\\{a_1+n,a_2+n,...,a_k+n\\} $$ ."
},
{
"from": "gpt... |
Suppose $A_1, A_2, A_3, \ldots, A_{20}$ is a 20 sides regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon but the sides are not the sides of the polygon? | Okay, so I need to find the number of non-isosceles (scalene) triangles that can be formed from the vertices of a 20-sided regular polygon. Additionally, none of the triangle's sides can be sides of the polygon. Hmm, let me start by understanding the problem step by step.
First, the polygon is regular, meaning all sid... | To determine the number of non-isosceles (scalene) triangles that can be formed from the vertices of a 20-sided regular polygon, where none of the triangle's sides are sides of the polygon, we follow these steps:
1. **Total number of triangles with no adjacent vertices**:
- The formula for choosing 3 non-consecutiv... | 1. **Calculate the total number of triangles:**
The total number of triangles that can be formed by choosing any 3 vertices from the 20 vertices of the polygon is given by the combination formula:
\[
\binom{20}{3} = \frac{20!}{3!(20-3)!} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140
\]
2. **C... | false | The provided solution incorrectly calculates the number of triangles by excluding adjacent vertices using an inappropriate method, leading to the erroneous conclusion of 640 scalene triangles. In contrast, the ground truth solution correctly computes the total number of triangles by taking all combinations of vertices ... | aops_forum | 1. **Calculate the total number of triangles:**
The total number of triangles that can be formed by choosing any 3 vertices from the 20 vertices of the polygon is given by the combination formula:
\[
\binom{20}{3} = \frac{20!}{3!(20-3)!} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140
\]
2. **C... | [{'content': 'Suppose $A_1, A_2, A_3, \\ldots, A_{20}$ is a 20 sides regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon but the sides are not the sides of the polygon?', 'role': 'user'}
{'content': '1. **Calculate the total number of triangle... | To solve the problem, we begin by calculating the total number of triangles that can be formed from the vertices of the 20-sided regular polygon \( A_1, A_2, A_3, \ldots, A_{20} \). We can choose any 3 vertices from the 20 vertices to form a triangle.
The total number of ways to choose 3 vertices from 20 is given by t... | 9,971 | [
{
"from": "human",
"value": "Suppose $A_1, A_2, A_3, \\ldots, A_{20}$ is a 20 sides regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon but the sides are not the sides of the polygon?"
},
{
"from": "gpt",
"value": "To so... |
Triangle $ABC$ has side lengths $AB=65$ , $BC=33$ , and $AC=56$ . Find the radius of the circle tangent to sides $AC$ and $BC$ and to the circumcircle of triangle $ABC$ . | Okay, so I need to find the radius of the circle that's tangent to sides AC and BC and also tangent to the circumcircle of triangle ABC. The triangle has sides AB=65, BC=33, and AC=56. Hmm, let's start by understanding the problem.
First, let me visualize triangle ABC. The sides are given as AB=65, BC=33, and AC=56. S... | Triangle \(ABC\) has side lengths \(AB = 65\), \(BC = 33\), and \(AC = 56\). We need to find the radius of the circle tangent to sides \(AC\) and \(BC\) and to the circumcircle of triangle \(ABC\).
First, we verify that triangle \(ABC\) is a right triangle by checking the Pythagorean theorem:
\[
33^2 + 56^2 = 1089 + 3... | To find the radius of the circle tangent to sides \( AC \) and \( BC \) and to the circumcircle of triangle \( ABC \), we need to follow these steps:
1. **Identify the type of circle**:
The circle described is the \( A \)-mixtilinear incircle of triangle \( ABC \).
2. **Calculate the semi-perimeter \( s \) of tria... | false | The provided solution calculates the radius of a certain circle associated with triangle ABC using the wrong method. Instead of finding the radius of the A-mixtilinear incircle as stated in the ground truth solution, the provided solution incorrectly arrives at a radius of 24 without utilizing the necessary geometric p... | aops_forum | To find the radius of the circle tangent to sides \( AC \) and \( BC \) and to the circumcircle of triangle \( ABC \), we need to follow these steps:
1. **Identify the type of circle**:
The circle described is the \( A \)-mixtilinear incircle of triangle \( ABC \).
2. **Calculate the semi-perimeter \( s \) of tria... | [{'content': 'Triangle $ABC$ has side lengths $AB=65$ , $BC=33$ , and $AC=56$ . Find the radius of the circle tangent to sides $AC$ and $BC$ and to the circumcircle of triangle $ABC$ .', 'role': 'user'}
{'content': "To find the radius of the circle tangent to sides \\( AC \\) and \\( BC \\) and to the circu... | To find the radius \( r \) of the circle that is tangent to sides \( AC \) and \( BC \) and also tangent to the circumcircle of triangle \( ABC \), we start by applying the formula for the radius of the excircle opposite to vertex \( A \) in triangle \( ABC \).
Given the triangle has side lengths:
- \( a = BC = 33 \)
... | 9,972 | [
{
"from": "human",
"value": "Triangle $ABC$ has side lengths $AB=65$ , $BC=33$ , and $AC=56$ . Find the radius of the circle tangent to sides $AC$ and $BC$ and to the circumcircle of triangle $ABC$ ."
},
{
"from": "gpt",
"value": "To find the radius \\( r \\) of the circle that is tan... |
Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product? $\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007$ | Okay, let's try to figure out this problem step by step. So, the question is about Palmer computing the product of the first 1,001 prime numbers, and we need to determine which of the given options is NOT a factor of that product. The options are A) 2,002; B) 3,003; C) 5,005; D) 6,006; and E) 7,007.
First, I need to ... | To determine which of the given options is NOT a factor of the product of the first 1,001 prime numbers, we need to factor each option into its prime components and check if all those primes (with their respective multiplicities) are included in the product. The product of the first 1,001 primes includes each prime num... | 1. Palmer computes the product of the first $1,001$ prime numbers. Let's denote this product as \( P \).
2. We need to determine which of the given numbers is NOT a factor of \( P \). The given numbers are:
- \( 2,002 \)
- \( 3,003 \)
- \( 5,005 \)
- \( 6,006 \)
- \( 7,007 \)
3. First, let's factorize e... | false | The solution provided incorrectly concludes that the answer is 'E' (7,007) instead of the correct answer 'E' being 7,007. Both the reasoning and the factorization presented are correct in the judge's evaluation which confirms that 7,007 cannot be a factor due to the presence of repeated prime 7. Hence the final answer ... | aops_forum | 1. Palmer computes the product of the first $1,001$ prime numbers. Let's denote this product as \( P \).
2. We need to determine which of the given numbers is NOT a factor of \( P \). The given numbers are:
- \( 2,002 \)
- \( 3,003 \)
- \( 5,005 \)
- \( 6,006 \)
- \( 7,007 \)
3. First, let's factorize e... | [{'content': "Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product? $\\text{(A) }2,002\\qquad\\text{(B) }3,003\\qquad\\text{(C) }5,005\\qquad\\text{(D) }6,006\\qquad\\text{(E) }7,007$ ", 'role': 'user'}
{'content': "1. Palmer computes th... | Palmer's product is the product of the first \(1001\) prime numbers. We need to determine which of the options provided is **not** a factor of this product. A number is a factor of Palmer's product if all the prime factors of that number are among the first \(1001\) prime numbers.
Let’s evaluate each option:
### Opti... | 9,973 | [
{
"from": "human",
"value": "Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product? $\\text{(A) }2,002\\qquad\\text{(B) }3,003\\qquad\\text{(C) }5,005\\qquad\\text{(D) }6,006\\qquad\\text{(E) }7,007$ "
},
{
"from": "gpt... |
Assume $ a_{1} \ge a_{2} \ge \dots \ge a_{107} > 0 $ satisfy $ \sum\limits_{k=1}^{107}{a_{k}} \ge M $ and $ b_{107} \ge b_{106} \ge \dots \ge b_{1} > 0 $ satisfy $ \sum\limits_{k=1}^{107}{b_{k}} \le M $ . Prove that for any $ m \in \{1,2, \dots, 107\} $ , the arithmetic mean of the following numbers $$ \fra... | Okay, so I need to prove that for any m in {1, 2, ..., 107}, the arithmetic mean of the ratios a₁/b₁, a₂/b₂, ..., aₘ/bₘ is at least M/N. Wait, the problem mentions N, but I don't see N defined here. Let me check the original problem again.
Hmm, the problem states that we have two sequences: a₁ ≥ a₂ ≥ ... ≥ a₁₀₇ > 0 wi... | To prove that for any \( m \in \{1, 2, \dots, 107\} \), the arithmetic mean of \( \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_m}{b_m} \) is at least \( \frac{M}{107} \), follow these steps:
1. **Ordering and Sum Constraints**:
- \( a_1 \geq a_2 \geq \dots \geq a_{107} > 0 \), with \( \sum_{k=1}^{107} a_k \geq ... | 1. **Define the Arithmetic Mean and Sequence**:
We define \( A_n \) as the arithmetic mean of the first \( n \) terms of the sequence:
\[
\frac{a_1}{b_1}, \frac{a_2}{b_2}, \cdots, \frac{a_n}{b_n}
\]
and define
\[
c_n = \frac{a_n}{b_n}
\]
2. **Lemma**:
We need to prove that \( A_n \geq c_{n+1... | false | The given solution is focused on proving that the arithmetic mean of fractions \\( \frac{a_i}{b_i} \\) is at least \\( \frac{M}{107} \\), while the ground truth solution aims to prove the same point but states it as \\( \frac{M}{N} \\). The core issue is that the comparison doesn't directly translate due to the differe... | aops_forum | 1. **Define the Arithmetic Mean and Sequence**:
We define \( A_n \) as the arithmetic mean of the first \( n \) terms of the sequence:
\[
\frac{a_1}{b_1}, \frac{a_2}{b_2}, \cdots, \frac{a_n}{b_n}
\]
and define
\[
c_n = \frac{a_n}{b_n}
\]
2. **Lemma**:
We need to prove that \( A_n \geq c_{n+1... | [{'content': 'Assume $ a_{1} \\ge a_{2} \\ge \\dots \\ge a_{107} > 0 $ satisfy $ \\sum\\limits_{k=1}^{107}{a_{k}} \\ge M $ and $ b_{107} \\ge b_{106} \\ge \\dots \\ge b_{1} > 0 $ satisfy $ \\sum\\limits_{k=1}^{107}{b_{k}} \\le M $ . Prove that for any $ m \\in \\{1,2, \\dots, 107\\} $ , the arithmetic mean of ... | To prove that the arithmetic mean of \( \frac{a_{1}}{b_{1}}, \frac{a_{2}}{b_{2}}, \ldots, \frac{a_{m}}{b_{m}} \) is greater than or equal to \( \frac{M}{N} \), we will denote this mean as \( A \):
\[
A = \frac{1}{m} \sum_{k=1}^{m} \frac{a_k}{b_k}
\]
We can also express the sums of \( a_k \) and \( b_k \) up to \( m \... | 9,974 | [
{
"from": "human",
"value": "Assume $ a_{1} \\ge a_{2} \\ge \\dots \\ge a_{107} > 0 $ satisfy $ \\sum\\limits_{k=1}^{107}{a_{k}} \\ge M $ and $ b_{107} \\ge b_{106} \\ge \\dots \\ge b_{1} > 0 $ satisfy $ \\sum\\limits_{k=1}^{107}{b_{k}} \\le M $ . Prove that for any $ m \\in \\{1,2, \\dots, 107\\} ... |
The sequence $ (x_n)$ is defined by $ x_1\equal{}1,x_2\equal{}a$ and $ x_n\equal{}(2n\plus{}1)x_{n\minus{}1}\minus{}(n^2\minus{}1)x_{n\minus{}2}$ $ \forall n \geq 3$ , where $ a \in N^*$ .For which value of $ a$ does the sequence have the property that $ x_i|x_j$ whenever $ i<j$ . | Okay, so I need to figure out for which value of 'a' the sequence defined by x₁=1, x₂=a, and the recurrence relation xₙ = (2n + 1)xₙ₋₁ − (n² − 1)xₙ₋₂ for all n ≥ 3 has the property that x_i divides x_j whenever i < j. Hmm, that's interesting. Let me start by understanding the problem step by step.
First, let me restat... | Given the sequence \( (x_n) \) defined by \( x_1 = 1 \), \( x_2 = a \), and the recurrence relation \( x_n = (2n + 1)x_{n-1} - (n^2 - 1)x_{n-2} \) for \( n \geq 3 \), we need to determine the value of \( a \) such that \( x_i \) divides \( x_j \) whenever \( i < j \).
1. **Initial Terms and Recurrence Calculation**:
... | 1. **Define the sequence and initial conditions:**
The sequence \( (x_n) \) is defined by:
\[
x_1 = 1, \quad x_2 = a
\]
and for \( n \geq 3 \):
\[
x_n = (2n+1)x_{n-1} - (n^2-1)x_{n-2}
\]
where \( a \in \mathbb{N}^* \).
2. **Determine the condition for \( x_i \mid x_j \) whenever \( i < j \):... | false | The original solution concludes with a single value of \( a = 2 \), while the ground truth indicates that both \( a = 2 \) and \( a = 4 \) satisfy the divisibility requirement for the sequence terms. Therefore, the solution is incomplete and does not fully encompass all valid solutions. | aops_forum | 1. **Define the sequence and initial conditions:**
The sequence \( (x_n) \) is defined by:
\[
x_1 = 1, \quad x_2 = a
\]
and for \( n \geq 3 \):
\[
x_n = (2n+1)x_{n-1} - (n^2-1)x_{n-2}
\]
where \( a \in \mathbb{N}^* \).
2. **Determine the condition for \( x_i \mid x_j \) whenever \( i < j \):... | [{'content': 'The sequence $ (x_n)$ is defined by $ x_1\\equal{}1,x_2\\equal{}a$ and $ x_n\\equal{}(2n\\plus{}1)x_{n\\minus{}1}\\minus{}(n^2\\minus{}1)x_{n\\minus{}2}$ $ \\forall n \\geq 3$ , where $ a \\in N^*$ .For which value of $ a$ does the sequence have the property that $ x_i|x_j$ whenever $ i<j$ .'... | To explore the sequence defined by \( x_1 = 1 \), \( x_2 = a \), and the recurrence relation
\[
x_n = (2n + 1)x_{n-1} - (n^2 - 1)x_{n-2} \quad \text{for } n \geq 3,
\]
we aim to determine the value of \( a \in \mathbb{N}^* \) such that \( x_i \mid x_j \) whenever \( i < j \).
First, we compute the initial terms of ... | 9,975 | [
{
"from": "human",
"value": "The sequence $ (x_n)$ is defined by $ x_1\\equal{}1,x_2\\equal{}a$ and $ x_n\\equal{}(2n\\plus{}1)x_{n\\minus{}1}\\minus{}(n^2\\minus{}1)x_{n\\minus{}2}$ $ \\forall n \\geq 3$ , where $ a \\in N^*$ .For which value of $ a$ does the sequence have the property that $ x_... |
Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$ . | Alright, let's tackle this problem: Compute the number of positive integers less than or equal to 10000 which are relatively prime to 2014. Hmm, okay. So, first, I need to recall what it means for two numbers to be relatively prime. If two numbers are relatively prime, their greatest common divisor (gcd) is 1. So, esse... | To compute the number of positive integers less than or equal to 10000 which are relatively prime to 2014, we use the following steps:
1. **Factorize 2014**:
\[
2014 = 2 \times 19 \times 53
\]
2. **Compute Euler's Totient Function (ETF) for 2014**:
\[
\phi(2014) = 2014 \left(1 - \frac{1}{2}\right) \lef... | To compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$, we use Euler's Totient Function, $\phi(n)$. The function $\phi(n)$ counts the number of integers up to $n$ that are relatively prime to $n$.
1. **Prime Factorization of 2014**:
\[
2014 = 2 \times 19 \ti... | true | The provided solution correctly computes the number of positive integers less than or equal to 10000 that are relatively prime to 2014, utilizing the Euler's Totient Function and implementing the inclusion-exclusion principle accurately. Both the step-by-step calculations and the final result align with the ground trut... | aops_forum | To compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$, we use Euler's Totient Function, $\phi(n)$. The function $\phi(n)$ counts the number of integers up to $n$ that are relatively prime to $n$.
1. **Prime Factorization of 2014**:
\[
2014 = 2 \times 19 \ti... | [{'content': 'Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$ .', 'role': 'user'}
{'content': "To compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$, we use Euler's Totient Function, $\\phi(n)$. The fu... | To determine the number of positive integers less than or equal to \( 10000 \) that are relatively prime to \( 2014 \), we can use Euler's Totient Function \( \phi(n) \) along with the principle of inclusion-exclusion. First, we'll find the prime factorization of \( 2014 \).
Starting with \( 2014 \):
\[
2014 = 2 \tim... | 9,976 | [
{
"from": "human",
"value": "Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$ ."
},
{
"from": "gpt",
"value": "To determine the number of positive integers less than or equal to \\( 10000 \\) that are relatively prime to \\( 2014 \\), ... |
Given the positive integer $m \geq 2$ , $n \geq 3$ . Define the following set $$ S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}. $$ Let $A$ be a subset of $S$ . If there does not exist positive integers $x_1, x_2, x_3, y_1, y_2, y_3$ such that $x_1 < x_2 < x_3, y_1 < y_2 < y... | Okay, let's try to tackle this problem. So, we have a set S which consists of ordered pairs (a, b) where a ranges from 1 to m and b from 1 to n. We need to find the largest subset A of S such that A doesn't contain a specific 3x3 grid pattern. The forbidden pattern involves five points arranged in a cross shape: the ce... | The largest possible number of elements in subset \( A \) that avoids the forbidden cross configuration is given by \( 2m + 2n - 4 \). This result is derived by constructing a subset consisting of all elements from two rows and two columns, adjusted to avoid overlap. This construction ensures that no three rows or colu... | 1. **Define the set \( S \) and subset \( A \):**
\[
S = \left\{(a, b) \mid a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}
\]
Let \( A \subseteq S \) such that there do not exist positive integers \( x_1, x_2, x_3, y_1, y_2, y_3 \) with \( x_1 < x_2 < x_3 \) and \( y_1 < y_2 < y_3 \) satisfyi... | true | The provided solution arrives at the same conclusion as the ground truth solution, which states that the maximum number of elements in set A that avoids the forbidden cross configuration is indeed \(2m + 2n - 4\). Both solutions correctly analyze the problem and justify their construction to ensure no three rows and co... | aops_forum | 1. **Define the set \( S \) and subset \( A \):**
\[
S = \left\{(a, b) \mid a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}
\]
Let \( A \subseteq S \) such that there do not exist positive integers \( x_1, x_2, x_3, y_1, y_2, y_3 \) with \( x_1 < x_2 < x_3 \) and \( y_1 < y_2 < y_3 \) satisfyi... | [{'content': 'Given the positive integer $m \\geq 2$ , $n \\geq 3$ . Define the following set $$ S = \\left\\{(a, b) | a \\in \\{1, 2, \\cdots, m\\}, b \\in \\{1, 2, \\cdots, n\\} \\right\\}. $$ Let $A$ be a subset of $S$ . If there does not exist positive integers $x_1, x_2, x_3, y_1, y_2, y_3$ such that $x_... | To solve this problem, we need to build a subset \( A \subseteq S \) that does not contain the specific configuration of points. The configuration involves five points in \( A \) for certain \( x \) and \( y \) coordinates.
We represent the points in \( S \) as \( (a, b) \) where \( a \in \{1, 2, \ldots, m\} \) and \... | 9,977 | [
{
"from": "human",
"value": "Given the positive integer $m \\geq 2$ , $n \\geq 3$ . Define the following set $$ S = \\left\\{(a, b) | a \\in \\{1, 2, \\cdots, m\\}, b \\in \\{1, 2, \\cdots, n\\} \\right\\}. $$ Let $A$ be a subset of $S$ . If there does not exist positive integers $x_1, x_2, x_3, y_1... |
Let $A,B,C$ be three points with integer coordinates in the plane and $K$ a circle with radius $R$ passing through $A,B,C$ . Show that $AB\cdot BC\cdot CA\ge 2R$ , and if the centre of $K$ is in the origin of the coordinates, show that $AB\cdot BC\cdot CA\ge 4R$ . | Okay, let me try to work through this problem step by step. So, the problem is about three points A, B, C with integer coordinates on a circle K of radius R. We need to show two things: first, that the product of the lengths of the sides AB, BC, and CA is at least 2R, and second, if the circle is centered at the origin... | To show that \(AB \cdot BC \cdot CA \geq 2R\) for three points \(A, B, C\) with integer coordinates on a circle \(K\) of radius \(R\), and to show that \(AB \cdot BC \cdot CA \geq 4R\) when the circle is centered at the origin, we proceed as follows:
### Part 1: General Case (Circle Not Necessarily Centered at the Ori... | 1. **Given:**
- Points \( A, B, C \) with integer coordinates.
- Circle \( K \) with radius \( R \) passing through \( A, B, C \).
- We need to show \( AB \cdot BC \cdot CA \ge 2R \).
- If the center of \( K \) is at the origin, show \( AB \cdot BC \cdot CA \ge 4R \).
2. **Area of Triangle:**
- Let \(\D... | true | The provided solution demonstrates the necessary steps and derivations to show that the product of the side lengths of the triangle formed by points with integer coordinates on a circle meets the conditions specified in the ground truth. Both cases, one for any circle and the other when the center is at the origin, are... | aops_forum | 1. **Given:**
- Points \( A, B, C \) with integer coordinates.
- Circle \( K \) with radius \( R \) passing through \( A, B, C \).
- We need to show \( AB \cdot BC \cdot CA \ge 2R \).
- If the center of \( K \) is at the origin, show \( AB \cdot BC \cdot CA \ge 4R \).
2. **Area of Triangle:**
- Let \(\D... | [{'content': 'Let $A,B,C$ be three points with integer coordinates in the plane and $K$ a circle with radius $R$ passing through $A,B,C$ . Show that $AB\\cdot BC\\cdot CA\\ge 2R$ , and if the centre of $K$ is in the origin of the coordinates, show that $AB\\cdot BC\\cdot CA\\ge 4R$ .', 'role': 'user'}
{'con... | Let \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) be the three points with integer coordinates. We denote the lengths of the sides of triangle \( ABC \) as follows:
\[
AB = d_{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2},
\]
\[
BC = d_{BC} = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2},
\]
\[
CA = d_{CA} = \sqrt{(... | 9,978 | [
{
"from": "human",
"value": "Let $A,B,C$ be three points with integer coordinates in the plane and $K$ a circle with radius $R$ passing through $A,B,C$ . Show that $AB\\cdot BC\\cdot CA\\ge 2R$ , and if the centre of $K$ is in the origin of the coordinates, show that $AB\\cdot BC\\cdot CA\\ge 4R$... |
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$ . Compute \[\sum_{n=1}^{\infty}\frac{\phi(n)}{5^n+1}.\] | Okay, let's tackle this problem: Compute the sum from n=1 to infinity of φ(n)/(5^n + 1). So, first, I need to recall what φ(n) is. φ(n) is Euler's totient function, which counts the number of integers up to n that are coprime to n. For example, φ(1) is 1, φ(2) is 1, φ(3) is 2, and so on. The challenge here is to sum th... | To compute the infinite series \(\sum_{n=1}^{\infty}\frac{\phi(n)}{5^n+1}\), where \(\phi(n)\) is Euler's totient function, we explored various analytical techniques and numerical approximations.
1. **Generating Function Approach**: We considered the generating function for \(\phi(n)\), which is known to be complex an... | 1. We start with the given series:
\[
\sum_{n=1}^{\infty}\frac{\phi(n)}{5^n+1}
\]
where \(\phi(n)\) is the Euler's totient function, which counts the number of positive integers up to \(n\) that are coprime to \(n\).
2. We use the series expansion for \(\frac{1}{5^n+1}\):
\[
\frac{1}{5^n+1} = \sum_{m... | false | The initial solution approximates a numerical value and conjectures a closed-form answer of \(\frac{7}{31}\), which is not the ground truth. The ground truth solution correctly evaluates the series using generating functions and transformations to arrive at \(\frac{65}{288}\). Therefore, the solutions are inconsistent,... | aops_forum | 1. We start with the given series:
\[
\sum_{n=1}^{\infty}\frac{\phi(n)}{5^n+1}
\]
where \(\phi(n)\) is the Euler's totient function, which counts the number of positive integers up to \(n\) that are coprime to \(n\).
2. We use the series expansion for \(\frac{1}{5^n+1}\):
\[
\frac{1}{5^n+1} = \sum_{m... | [{'content': 'Let $\\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$ . Compute \\[\\sum_{n=1}^{\\infty}\\frac{\\phi(n)}{5^n+1}.\\]', 'role': 'user'}
{'content': "1. We start with the given series:\n \\[\n \\sum_{n=1}^{\\infty}\\frac{\\phi(n)}{5^n+1}\n \\]\n ... | To compute the series
\[
S = \sum_{n=1}^{\infty}\frac{\phi(n)}{5^n+1},
\]
we will utilize properties of the Euler φ-function and the fact that we can express it in terms of a Dirichlet series.
### Step 1: Representing S as an integral
We can rewrite the series for \( S \) using the formula for the Euler totient fun... | 9,979 | [
{
"from": "human",
"value": "Let $\\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$ . Compute \\[\\sum_{n=1}^{\\infty}\\frac{\\phi(n)}{5^n+1}.\\]"
},
{
"from": "gpt",
"value": "To compute the series \n\n\\[\nS = \\sum_{n=1}^{\\infty}\\frac{\\... |
Let $n$ and $k$ be two integers which are greater than $1$ . Let $a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m$ be non-negative real numbers such that
i) $a_1\ge a_2\ge\ldots\ge a_n$ and $a_1+a_2+\ldots+a_n=1$ ;
ii) For any integer $m\in\{1,2,\ldots,n\}$ , we have that $c_1+c_2+\ldots+c_m\le m^k$ .
Find the maximu... | Okay, let's try to tackle this problem step by step. So, we have two integers n and k, both greater than 1. We have non-negative real numbers a_1, a_2, ..., a_n and c_1, c_2, ..., c_n. The conditions are:
i) The a's are ordered in non-increasing order, so a_1 is the largest, down to a_n, and their sum is 1.
ii) For a... | To solve the problem of finding the maximum value of \( c_1a_1^k + c_2a_2^k + \ldots + c_na_n^k \) under the given constraints, we analyze the interaction between the sequences \( \{a_i\} \) and \( \{c_i\} \).
### Key Steps and Analysis:
1. **Constraints Understanding**:
- The sequence \( \{a_i\} \) is non-increas... | To find the maximum of \( c_1a_1^k + c_2a_2^k + \ldots + c_na_n^k \), we will use the given conditions and properties of the sequences \( \{a_i\} \) and \( \{c_i\} \).
1. **Given Conditions:**
- \( a_1 \ge a_2 \ge \ldots \ge a_n \) and \( a_1 + a_2 + \ldots + a_n = 1 \).
- For any integer \( m \in \{1, 2, \ldots... | true | Both the provided solution and the ground truth solution correctly analyze the constraints and arrive at the conclusion that the maximum achievable value for the given expression is 1 under the conditions specified. They both go through valid reasoning regarding the arrangement of terms and optimization techniques, lea... | aops_forum | To find the maximum of \( c_1a_1^k + c_2a_2^k + \ldots + c_na_n^k \), we will use the given conditions and properties of the sequences \( \{a_i\} \) and \( \{c_i\} \).
1. **Given Conditions:**
- \( a_1 \ge a_2 \ge \ldots \ge a_n \) and \( a_1 + a_2 + \ldots + a_n = 1 \).
- For any integer \( m \in \{1, 2, \ldots... | [{'content': 'Let $n$ and $k$ be two integers which are greater than $1$ . Let $a_1,a_2,\\ldots,a_n,c_1,c_2,\\ldots,c_m$ be non-negative real numbers such that\ni) $a_1\\ge a_2\\ge\\ldots\\ge a_n$ and $a_1+a_2+\\ldots+a_n=1$ ;\nii) For any integer $m\\in\\{1,2,\\ldots,n\\}$ , we have that $c_1+c_2+\\ldots+c... | To find the maximum of the expression
\[
S = c_1 a_1^k + c_2 a_2^k + \ldots + c_n a_n^k,
\]
given the conditions stated, we consider the constraints imposed by the conditions on \(c_i\) and \(a_i\).
1. **Understanding the values of \(c_i\):**
The conditions indicate that for any integer \(m\):
\[
c_1 + c_... | 9,980 | [
{
"from": "human",
"value": "Let $n$ and $k$ be two integers which are greater than $1$ . Let $a_1,a_2,\\ldots,a_n,c_1,c_2,\\ldots,c_m$ be non-negative real numbers such that\ni) $a_1\\ge a_2\\ge\\ldots\\ge a_n$ and $a_1+a_2+\\ldots+a_n=1$ ;\nii) For any integer $m\\in\\{1,2,\\ldots,n\\}$ , we ha... |
Solve for real numbers $x$ and $y$ ,
\begin{eqnarray*} xy^2 &=& 15x^2 + 17xy +15y^2 ; x^2y &=& 20x^2 + 3y^2. \end{eqnarray*} | Okay, so I need to solve this system of equations for real numbers x and y. Let me write them down again to make sure I have them right:
First equation: xy² = 15x² + 17xy + 15y²
Second equation: x²y = 20x² + 3y²
Hmm. Both equations look a bit complicated. Let me see if I can manipulate them to express one variable i... | To solve the system of equations for real numbers \( x \) and \( y \):
\[
\begin{cases}
xy^2 = 15x^2 + 17xy + 15y^2 \\
x^2y = 20x^2 + 3y^2
\end{cases}
\]
we start by analyzing the first equation. By setting \( t = \frac{x}{y} \) (assuming \( y \neq 0 \)), we transform the first equation into:
\[
t y = 15t^2 + 17t + ... | We are given the system of equations:
\[
\begin{cases}
xy^2 = 15x^2 + 17xy + 15y^2 \\
x^2y = 20x^2 + 3y^2
\end{cases}
\]
1. **Rewrite the second equation:**
\[
x^2y = 20x^2 + 3y^2
\]
Divide both sides by \(x\) (assuming \(x \neq 0\)):
\[
xy = 20x + \frac{3y^2}{x}
\]
Multiply both sides by \(x\)... | false | The given solution includes the points (0, 0) and (19, 95) as solutions to the original system of equations. However, the ground truth solution also identifies (0, 0) but specifies an additional solution as (-19, -2), which is not mentioned in the provided solution. Therefore, the provided solution is incorrect as it f... | aops_forum | We are given the system of equations:
\[
\begin{cases}
xy^2 = 15x^2 + 17xy + 15y^2 \\
x^2y = 20x^2 + 3y^2
\end{cases}
\]
1. **Rewrite the second equation:**
\[
x^2y = 20x^2 + 3y^2
\]
Divide both sides by \(x\) (assuming \(x \neq 0\)):
\[
xy = 20x + \frac{3y^2}{x}
\]
Multiply both sides by \(x\)... | [{'content': 'Solve for real numbers $x$ and $y$ ,\r\n\\begin{eqnarray*} xy^2 &=& 15x^2 + 17xy +15y^2 ; x^2y &=& 20x^2 + 3y^2. \\end{eqnarray*}', 'role': 'user'}
{'content': 'We are given the system of equations:\n\\[\n\\begin{cases}\nxy^2 = 15x^2 + 17xy + 15y^2 \\\\\nx^2y = 20x^2 + 3y^2\n\\end{cases}\n\\]\n\n1.... | To solve the equations given by
\[
xy^2 = 15x^2 + 17xy + 15y^2
\]
and
\[
x^2y = 20x^2 + 3y^2,
\]
we will analyze and manipulate these equations.
First, let's reorganize the first equation:
\[
xy^2 - 17xy - 15x^2 - 15y^2 = 0.
\]
This is a quadratic equation in terms of \( y \):
\[
y^2 x - 17xy - (15x^2 + 15y^... | 9,981 | [
{
"from": "human",
"value": "Solve for real numbers $x$ and $y$ ,\r\n\\begin{eqnarray*} xy^2 &=& 15x^2 + 17xy +15y^2 ; x^2y &=& 20x^2 + 3y^2. \\end{eqnarray*}"
},
{
"from": "gpt",
"value": "To solve the equations given by \n\n\\[\nxy^2 = 15x^2 + 17xy + 15y^2\n\\]\n\nand \n\n\\[\nx^2y = 20x... |
Let $n$ be a positive integer. An $n\times n$ board consisting of $n^2$ cells, each being a unit square colored either black or white, is called *convex* if for every black colored cell, both the cell directly to the left of it and the cell directly above it are also colored black. We define the *beauty* of a boa... | Okay, so I have this problem about a convex n×n board where the beauty is defined as the number of pairs of cells (u, v) such that u is black, v is white, and they are in the same row or column. I need to find the maximum possible beauty of such a convex board. Let me try to unpack this step by step.
First, let me mak... | To determine the maximum possible beauty of a convex \( n \times n \) board, we need to consider the structure of the black cells under the convexity condition. The convexity condition implies that the set of black cells forms a Young diagram, where each row and column of black cells is non-increasing in length from th... | 1. **Define the problem and notation:**
We are given an \( n \times n \) board where each cell is either black or white. The board is called *convex* if for every black cell, the cell directly to the left and the cell directly above it are also black. We need to determine the maximum possible beauty of such a board,... | false | The provided solution outlines a method for determining the maximum beauty of a convex board but diverges significantly from the logical structure and stepwise approach used in the ground truth solution. The original solution asserts a beauty value dependent on floor and ceiling functions, while the ground truth soluti... | aops_forum | 1. **Define the problem and notation:**
We are given an \( n \times n \) board where each cell is either black or white. The board is called *convex* if for every black cell, the cell directly to the left and the cell directly above it are also black. We need to determine the maximum possible beauty of such a board,... | [{'content': 'Let $n$ be a positive integer. An $n\\times n$ board consisting of $n^2$ cells, each being a unit square colored either black or white, is called *convex* if for every black colored cell, both the cell directly to the left of it and the cell directly above it are also colored black. We define the *b... | To maximize the beauty of a convex \( n \times n \) board, we need to strategically choose which cells are black and which are white, considering the constraints of convexity and how beauty is defined.
Given a convex board definition, if a cell at the position \((i,j)\) is black, then:
- The cell \((i-1,j)\) (immediat... | 9,982 | [
{
"from": "human",
"value": "Let $n$ be a positive integer. An $n\\times n$ board consisting of $n^2$ cells, each being a unit square colored either black or white, is called *convex* if for every black colored cell, both the cell directly to the left of it and the cell directly above it are also colo... |
Let $\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$ . It is known that $1<\alpha<2$ . We define the sequence of polynomials $\left\{{p_n(x)}\right\}_{n\ge0}$ by taking $p_0(x)=x$ and setting
\begin{align*}
p_{n+1}(x)=(p_n(x))^2-\alpha
\end{align*}
How many distinct real roots does $p_{10}(x)$... | Okay, so I need to figure out how many distinct real roots the polynomial \( p_{10}(x) \) has. The problem gives me a recursive definition for these polynomials. Let me start by understanding the setup.
First, \( \alpha \) is the unique real root of \( x^3 - 2x^2 + x - 1 \), and it's between 1 and 2. The polynomials \... | Given the polynomial sequence defined by \( p_0(x) = x \) and \( p_{n+1}(x) = (p_n(x))^2 - \alpha \), where \( \alpha \) is the unique real root of \( x^3 - 2x^2 + x - 1 \), we need to determine the number of distinct real roots of \( p_{10}(x) \).
1. **Initial Polynomials and Roots**:
- \( p_0(x) = x \) has 1 real... | 1. **Define the sequence of polynomials**:
We start with \( p_0(x) = x \) and the recursive relation:
\[
p_{n+1}(x) = (p_n(x))^2 - \alpha
\]
2. **Analyze the polynomial \( p_1(x) \)**:
\[
p_1(x) = (p_0(x))^2 - \alpha = x^2 - \alpha
\]
3. **Analyze the polynomial \( p_2(x) \)**:
\[
p_2(x) = ... | false | The solution incorrectly assumes that the number of distinct real roots is equal to the degree of the polynomial for all iterations. Based on the provided ground truth, the recursive nature of the polynomial sequence and the uniqueness of the real root for the parameter \\alpha affects the distinctiveness of roots. The... | aops_forum | 1. **Define the sequence of polynomials**:
We start with \( p_0(x) = x \) and the recursive relation:
\[
p_{n+1}(x) = (p_n(x))^2 - \alpha
\]
2. **Analyze the polynomial \( p_1(x) \)**:
\[
p_1(x) = (p_0(x))^2 - \alpha = x^2 - \alpha
\]
3. **Analyze the polynomial \( p_2(x) \)**:
\[
p_2(x) = ... | [{'content': 'Let $\\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$ . It is known that $1<\\alpha<2$ . We define the sequence of polynomials $\\left\\{{p_n(x)}\\right\\}_{n\\ge0}$ by taking $p_0(x)=x$ and setting\n\\begin{align*}\np_{n+1}(x)=(p_n(x))^2-\\alpha\n\\end{align*}\nHow many distinct ... | To solve for how many distinct real roots \( p_{10}(x) \) has, we will analyze the behavior of the sequence of polynomials defined as \( p_0(x) = x \) and
\[
p_{n+1}(x) = (p_n(x))^2 - \alpha
\]
for \( n \ge 0 \).
1. **Calculating the First Few Polynomials**:
- For \( n = 0 \):
\[
p_0(x) = x.
\]
... | 9,983 | [
{
"from": "human",
"value": "Let $\\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$ . It is known that $1<\\alpha<2$ . We define the sequence of polynomials $\\left\\{{p_n(x)}\\right\\}_{n\\ge0}$ by taking $p_0(x)=x$ and setting\n\\begin{align*}\np_{n+1}(x)=(p_n(x))^2-\\alpha\n\\end{... |
Find all pairs $(a,b)$ of different positive integers that satisfy the equation $W(a)=W(b)$ , where $W(x)=x^{4}-3x^{3}+5x^{2}-9x$ . | Okay, so I need to find all pairs of different positive integers (a, b) such that W(a) = W(b), where W(x) = x⁴ - 3x³ + 5x² - 9x. Let me start by understanding the problem and figuring out a strategy.
First, since we're dealing with positive integers, both a and b are in the set {1, 2, 3, ...}. They have to be differen... | To find all pairs \((a, b)\) of different positive integers that satisfy the equation \(W(a) = W(b)\), where \(W(x) = x^4 - 3x^3 + 5x^2 - 9x\), we start by evaluating the polynomial for small positive integers:
- \(W(1) = 1^4 - 3 \cdot 1^3 + 5 \cdot 1^2 - 9 \cdot 1 = -6\)
- \(W(2) = 2^4 - 3 \cdot 2^3 + 5 \cdot 2^2 - 9... | 1. Given the function \( W(x) = x^4 - 3x^3 + 5x^2 - 9x \), we need to find all pairs \((a, b)\) of different positive integers such that \( W(a) = W(b) \).
2. Suppose \( W(a) = W(b) \). Since \( a \) and \( b \) are different, we can write:
\[
a^4 - 3a^3 + 5a^2 - 9a = b^4 - 3b^3 + 5b^2 - 9b
\]
Rearranging ... | false | The provided solution correctly identifies the pair (1, 2) but does not mention (2, 1) as another valid pair of different positive integers satisfying W(1) = W(2). Ground truth solution includes this important detail, marking the provided solution as incomplete. | aops_forum | 1. Given the function \( W(x) = x^4 - 3x^3 + 5x^2 - 9x \), we need to find all pairs \((a, b)\) of different positive integers such that \( W(a) = W(b) \).
2. Suppose \( W(a) = W(b) \). Since \( a \) and \( b \) are different, we can write:
\[
a^4 - 3a^3 + 5a^2 - 9a = b^4 - 3b^3 + 5b^2 - 9b
\]
Rearranging ... | [{'content': 'Find all pairs $(a,b)$ of different positive integers that satisfy the equation $W(a)=W(b)$ , where $W(x)=x^{4}-3x^{3}+5x^{2}-9x$ .', 'role': 'user'}
{'content': "1. Given the function \\( W(x) = x^4 - 3x^3 + 5x^2 - 9x \\), we need to find all pairs \\((a, b)\\) of different positive integers such th... | To solve the equation \( W(a) = W(b) \) for different positive integers \( a \) and \( b \), where
\[
W(x) = x^4 - 3x^3 + 5x^2 - 9x,
\]
we start by analyzing the function \( W(x) \). The requirement \( W(a) = W(b) \) means that the polynomial \( W(x) \) must take the same value for two different inputs \( a \) and \(... | 9,984 | [
{
"from": "human",
"value": "Find all pairs $(a,b)$ of different positive integers that satisfy the equation $W(a)=W(b)$ , where $W(x)=x^{4}-3x^{3}+5x^{2}-9x$ ."
},
{
"from": "gpt",
"value": "To solve the equation \\( W(a) = W(b) \\) for different positive integers \\( a \\) and \\( b \\), w... |
For all real number $x$ consider the family $F(x)$ of all sequences $(a_{n})_{n\geq 0}$ satisfying the equation \[a_{n+1}=x-\frac{1}{a_{n}}\quad (n\geq 0).\] A positive integer $p$ is called a *minimal period* of the family $F(x)$ if
(a) each sequence $\left(a_{n}\right)\in F(x)$ is periodic with the per... | Alright, let's try to tackle this problem step by step. The question is about proving or disproving that for every positive integer P, there exists a real number x such that the family F(x) of sequences defined by the recurrence relation a_{n+1} = x - 1/a_n has a minimal period p greater than P.
First, let me make su... | To determine whether for each positive integer \( P \) there exists a real number \( x = x(P) \) such that the family \( F(x) \) has a minimal period \( p > P \), we analyze the recurrence relation \( a_{n+1} = x - \frac{1}{a_n} \).
1. **Understanding Minimal Period**: A minimal period \( p \) for the family \( F(x) \... | To prove or disprove that for each positive integer \( P \) there exists a real number \( x = x(P) \) such that the family \( F(x) \) has the minimal period \( p > P \), we need to analyze the given recurrence relation and the conditions for periodicity.
1. **Define the recurrence relation:**
\[
a_{n+1} = x - \f... | false | The provided solution accurately derives a value of x to ensure periodicity of the recurrence relation for sequences. However, the ground truth discusses a different approach using polynomials and does not explicitly assert that for every positive integer P there exists x fulfilling the requirement of minimal period p ... | aops_forum | To prove or disprove that for each positive integer \( P \) there exists a real number \( x = x(P) \) such that the family \( F(x) \) has the minimal period \( p > P \), we need to analyze the given recurrence relation and the conditions for periodicity.
1. **Define the recurrence relation:**
\[
a_{n+1} = x - \f... | [{'content': 'For all real number $x$ consider the family $F(x)$ of all sequences $(a_{n})_{n\\geq 0}$ satisfying the equation \\[a_{n+1}=x-\\frac{1}{a_{n}}\\quad (n\\geq 0).\\] A positive integer $p$ is called a *minimal period* of the family $F(x)$ if\r\n\r\n(a) each sequence $\\left(a_{n}\\right)\\in F(x)... | To determine whether for each positive integer \( P \) there exists a real number \( x = x(P) \) such that the family \( F(x) \) has a minimal period \( p > P \), we will explore the periodic behavior of the sequences defined by the recurrence relation:
\[
a_{n+1} = x - \frac{1}{a_n}.
\]
We start by noting that if th... | 9,985 | [
{
"from": "human",
"value": "For all real number $x$ consider the family $F(x)$ of all sequences $(a_{n})_{n\\geq 0}$ satisfying the equation \\[a_{n+1}=x-\\frac{1}{a_{n}}\\quad (n\\geq 0).\\] A positive integer $p$ is called a *minimal period* of the family $F(x)$ if\r\n\r\n(a) each sequence $\\... |
A sequence $a_n$ is defined by $$ a_0=0,\qquad a_1=3; $$ $$ a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2. $$ Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$ .
| Okay, let's see. I need to find the least positive integer h such that a_{n+h} - a_n is divisible by 1999 for all n ≥ 0. The sequence is defined by a_0 = 0, a_1 = 3, and for n ≥ 2, a_n = 8a_{n-1} + 9a_{n-2} + 16. Hmm, this seems like a linear recurrence relation with constant coefficients and a constant nonhomogeneous ... | To find the least positive integer \( h \) such that \( a_{n+h} - a_n \) is divisible by 1999 for all \( n \ge 0 \), we start by solving the given recurrence relation:
\[ a_0 = 0, \quad a_1 = 3, \]
\[ a_n = 8a_{n-1} + 9a_{n-2} + 16 \text{ for } n \ge 2. \]
The homogeneous part of the recurrence relation is solved by ... | To solve the problem, we need to find the least positive integer \( h \) such that \( a_{n+h} - a_n \) is divisible by \( 1999 \) for all \( n \geq 0 \).
1. **Define the sequence transformation:**
Given the sequence \( a_n \) defined by:
\[
a_0 = 0, \quad a_1 = 3, \quad a_n = 8a_{n-1} + 9a_{n-2} + 16 \text{ f... | false | The solution provided incorrectly calculates the least positive integer h for which a_{n+h} - a_n is divisible by 1999. The work shows that the order of 9 modulo 1999 is 999, thus the least even h should be 1998, not 18. The solution arrives at the correct formula but ultimately concludes with an incorrect evaluation o... | aops_forum | To solve the problem, we need to find the least positive integer \( h \) such that \( a_{n+h} - a_n \) is divisible by \( 1999 \) for all \( n \geq 0 \).
1. **Define the sequence transformation:**
Given the sequence \( a_n \) defined by:
\[
a_0 = 0, \quad a_1 = 3, \quad a_n = 8a_{n-1} + 9a_{n-2} + 16 \text{ f... | [{'content': 'A sequence $a_n$ is defined by $$ a_0=0,\\qquad a_1=3; $$ $$ a_n=8a_{n-1}+9a_{n-2}+16\\text{ for }n\\ge2. $$ Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\\ge0$ .\n', 'role': 'user'}
{'content': "To solve the problem, we need to find the least... | The sequence \( a_n \) is defined as follows:
\[
a_0 = 0, \quad a_1 = 3,
\]
\[
a_n = 8a_{n-1} + 9a_{n-2} + 16 \quad \text{for } n \ge 2.
\]
To find the least positive integer \( h \) such that \( a_{n+h} - a_n \) is divisible by \( 1999 \) for all \( n \ge 0 \), we start by solving the recurrence relation.
We first ... | 9,986 | [
{
"from": "human",
"value": "A sequence $a_n$ is defined by $$ a_0=0,\\qquad a_1=3; $$ $$ a_n=8a_{n-1}+9a_{n-2}+16\\text{ for }n\\ge2. $$ Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\\ge0$ .\n"
},
{
"from": "gpt",
"value": "The seque... |
If $a, b$ are positive reals such that $a+b<2$ . Prove that $$ \frac{1}{1+a^2}+\frac{1}{1+b^2} \le \frac{2}{1+ab} $$ and determine all $a, b$ yielding equality.
*Proposed by Gottfried Perz* | Okay, let's see. I have this inequality to prove: if a and b are positive real numbers such that a + b < 2, then 1/(1 + a²) + 1/(1 + b²) ≤ 2/(1 + ab). And I need to determine all a and b where equality holds. Hmm, interesting. Alright, let's start by understanding the problem and then try different approaches.
First, ... | Given positive real numbers \(a\) and \(b\) such that \(a + b < 2\), we need to prove the inequality:
\[
\frac{1}{1+a^2} + \frac{1}{1+b^2} \le \frac{2}{1+ab}
\]
and determine all \(a, b\) yielding equality.
First, we start by cross-multiplying to eliminate the denominators. The inequality can be rewritten as:
\[
\f... | 1. Given \(a, b > 0\) and \(a + b < 2\), we need to prove that:
\[
\frac{1}{1 + a^2} + \frac{1}{1 + b^2} \leq \frac{2}{1 + ab}
\]
and determine all \(a, b\) yielding equality.
2. We start by considering a more general inequality:
\[
\frac{1}{k + a^2} + \frac{1}{k + b^2} \leq \frac{2}{k + ab}
\]
... | true | The solution provided accurately follows the steps to prove the given inequality and correctly identifies the conditions for equality. The reasoning aligns well with the ground truth solution, demonstrating a solid understanding of the mathematical principles involved. Both solutions agree on the critical outcome that ... | aops_forum | 1. Given \(a, b > 0\) and \(a + b < 2\), we need to prove that:
\[
\frac{1}{1 + a^2} + \frac{1}{1 + b^2} \leq \frac{2}{1 + ab}
\]
and determine all \(a, b\) yielding equality.
2. We start by considering a more general inequality:
\[
\frac{1}{k + a^2} + \frac{1}{k + b^2} \leq \frac{2}{k + ab}
\]
... | [{'content': 'If $a, b$ are positive reals such that $a+b<2$ . Prove that $$ \\frac{1}{1+a^2}+\\frac{1}{1+b^2} \\le \\frac{2}{1+ab} $$ and determine all $a, b$ yielding equality.\n\n*Proposed by Gottfried Perz*', 'role': 'user'}
{'content': '1. Given \\(a, b > 0\\) and \\(a + b < 2\\), we need to prove that:\... | To prove the inequality
\[
\frac{1}{1+a^2}+\frac{1}{1+b^2} \leq \frac{2}{1+ab}
\]
for positive reals \( a \) and \( b \) such that \( a + b < 2 \), we start by using a common approach involving the Cauchy-Schwarz inequality.
Recall that the Cauchy-Schwarz inequality states that:
\[
(x_1^2 + x_2^2)(y_1^2 + y_2^2) ... | 9,987 | [
{
"from": "human",
"value": "If $a, b$ are positive reals such that $a+b<2$ . Prove that $$ \\frac{1}{1+a^2}+\\frac{1}{1+b^2} \\le \\frac{2}{1+ab} $$ and determine all $a, b$ yielding equality.\n\n*Proposed by Gottfried Perz*"
},
{
"from": "gpt",
"value": "To prove the inequality \n\n\\... |
Peter has $2022$ pieces of magnetic railroad cars, which are of two types: some have the front with north and the rear with south magnetic polarity, and some have the rear with north and the rear with south magnetic polarity (on these railroad cars the front and the rear can be distinguished). Peter wants to decide w... | Okay, so Peter has 2022 magnetic railroad cars, and they come in two types. Let me try to understand the problem first. The two types are:
1. Front has north polarity, rear has south polarity.
2. Rear has north polarity, front has south polarity.
Wait, actually, the problem says: "some have the front with north and t... | To determine whether Peter has an equal number of two types of magnetic railroad cars, we need to analyze the minimum number of connection attempts (tries) required. Each try involves testing two cars in a specific orientation to see if they connect, which reveals whether they are of the same type or different types.
... | 1. **Understanding the Problem:**
Peter has 2022 magnetic railroad cars of two types:
- Type A: Front with north and rear with south magnetic polarity.
- Type B: Rear with north and rear with south magnetic polarity.
Peter wants to determine if there is an equal number of both types of cars. He can test... | false | The provided solution claims the minimum number of tries required to determine if Peter has an equal number of magnetic railroad cars is 2021. However, the ground truth states that the actual minimum number of tries needed is 2014, calculated using a specific mathematical formula and approach. Thus, the two answers do ... | aops_forum | 1. **Understanding the Problem:**
Peter has 2022 magnetic railroad cars of two types:
- Type A: Front with north and rear with south magnetic polarity.
- Type B: Rear with north and rear with south magnetic polarity.
Peter wants to determine if there is an equal number of both types of cars. He can test... | [{'content': 'Peter has $2022$ pieces of magnetic railroad cars, which are of two types: some have the front with north and the rear with south magnetic polarity, and some have the rear with north and the rear with south magnetic polarity (on these railroad cars the front and the rear can be distinguished). Peter wan... | To determine whether Peter has the same number of two types of magnetic railroad cars—let's denote them as Type A (front North, rear South) and Type B (front South, rear North)—we can use a strategy that allows us to pair the cars and infer their quantities based on successful pairings.
Let \( a \) be the number of Ty... | 9,988 | [
{
"from": "human",
"value": "Peter has $2022$ pieces of magnetic railroad cars, which are of two types: some have the front with north and the rear with south magnetic polarity, and some have the rear with north and the rear with south magnetic polarity (on these railroad cars the front and the rear can b... |
$4$ equilateral triangles of side length $1$ are drawn on the interior of a unit square, each one of which shares a side with one of the $4$ sides of the unit square. What is the common area enclosed by all $4$ equilateral triangles? | Okay, so I need to find the common area enclosed by all four equilateral triangles that are drawn inside a unit square. Each triangle shares a side with one of the four sides of the square. Let me start by visualizing the problem.
First, a unit square has sides of length 1. Each equilateral triangle is drawn on the in... | To determine the common area enclosed by all four equilateral triangles drawn inside a unit square, we analyze the intersection of these triangles. Each triangle is positioned such that one side coincides with a side of the square, and the other two sides extend inward.
Key steps:
1. **Equilateral Triangles**: Each t... | 1. **Define the square and the equilateral triangles:**
- Consider a unit square \(ABCD\) with vertices \(A(0,0)\), \(B(1,0)\), \(C(1,1)\), and \(D(0,1)\).
- Four equilateral triangles are drawn on the interior of the square, each sharing a side with one of the sides of the square.
2. **Determine the vertices of... | false | The solution presented calculates the area of the overlap of the equilateral triangles inside the unit square incorrectly and concludes with a negative area, which is not possible. The ground truth solution instead arrived at an area of \( 2\sqrt{3} - 3 \), which indicates the overlapping region properly. The negative ... | aops_forum | 1. **Define the square and the equilateral triangles:**
- Consider a unit square \(ABCD\) with vertices \(A(0,0)\), \(B(1,0)\), \(C(1,1)\), and \(D(0,1)\).
- Four equilateral triangles are drawn on the interior of the square, each sharing a side with one of the sides of the square.
2. **Determine the vertices of... | [{'content': ' $4$ equilateral triangles of side length $1$ are drawn on the interior of a unit square, each one of which shares a side with one of the $4$ sides of the unit square. What is the common area enclosed by all $4$ equilateral triangles?', 'role': 'user'}
{'content': "1. **Define the square and the e... | Given four equilateral triangles, each with a side length of \(1\), inscribed in a unit square such that each triangle shares a side with one of the square's sides, we can find the area common to all four triangles.
First, we denote the vertices of the unit square as the following coordinates:
- Bottom-left corner: \... | 9,989 | [
{
"from": "human",
"value": " $4$ equilateral triangles of side length $1$ are drawn on the interior of a unit square, each one of which shares a side with one of the $4$ sides of the unit square. What is the common area enclosed by all $4$ equilateral triangles?"
},
{
"from": "gpt",
"val... |
Given a prime number $p > 2$ . Find the least $n\in Z_+$ , for which every set of $n$ perfect squares not divisible by $p$ contains nonempty subset with product of all it's elements equal to $1\ (\text{mod}\ p)$ | Okay, let's try to tackle this problem. So, we have a prime number p greater than 2, and we need to find the smallest positive integer n such that any set of n perfect squares not divisible by p contains a nonempty subset whose product is congruent to 1 modulo p. Hmm, alright. Let's break this down step by step.
First... | Given a prime number \( p > 2 \), we need to find the least positive integer \( n \) such that every set of \( n \) perfect squares not divisible by \( p \) contains a nonempty subset with the product of all its elements equal to \( 1 \pmod{p} \).
1. **Quadratic Residues**: The set of perfect squares not divisible by ... | 1. **Base Case for \( p = 3 \):**
- For \( p = 3 \), the conclusion is straightforward. We can take a one-element subset since any perfect square modulo 3 is either 0 or 1. Since we are considering perfect squares not divisible by \( p \), the only possible residue is 1. Thus, any single perfect square will satisfy ... | true | Both the provided solution and the ground truth solution arrive at the same conclusion: the least positive integer n such that every set of n perfect squares not divisible by a prime p (where p > 2) contains a nonempty subset whose product is 1 modulo p is indeed \(\frac{p-1}{2}\. The line of reasoning and the mathemat... | aops_forum | 1. **Base Case for \( p = 3 \):**
- For \( p = 3 \), the conclusion is straightforward. We can take a one-element subset since any perfect square modulo 3 is either 0 or 1. Since we are considering perfect squares not divisible by \( p \), the only possible residue is 1. Thus, any single perfect square will satisfy ... | [{'content': "Given a prime number $p > 2$ . Find the least $n\\in Z_+$ , for which every set of $n$ perfect squares not divisible by $p$ contains nonempty subset with product of all it's elements equal to $1\\ (\\text{mod}\\ p)$ ", 'role': 'user'}
{'content': "1. **Base Case for \\( p = 3 \\):**\n - For \\( ... | To solve the problem, we need to find the least positive integer \( n \) such that in any set of \( n \) perfect squares that are not divisible by a given odd prime \( p \), there exists a non-empty subset whose product is congruent to \( 1 \mod p \).
First, let's analyze the perfect squares modulo \( p \). Since \( p... | 9,990 | [
{
"from": "human",
"value": "Given a prime number $p > 2$ . Find the least $n\\in Z_+$ , for which every set of $n$ perfect squares not divisible by $p$ contains nonempty subset with product of all it's elements equal to $1\\ (\\text{mod}\\ p)$ "
},
{
"from": "gpt",
"value": "To solve the... |
How many ways are there to divide a set with 6 elements into 3 disjoint subsets? $\textbf{(A)}\ 90 \qquad\textbf{(B)}\ 105 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\ 180 \qquad\textbf{(E)}\ 243$ | Okay, let me try to figure out how to solve this problem. The question is asking: How many ways are there to divide a set with 6 elements into 3 disjoint subsets? The options are given from A to E.
First, I need to understand exactly what the question is asking. So, we have a set with 6 elements, and we need to split... | To determine the number of ways to divide a set with 6 elements into 3 disjoint subsets, we need to consider whether the subsets are labeled or unlabeled and whether they can be empty.
1. **Stirling Numbers of the Second Kind**: These count the number of ways to partition a set of \( n \) elements into \( k \) non-em... | To solve the problem of dividing a set with 6 elements into 3 disjoint subsets, we need to consider the combinatorial approach carefully. The problem can be approached using the concept of Stirling numbers of the second kind, which count the number of ways to partition a set of \( n \) elements into \( k \) non-empty s... | false | The provided solution incorrectly identifies the number of ways to divide 6 elements into 3 subsets, leading to a conclusion of 105, while the ground truth correctly evaluates the count to be 15. The mistake lies in misunderstanding the partitioning rules and the implications of labeling versus unlabeled subsets. | aops_forum | To solve the problem of dividing a set with 6 elements into 3 disjoint subsets, we need to consider the combinatorial approach carefully. The problem can be approached using the concept of Stirling numbers of the second kind, which count the number of ways to partition a set of \( n \) elements into \( k \) non-empty s... | [{'content': 'How many ways are there to divide a set with 6 elements into 3 disjoint subsets? $\\textbf{(A)}\\ 90 \\qquad\\textbf{(B)}\\ 105 \\qquad\\textbf{(C)}\\ 120 \\qquad\\textbf{(D)}\\ 180 \\qquad\\textbf{(E)}\\ 243$ ', 'role': 'user'}
{'content': 'To solve the problem of dividing a set with 6 elements ... | To find the number of ways to divide a set of \( n \) elements into \( k \) disjoint subsets, we can use the Stirling numbers of the second kind, denoted as \( S(n, k) \). These numbers count the ways to partition a set of \( n \) elements into \( k \) non-empty subsets. Since we want to divide a set of \( 6 \) element... | 9,991 | [
{
"from": "human",
"value": "How many ways are there to divide a set with 6 elements into 3 disjoint subsets? $\\textbf{(A)}\\ 90 \\qquad\\textbf{(B)}\\ 105 \\qquad\\textbf{(C)}\\ 120 \\qquad\\textbf{(D)}\\ 180 \\qquad\\textbf{(E)}\\ 243$ "
},
{
"from": "gpt",
"value": "To find the numbe... |
Determine the largest natural number $ N $ having the following property: every $ 5\times 5 $ array consisting of pairwise distinct natural numbers from $ 1 $ to $ 25 $ contains a $ 2\times 2 $ subarray of numbers whose sum is, at least, $ N. $ *Demetres Christofides* and *Silouan Brazitikos* | Okay, so I need to find the largest natural number N such that every 5x5 array with distinct numbers from 1 to 25 has at least one 2x2 subarray where the sum of its numbers is at least N. Hmm, interesting problem. Let me start by understanding what exactly is being asked here.
First, the array is 5x5, filled with numb... | To determine the largest natural number \( N \) such that every \( 5 \times 5 \) array with distinct natural numbers from 1 to 25 contains a \( 2 \times 2 \) subarray with a sum of at least \( N \), we need to consider various arrangements and sums of subarrays.
1. **Understanding the Problem**:
- A \( 5 \times 5 \... | To determine the largest natural number \( N \) such that every \( 5 \times 5 \) array consisting of pairwise distinct natural numbers from \( 1 \) to \( 25 \) contains a \( 2 \times 2 \) subarray of numbers whose sum is at least \( N \), we need to analyze the properties of such arrays and subarrays.
1. **Sum of All ... | false | The provided solution claims that the largest natural number `N` such that every `2x2` subarray has a sum of at least `N` is `61`. However, the ground truth solution correctly identifies that with an optimal arrangement, the largest `N` cannot be greater than `45`. Therefore, the provided solution is incorrect. | aops_forum | To determine the largest natural number \( N \) such that every \( 5 \times 5 \) array consisting of pairwise distinct natural numbers from \( 1 \) to \( 25 \) contains a \( 2 \times 2 \) subarray of numbers whose sum is at least \( N \), we need to analyze the properties of such arrays and subarrays.
1. **Sum of All ... | [{'content': 'Determine the largest natural number $ N $ having the following property: every $ 5\\times 5 $ array consisting of pairwise distinct natural numbers from $ 1 $ to $ 25 $ contains a $ 2\\times 2 $ subarray of numbers whose sum is, at least, $ N. $ *Demetres Christofides* and *Silouan Brazitikos*... | To solve this problem, we need to analyze a \( 5 \times 5 \) array filled with distinct natural numbers from \( 1 \) to \( 25 \). Each number appears exactly once in the array, and we aim to find the largest natural number \( N \) such that every \( 2 \times 2 \) subarray of the matrix has a sum that is at least \( N \... | 9,992 | [
{
"from": "human",
"value": "Determine the largest natural number $ N $ having the following property: every $ 5\\times 5 $ array consisting of pairwise distinct natural numbers from $ 1 $ to $ 25 $ contains a $ 2\\times 2 $ subarray of numbers whose sum is, at least, $ N. $ *Demetres Christofide... |
If $\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}}=\sqrt{m}-\sqrt{n}$ , where $m$ and $n$ are positive integers, what is the value of $m+n$ ? | Okay, let me try to solve this problem. The equation given is:
1 over the square root of (2011 plus the square root of (2011 squared minus 1)) equals the square root of m minus the square root of n, where m and n are positive integers. We need to find m + n.
Hmm, so the left side is a complicated expression involving... | Given the equation:
\[
\frac{1}{\sqrt{2011 + \sqrt{2011^2 - 1}}} = \sqrt{m} - \sqrt{n}
\]
where \( m \) and \( n \) are positive integers, we need to find the value of \( m + n \).
First, we rationalize the denominator on the left-hand side. Let \( L \) be the left-hand side expression:
\[
L = \frac{1}{\sqrt{2011 +... | We start with the given equation:
\[
\frac{1}{\sqrt{2011 + \sqrt{2011^2 - 1}}} = \sqrt{m} - \sqrt{n}
\]
1. **Square both sides**:
\[
\left( \frac{1}{\sqrt{2011 + \sqrt{2011^2 - 1}}} \right)^2 = (\sqrt{m} - \sqrt{n})^2
\]
\[
\frac{1}{2011 + \sqrt{2011^2 - 1}} = m - 2\sqrt{mn} + n
\]
2. **Rationalize the denominator on... | true | The provided solution accurately followed the steps to rationalize the left-hand side of the equation and established the correct relationships between the rational and irrational components. Both the calculation and reasoning were consistent with the ground truth solution, leading to the correct final answer of \( m +... | aops_forum | We start with the given equation:
\[
\frac{1}{\sqrt{2011 + \sqrt{2011^2 - 1}}} = \sqrt{m} - \sqrt{n}
\]
1. **Square both sides**:
\[
\left( \frac{1}{\sqrt{2011 + \sqrt{2011^2 - 1}}} \right)^2 = (\sqrt{m} - \sqrt{n})^2
\]
\[
\frac{1}{2011 + \sqrt{2011^2 - 1}} = m - 2\sqrt{mn} + n
\]
2. **Rationalize the denominator on... | [{'content': ' If $\\frac{1}{\\sqrt{2011+\\sqrt{2011^2-1}}}=\\sqrt{m}-\\sqrt{n}$ , where $m$ and $n$ are positive integers, what is the value of $m+n$ ?', 'role': 'user'}
{'content': 'We start with the given equation:\n\\[\n\\frac{1}{\\sqrt{2011 + \\sqrt{2011^2 - 1}}} = \\sqrt{m} - \\sqrt{n}\n\\]\n\n1. **Square... | To solve the equation
\[
\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}} = \sqrt{m} - \sqrt{n},
\]
let's start by simplifying the left-hand side. We denote:
\[
x = \sqrt{2011+\sqrt{2011^2-1}}.
\]
We first identify \( \sqrt{2011^2 - 1} \):
\[
\sqrt{2011^2 - 1} = \sqrt{2011^2 - 1^2} = \sqrt{(2011-1)(2011+1)} = \sqrt{2010 \cd... | 9,993 | [
{
"from": "human",
"value": " If $\\frac{1}{\\sqrt{2011+\\sqrt{2011^2-1}}}=\\sqrt{m}-\\sqrt{n}$ , where $m$ and $n$ are positive integers, what is the value of $m+n$ ?"
},
{
"from": "gpt",
"value": "To solve the equation \n\n\\[\n\\frac{1}{\\sqrt{2011+\\sqrt{2011^2-1}}} = \\sqrt{m} - \\sq... |
There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$ .
Julia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased? | Okay, let's see. The problem says there are 200 numbers on a blackboard: 1!, 2!, 3!, all the way up to 200!. Julia erases one of them, and when she multiplies the remaining 199 numbers together, the product is a perfect square. We need to figure out which number she erased.
First, I need to recall that a perfect squar... | The number Julia erased is $\boxed{100!}$.
**Step-by-Step Explanation:**
1. **Understanding the Problem:**
We need to determine which single factorial erased from the list $1!, 2!, \ldots, 200!$ results in the product of the remaining 199 factorials being a perfect square.
2. **Key Insight:**
A number is ... | 1. **Define the product \( N \):**
\[
N = 1! \cdot 2! \cdot 3! \cdot \ldots \cdot 199! \cdot 200!
\]
We need to determine which number, when erased, makes the product of the remaining numbers a perfect square.
2. **Factorize \( N \):**
\[
N = \prod_{k=1}^{200} k!
\]
We can split this product in... | true | The solution accurately follows the logical steps needed to determine that erasing $100!$ from the product results in all prime factors having even exponents. This confirms that the solution matches the ground truth reasoning. | aops_forum | 1. **Define the product \( N \):**
\[
N = 1! \cdot 2! \cdot 3! \cdot \ldots \cdot 199! \cdot 200!
\]
We need to determine which number, when erased, makes the product of the remaining numbers a perfect square.
2. **Factorize \( N \):**
\[
N = \prod_{k=1}^{200} k!
\]
We can split this product in... | [{'content': 'There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$ .\nJulia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased?', 'role': 'user'}
{'content': '1. **Define the product \\( N \\):... | To determine which factorial number was erased such that the product of the remaining numbers on the blackboard is a perfect square, we begin by analyzing the factorials \( n! \) for \( n = 1, 2, ..., 200 \).
A product is a perfect square if all primes in its prime factorization have even exponents. Hence, we will ana... | 9,994 | [
{
"from": "human",
"value": "There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$ .\nJulia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased?"
},
{
"from": "gpt",
"value... |
Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\frac{\pi}2]$ . Find the largest value of the sum of the $\binom n2$ angles between line.
*By Aliakbar Daemi* | Okay, so I need to find the largest possible value of the sum of all the angles between n lines in a plane, where no two lines are parallel and no three are concurrent. Each angle is between 0 and π/2. Hmm, let me think about how to approach this.
First, let me recall that if there are n lines with no two parallel and... | To find the largest possible value of the sum of all the angles between \( n \) lines in a plane, where no two lines are parallel and no three are concurrent, we need to consider the optimal arrangement of lines to maximize the sum of their angles. Each angle is between 0 and \(\frac{\pi}{2}\).
1. **Key Observations**... | 1. **Initial Setup and Assumptions:**
- We are given \( n \) lines in a plane such that no two lines are parallel and no three lines are concurrent.
- For each pair of lines, the angle between them is a real number in \([0, \frac{\pi}{2}]\).
- We aim to find the largest value of the sum of the \(\binom{n}{2}\)... | false | The provided solution gives a formula for the maximum sum of angles as \\(\left\lfloor \frac{n^2}{4} \right\rfloor \cdot \frac{\pi}{2}\\ which does not align with the ground truth solution that ultimately presents a formula specific to the parity of \( n \) (odd/even). The ground truth is more detailed and specific reg... | aops_forum | 1. **Initial Setup and Assumptions:**
- We are given \( n \) lines in a plane such that no two lines are parallel and no three lines are concurrent.
- For each pair of lines, the angle between them is a real number in \([0, \frac{\pi}{2}]\).
- We aim to find the largest value of the sum of the \(\binom{n}{2}\)... | [{'content': 'Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\\frac{\\pi}2]$ . Find the largest value of the sum of the $\\binom n2$ angles between line.\r\n*By Aliakbar Daemi*', 'role': 'user'}
{'content': '1. **Ini... | To find the largest value of the sum of the angles between \( n \) lines in the plane, where no two lines are parallel and no three lines are concurrent, we represent each line by unit vectors in the plane. For two lines represented by unit vectors \( \mathbf{u}_i \) and \( \mathbf{u}_j \), the angle \( \theta_{ij} \) ... | 9,995 | [
{
"from": "human",
"value": "Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\\frac{\\pi}2]$ . Find the largest value of the sum of the $\\binom n2$ angles between line.\r\n*By Aliakbar Daemi*"
},
{
... |
The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$ | Okay, so I need to construct points E on BC and F on AC in triangle ABC, given a point D on AB. The conditions are that the midpoints of DE and DF are collinear with B, and the midpoints of DE and EF are collinear with C. Hmm, that's a bit complex, but let's break it down step by step.
First, let me visualize triangle... | To construct points \( E \) on \( BC \) and \( F \) on \( AC \) in triangle \( ABC \) given point \( D \) on \( AB \), follow these steps:
**Construction Steps:**
1. **Locate Point \( D \):** Let \( D \) be a given point on side \( AB \).
2. **Construct Point \( F' \) on \( AC \):** Using the same length as \( AD \)... | 1. **Construct the reflection \( X \) of the point \( D \) with respect to the point \( C \).**
- To find the reflection \( X \), we use the fact that \( C \) is the midpoint of \( DX \). Therefore, \( X \) is such that \( \overrightarrow{CX} = \overrightarrow{CD} \). This gives us the coordinates of \( X \) as \( X... | false | The provided solution follows a different approach than the ground truth solution. While both aim to construct points E and F in triangle ABC given D, the methodologies differ significantly. The ground truth solution focuses on reflections and parallel lines, which is a different geometric construction than the horizon... | aops_forum | 1. **Construct the reflection \( X \) of the point \( D \) with respect to the point \( C \).**
- To find the reflection \( X \), we use the fact that \( C \) is the midpoint of \( DX \). Therefore, \( X \) is such that \( \overrightarrow{CX} = \overrightarrow{CD} \). This gives us the coordinates of \( X \) as \( X... | [{'content': 'The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$ ', 'role': 'user'}
{'content': '1. **Construct t... | To solve this problem, we will denote the points of triangle \(ABC\) with coordinates as follows:
- Let the coordinates of point \(A\) be \(A = (0, 0)\).
- Let the coordinates of point \(B\) be \(B = (b_x, b_y)\).
- Let the coordinates of point \(C\) be \(C = (c_x, c_y)\).
- Let point \(D\) on side \(AB\) have coordin... | 9,996 | [
{
"from": "human",
"value": "The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$ "
},
{
"from":... |
Let $\omega$ be the incircle of triangle $ABC$ . For any positive real number $\lambda$ , let $\omega_{\lambda}$ be the circle concentric with $\omega$ that has radius $\lambda$ times that of $\omega$ . Let $X$ be the intersection between a trisector of $\angle B$ closer to $BC$ and a trisector of $\a... | Okay, so I need to show that the circumcircle of triangle XYZ lies inside the circle ω_{1−ε}, which is concentric with the incircle ω of triangle ABC and has a radius of (1−ε) times that of ω. Here, ε is given as 1/2024.
First, let me try to understand the problem step by step.
We have triangle ABC with incircle ω.... | To show that the circumcircle of triangle \(XYZ\) lies inside \(\omega_{1-\epsilon}\) with \(\epsilon = \frac{1}{2024}\), we need to demonstrate that the circumradius of \(XYZ\) is less than \((1 - \epsilon)r\), where \(r\) is the inradius of triangle \(ABC\).
1. **Understanding the Problem**:
- The incircle \(\ome... | 1. **Define the incircle and its properties:**
Let $\omega$ be the incircle of triangle $ABC$ with center $I$ and radius $r$. The circle $\omega_{\lambda}$ is concentric with $\omega$ and has radius $\lambda r$.
2. **Define the points $X$, $Y$, and $Z$:**
The point $X$ is the intersection of the trisector of $\a... | false | The given solution concludes that the circumcircle of triangle XYZ lies inside the circle ω_{1-ε} for ε = 1/2024, stating a complete reasoning process. However, it fails to provide a definitive numerical result or reference to a need for a box notation that strictly matches the ground truth, which implies only the asse... | aops_forum | 1. **Define the incircle and its properties:**
Let $\omega$ be the incircle of triangle $ABC$ with center $I$ and radius $r$. The circle $\omega_{\lambda}$ is concentric with $\omega$ and has radius $\lambda r$.
2. **Define the points $X$, $Y$, and $Z$:**
The point $X$ is the intersection of the trisector of $\a... | [{'content': 'Let $\\omega$ be the incircle of triangle $ABC$ . For any positive real number $\\lambda$ , let $\\omega_{\\lambda}$ be the circle concentric with $\\omega$ that has radius $\\lambda$ times that of $\\omega$ . Let $X$ be the intersection between a trisector of $\\angle B$ closer to $BC$ a... | To show that the circumcircle of triangle \( XYZ \) lies inside the circle \( \omega_{1 - \epsilon} \), we need to establish relevant geometric properties of the points \( X, Y, Z \) and their relation to the incircle \( \omega \) and the modified circle \( \omega_{1 - \epsilon} \).
1. **Understanding the Points X, Y,... | 9,997 | [
{
"from": "human",
"value": "Let $\\omega$ be the incircle of triangle $ABC$ . For any positive real number $\\lambda$ , let $\\omega_{\\lambda}$ be the circle concentric with $\\omega$ that has radius $\\lambda$ times that of $\\omega$ . Let $X$ be the intersection between a trisector of $\\a... |
Let \(P(x)=x^2+\dfrac x 2 +b\) and \(Q(x)=x^2+cx+d\) be two polynomials with real coefficients such that \(P(x)Q(x)=Q(P(x))\) for all real \(x\). Find all real roots of \(P(Q(x))=0\). | Okay, let's try to tackle this problem step by step. The question is about finding all real roots of \( P(Q(x)) = 0 \) given that \( P(x)Q(x) = Q(P(x)) \) for all real \( x \). The polynomials are defined as \( P(x) = x^2 + \dfrac{x}{2} + b \) and \( Q(x) = x^2 + cx + d \).
First, I need to understand what the conditi... | Given the polynomials \( P(x) = x^2 + \dfrac{x}{2} + b \) and \( Q(x) = x^2 + cx + d \) with real coefficients such that \( P(x)Q(x) = Q(P(x)) \) for all real \( x \), we need to find all real roots of \( P(Q(x)) = 0 \).
First, we expand both sides of the equation \( P(x)Q(x) = Q(P(x)) \) and equate the coefficients o... | 1. **Given Polynomials:**
\[
P(x) = x^2 + \frac{x}{2} + b \quad \text{and} \quad Q(x) = x^2 + cx + d
\]
We are given that \(P(x)Q(x) = Q(P(x))\) for all real \(x\).
2. **Setting \(P(x) = 0\):**
\[
P(x) = 0 \implies x^2 + \frac{x}{2} + b = 0
\]
Let the roots of \(P(x) = 0\) be \(\alpha\) and \(\... | false | The given solution correctly derives the polynomials based on the condition that \, P(x)Q(x) = Q(P(x)) \ && then expands and tries to solve \ P(Q(x)) = 0 \ but fails to identify the roots correctly. It states that the roots are both \ -1 \ and \ \frac{1}{2} \ but this is incorrect as the final correct expression for th... | aops_forum | 1. **Given Polynomials:**
\[
P(x) = x^2 + \frac{x}{2} + b \quad \text{and} \quad Q(x) = x^2 + cx + d
\]
We are given that \(P(x)Q(x) = Q(P(x))\) for all real \(x\).
2. **Setting \(P(x) = 0\):**
\[
P(x) = 0 \implies x^2 + \frac{x}{2} + b = 0
\]
Let the roots of \(P(x) = 0\) be \(\alpha\) and \(\... | [{'content': 'Let \\(P(x)=x^2+\\dfrac x 2 +b\\) and \\(Q(x)=x^2+cx+d\\) be two polynomials with real coefficients such that \\(P(x)Q(x)=Q(P(x))\\) for all real \\(x\\). Find all real roots of \\(P(Q(x))=0\\).', 'role': 'user'}
{'content': '1. **Given Polynomials:**\n \\[\n P(x) = x^2 + \\frac{x}{2} + b \\quad \\te... | Let \( P(x) = x^2 + \frac{1}{2}x + b \) and \( Q(x) = x^2 + cx + d \). We are given the condition \( P(x)Q(x) = Q(P(x)) \) for all real \( x \).
First, we will expand both sides and compare coefficients.
### Step 1: Expand \( P(x) Q(x) \)
\[
P(x) Q(x) = \left( x^2 + \frac{1}{2} x + b \right) \left( x^2 + cx + d \rig... | 9,998 | [
{
"from": "human",
"value": "Let \\(P(x)=x^2+\\dfrac x 2 +b\\) and \\(Q(x)=x^2+cx+d\\) be two polynomials with real coefficients such that \\(P(x)Q(x)=Q(P(x))\\) for all real \\(x\\). Find all real roots of \\(P(Q(x))=0\\)."
},
{
"from": "gpt",
"value": "Let \\( P(x) = x^2 + \\frac{1}{2}x + b \\... |
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$ . Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square. | Okay, so I need to show that if a and b are positive integers such that ab + 1 divides a² + b², then the quotient (a² + b²)/(ab + 1) is a perfect square. Hmm, let's start by understanding the problem.
First, let's restate the condition: ab + 1 divides a² + b². That means there exists some integer k such that (a² + b²)... | To prove that \(\frac{a^2 + b^2}{ab + 1}\) is a perfect square when \(a\) and \(b\) are positive integers with \(ab + 1\) dividing \(a^2 + b^2\), we employ the method of Vieta jumping:
1. **Set Up the Equation**: Given \(ab + 1 \mid a^2 + b^2\), let \(k = \frac{a^2 + b^2}{ab + 1}\). This implies:
\[
a^2 - kab + ... | 1. **Given Condition**: We are given that \(a \cdot b + 1\) divides \(a^2 + b^2\). We need to show that \(\frac{a^2 + b^2}{a \cdot b + 1}\) is a perfect square.
2. **Assumption of Minimality**: Let \((a, b)\) be the smallest solution in the sense that \(a + b\) is minimized. Assume without loss of generality that \(a ... | false | While both solutions correctly attempt to prove that the expression is a perfect square under the given condition, they arrive at different final conclusions regarding the format of the expression. The first solution claims that the expression itself is a perfect square without specifying its value, while the ground tr... | aops_forum | 1. **Given Condition**: We are given that \(a \cdot b + 1\) divides \(a^2 + b^2\). We need to show that \(\frac{a^2 + b^2}{a \cdot b + 1}\) is a perfect square.
2. **Assumption of Minimality**: Let \((a, b)\) be the smallest solution in the sense that \(a + b\) is minimized. Assume without loss of generality that \(a ... | [{'content': 'Let $ a$ and $ b$ be two positive integers such that $ a \\cdot b \\plus{} 1$ divides $ a^{2} \\plus{} b^{2}$ . Show that $ \\frac {a^{2} \\plus{} b^{2}}{a \\cdot b \\plus{} 1}$ is a perfect square.', 'role': 'user'}
{'content': "1. **Given Condition**: We are given that \\(a \\cdot b + 1\\) div... | Let \( d = a \cdot b + 1 \). We start from the condition that \( d \) divides \( a^2 + b^2 \). This means there exists an integer \( k \) such that
\[
a^2 + b^2 = k \cdot d = k(a \cdot b + 1).
\]
We can rearrange this expression as follows:
\[
a^2 + b^2 = k(a \cdot b) + k.
\]
Simplifying, we get:
\[
a^2 + b^2 - k ... | 9,999 | [
{
"from": "human",
"value": "Let $ a$ and $ b$ be two positive integers such that $ a \\cdot b \\plus{} 1$ divides $ a^{2} \\plus{} b^{2}$ . Show that $ \\frac {a^{2} \\plus{} b^{2}}{a \\cdot b \\plus{} 1}$ is a perfect square."
},
{
"from": "gpt",
"value": "Let \\( d = a \\cdot b + 1 \... |
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