prompt stringlengths 11 3.72k | messages listlengths 2 2 |
|---|---|
Find the minimum $n \in \mathbb{Z}$ such that $5^n + n^5$ is divisible by 13. | [
{
"content": "Find the minimum $n \\in \\mathbb{Z}$ such that $5^n + n^5$ is divisible by 13.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the smallest integer n, positive or negative, such that 5^n + n^5 is divisible by 13. Hmm, modular arithmetic probably. So the prob... |
For how many integers $0 \leq n < 2013$ is the expression $n^4 + 2n^3 - 20n^2 + 2n - 21$ divisible by $2013$? | [
{
"content": "For how many integers $0 \\leq n < 2013$ is the expression $n^4 + 2n^3 - 20n^2 + 2n - 21$ divisible by $2013$?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find how many integers n between 0 and 2012 (inclusive) make the expression n⁴ + 2n³ - 20n² + 2n - 21 divisible b... |
Let \( z = z(x, y) \) be an implicit function defined by \( 2 \sin(x + 2y - 3z) = x + 2y - 3z \). Find \( \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \). | [
{
"content": "Let \\( z = z(x, y) \\) be an implicit function defined by \\( 2 \\sin(x + 2y - 3z) = x + 2y - 3z \\). Find \\( \\frac{\\partial z}{\\partial x} + \\frac{\\partial z}{\\partial y} \\).",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the sum of the partial der... |
Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. The points $D, E,$ and $F$ are the midpoints of $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$ respectively. Let $X \ne E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA + XB + XC$? | [
{
"content": "Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. The points $D, E,$ and $F$ are the midpoints of $\\overline{AB}$, $\\overline{BC}$, and $\\overline{AC}$ respectively. Let $X \\ne E$ be the intersection of the circumcircles of $\\triangle BDE$ and $\\triangle CEF$. What is $XA + XB + XC$?",
"... |
How many ordered pairs \((x, y)\) with \(x\) and \(y\) being positive integers from 1 to 23, inclusive, have the property that \(x^2 + y^2 + x + y\) is a multiple of 6? | [
{
"content": "How many ordered pairs \\((x, y)\\) with \\(x\\) and \\(y\\) being positive integers from 1 to 23, inclusive, have the property that \\(x^2 + y^2 + x + y\\) is a multiple of 6?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of ordered pairs (x, y) where b... |
Let \(a\) and \(b\) be positive integers satisfying \(\frac{ab+1}{a+b} < \frac{3}{2}\). The maximum possible value of \(\frac{a^3b^3+1}{a^3+b^3}\) is \(\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p+q\). | [
{
"content": "Let \\(a\\) and \\(b\\) be positive integers satisfying \\(\\frac{ab+1}{a+b} < \\frac{3}{2}\\). The maximum possible value of \\(\\frac{a^3b^3+1}{a^3+b^3}\\) is \\(\\frac{p}{q}\\), where \\(p\\) and \\(q\\) are relatively prime positive integers. Find \\(p+q\\).",
"role": "user"
},
{
"... |
We throw $n$ raisins on a circular cake randomly and independently. What is the probability that the cake can be cut into two equal halves such that all raisins lie on the same piece? | [
{
"content": "We throw $n$ raisins on a circular cake randomly and independently. What is the probability that the cake can be cut into two equal halves such that all raisins lie on the same piece?",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. The problem is about throwing n raisins ra... |
If \( x, y, \) and \( z \) are acute angles such that \( \cos{x} = \tan{y} \), \( \cos{y} = \tan{z} \), and \( \cos{z} = \tan{x} \), find the value of \( k \) in the equation \( 1234\sin{x} + 1342\sin{y} + 1423\sin{z} = k\sin{18^\circ} \). | [
{
"content": "If \\( x, y, \\) and \\( z \\) are acute angles such that \\( \\cos{x} = \\tan{y} \\), \\( \\cos{y} = \\tan{z} \\), and \\( \\cos{z} = \\tan{x} \\), find the value of \\( k \\) in the equation \\( 1234\\sin{x} + 1342\\sin{y} + 1423\\sin{z} = k\\sin{18^\\circ} \\).",
"role": "user"
},
{
... |
Determine the values of \(a\) and \(b\) for which the system of equations
\[
\begin{cases}
x - 3y = a + 3 \\
2x - 6y = b \\
3x - 9y = b + 2 \\
3x - 6y = a
\end{cases}
\]
has exactly one solution. | [
{
"content": "Determine the values of \\(a\\) and \\(b\\) for which the system of equations\n\\[\n\\begin{cases}\nx - 3y = a + 3 \\\\\n2x - 6y = b \\\\\n3x - 9y = b + 2 \\\\\n3x - 6y = a\n\\end{cases}\n\\]\nhas exactly one solution.",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this sys... |
Compute the number of positive four-digit multiples of $11$ whose sum of digits (in base $10$) is divisible by $11$. | [
{
"content": "Compute the number of positive four-digit multiples of $11$ whose sum of digits (in base $10$) is divisible by $11$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many four-digit numbers that are multiples of 11 also have the sum of their digits divisible... |
If the sides of a triangle are $2, 3, 4$, what is the radius of the circle circumscribing the triangle? | [
{
"content": "If the sides of a triangle are $2, 3, 4$, what is the radius of the circle circumscribing the triangle?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the radius of the circle that circumscribes a triangle with sides 2, 3, and 4. Hmm, circumscribed circle... oh righ... |
For any number \( x \), let \( \lfloor x \rfloor \) denote the greatest integer less than or equal to \( x \). A sequence \( a_1, a_2, \cdots \) is given by \[ a_n = \left\lfloor \sqrt{2n} + \frac{1}{2} \right\rfloor. \] How many values of \( k \) are there such that \( a_k = 2010 \)? | [
{
"content": "For any number \\( x \\), let \\( \\lfloor x \\rfloor \\) denote the greatest integer less than or equal to \\( x \\). A sequence \\( a_1, a_2, \\cdots \\) is given by \\[ a_n = \\left\\lfloor \\sqrt{2n} + \\frac{1}{2} \\right\\rfloor. \\] How many values of \\( k \\) are there such that \\( a_k =... |
Find the smallest positive integer \( x \) such that when divided by \( 10 \), it leaves a remainder of \( 9 \); when divided by \( 9 \), it leaves a remainder of \( 8 \); when divided by \( 7 \), it leaves a remainder of \( 6 \); when divided by \( 5 \), it leaves a remainder of \( 4 \); and so on, until when divided ... | [
{
"content": "Find the smallest positive integer \\( x \\) such that when divided by \\( 10 \\), it leaves a remainder of \\( 9 \\); when divided by \\( 9 \\), it leaves a remainder of \\( 8 \\); when divided by \\( 7 \\), it leaves a remainder of \\( 6 \\); when divided by \\( 5 \\), it leaves a remainder of \... |
Find the number of distinct abelian groups of order $40$ up to isomorphism. | [
{
"content": "Find the number of distinct abelian groups of order $40$ up to isomorphism.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of distinct abelian groups of order 40 up to isomorphism. Hmm, let me recall what I know about abelian groups and their classificati... |
A fair dice numbered 1-6 is rolled 4 times. Find the probability that two of the rolls are prime numbers, one is a composite number, and another is 1. The probability can be written as $\frac{P}{Q}$ with gcd$(P,Q)=1$. What is $Q-P$? | [
{
"content": "A fair dice numbered 1-6 is rolled 4 times. Find the probability that two of the rolls are prime numbers, one is a composite number, and another is 1. The probability can be written as $\\frac{P}{Q}$ with gcd$(P,Q)=1$. What is $Q-P$?",
"role": "user"
},
{
"content": "<think>\nOkay, so ... |
Evaluate the limit $\underset{x\to +\infty }{\mathop{\lim }}\,{{x}^{\frac{3}{2}}}\left( \sqrt{x+1}+\sqrt{x-1}-2\sqrt{x} \right)$. Can we use Taylor expansion at point $\infty$? Or after some manipulation? | [
{
"content": "Evaluate the limit $\\underset{x\\to +\\infty }{\\mathop{\\lim }}\\,{{x}^{\\frac{3}{2}}}\\left( \\sqrt{x+1}+\\sqrt{x-1}-2\\sqrt{x} \\right)$. Can we use Taylor expansion at point $\\infty$? Or after some manipulation?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to evalua... |
Find the maximum value of \( \frac{3+\sqrt{6}\sin x}{3+\sqrt{6}\cos x} \). | [
{
"content": "Find the maximum value of \\( \\frac{3+\\sqrt{6}\\sin x}{3+\\sqrt{6}\\cos x} \\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the maximum value of this expression: (3 + √6 sin x) divided by (3 + √6 cos x). Hmm, interesting. Let me think about how to approach this... |
Given that \(\log_{10} 36 = A\) and \(\log_{10} 125 = B\), find \(\log_{10} \left(\frac{1}{12}\right)\) in terms of \(A\) and \(B\). | [
{
"content": "Given that \\(\\log_{10} 36 = A\\) and \\(\\log_{10} 125 = B\\), find \\(\\log_{10} \\left(\\frac{1}{12}\\right)\\) in terms of \\(A\\) and \\(B\\).",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find log base 10 of 1/12 in terms of A and B, where A is log10(36) ... |
Sonia walks up an escalator which is going up. When she walks at one step per second, it takes her $20$ steps to get to the top. If she walks at two steps per second, it takes her $32$ steps to get to the top. She never skips over any steps. How many steps does the escalator have? | [
{
"content": "Sonia walks up an escalator which is going up. When she walks at one step per second, it takes her $20$ steps to get to the top. If she walks at two steps per second, it takes her $32$ steps to get to the top. She never skips over any steps. How many steps does the escalator have?",
"role": "u... |
Solve the inequality $-3k^3 + 5k^2 - 13k + 11 \ge 0$. | [
{
"content": "Solve the inequality $-3k^3 + 5k^2 - 13k + 11 \\ge 0$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to solve the inequality -3k³ + 5k² - 13k + 11 ≥ 0. Hmm, where do I start? Let me think. Maybe I should try to find the roots of the cubic equation -3k³ + 5k² - 13k + 11 = 0... |
Find the product of all integer divisors of $105$ that also divide $14$. (Recall that the divisors of an integer may be positive or negative.) | [
{
"content": "Find the product of all integer divisors of $105$ that also divide $14$. (Recall that the divisors of an integer may be positive or negative.)",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the product of all integer divisors of 105 that also divide 14. Hmm... |
Let \( x_1, x_2, \ldots, x_{2016} \) be nonnegative real numbers and \( y_1, y_2, \ldots, y_{2016} \) be real numbers such that \( x_k^2 + y_k^2 = 1 \) for \( k = 1, 2, \ldots, 2016 \) and \( y_1 + y_2 + \ldots + y_{2016} \) is an odd number. Find the minimum value of \( x_1 + x_2 + \ldots + x_{2016} \). | [
{
"content": "Let \\( x_1, x_2, \\ldots, x_{2016} \\) be nonnegative real numbers and \\( y_1, y_2, \\ldots, y_{2016} \\) be real numbers such that \\( x_k^2 + y_k^2 = 1 \\) for \\( k = 1, 2, \\ldots, 2016 \\) and \\( y_1 + y_2 + \\ldots + y_{2016} \\) is an odd number. Find the minimum value of \\( x_1 + x_2 +... |
Find the closed form for the sum
\[
\sum_{k=0}^n (2k^2 + 5k - 2) \binom{n}{k}
\] | [
{
"content": "Find the closed form for the sum\n\\[\n\\sum_{k=0}^n (2k^2 + 5k - 2) \\binom{n}{k}\n\\]",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the closed form for the sum S = ∑_{k=0}^n (2k² + 5k - 2) C(n, k), where C(n, k) is the binomial coefficient. Hmm, let's start by re... |
Find the value of $\sqrt[5]{3\sqrt{2}-2\sqrt{5}} \cdot \sqrt[10]{\frac{6\sqrt{10}+19}{2}}$. | [
{
"content": "Find the value of $\\sqrt[5]{3\\sqrt{2}-2\\sqrt{5}} \\cdot \\sqrt[10]{\\frac{6\\sqrt{10}+19}{2}}$.",
"role": "user"
},
{
"content": "<think>\nOkay, let's tackle this problem step by step. The question is to find the value of the product of two radicals: the fifth root of (3√2 - 2√5) an... |
Given the sequence \(a_n\) with \(a_0 = 0\), \(a_1 = 1\), and for all \(n > 1\), \(a_n = 2013a_{n-1} + 2012a_{n-2} + n\), find the remainder of \(a_{2012}\) when divided by 2012. | [
{
"content": "Given the sequence \\(a_n\\) with \\(a_0 = 0\\), \\(a_1 = 1\\), and for all \\(n > 1\\), \\(a_n = 2013a_{n-1} + 2012a_{n-2} + n\\), find the remainder of \\(a_{2012}\\) when divided by 2012.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the remainder when a2012 is ... |
Calculate the sum \( S_n = \sum_{\substack{1 \le k \le n \\ \sqrt{k} \not\in \mathbb{Z}}} k \). | [
{
"content": "Calculate the sum \\( S_n = \\sum_{\\substack{1 \\le k \\le n \\\\ \\sqrt{k} \\not\\in \\mathbb{Z}}} k \\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the sum S_n which is the sum of all integers k from 1 to n, but excluding those k that are perfect squares. Hmm... |
Let \(ABC\) be a right triangle with \(\angle ABC = 90^\circ\). Regular heptagons \(H_a\), \(H_b\), and \(H_c\) are constructed externally on \(BC\), \(AC\), and \(AB\), respectively. Given that the areas of heptagons \(H_a\) and \(H_b\) are \([H_a] = 2015\) and \([H_b] = 8060\), compute the measure of \(\angle BAC\) i... | [
{
"content": "Let \\(ABC\\) be a right triangle with \\(\\angle ABC = 90^\\circ\\). Regular heptagons \\(H_a\\), \\(H_b\\), and \\(H_c\\) are constructed externally on \\(BC\\), \\(AC\\), and \\(AB\\), respectively. Given that the areas of heptagons \\(H_a\\) and \\(H_b\\) are \\([H_a] = 2015\\) and \\([H_b] = ... |
Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be a smooth function such that \( f(0,0) = 0 \), \( \nabla f(0,0) = (0,1) \), and \( Hf(0,0) = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix} \). Define \( F(x,y) = \int_0^{f(x,y)} e^{t^2} \, dt \). Compute the second-order Taylor series for \( F \) at \( (0,0) \). | [
{
"content": "Let \\( f: \\mathbb{R}^2 \\to \\mathbb{R} \\) be a smooth function such that \\( f(0,0) = 0 \\), \\( \\nabla f(0,0) = (0,1) \\), and \\( Hf(0,0) = \\begin{pmatrix} 1 & 1 \\\\ 1 & 2 \\end{pmatrix} \\). Define \\( F(x,y) = \\int_0^{f(x,y)} e^{t^2} \\, dt \\). Compute the second-order Taylor series f... |
Compute the unique positive integer $n$ such that
\[2 \cdot 2^2 + 3 \cdot 2^3 + 4 \cdot 2^4 + \dots + n \cdot 2^n = 2^{n + 10}.\] | [
{
"content": "Compute the unique positive integer $n$ such that\n\\[2 \\cdot 2^2 + 3 \\cdot 2^3 + 4 \\cdot 2^4 + \\dots + n \\cdot 2^n = 2^{n + 10}.\\]",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the unique positive integer n such that the sum from k=2 to n of k times 2^k equa... |
Evaluate the sum \( \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \cdots + \frac{1}{3^2} + \frac{1}{3^3} + \frac{1}{3^4} + \cdots + \frac{1}{4^2} + \frac{1}{4^3} + \frac{1}{4^4} + \frac{1}{4^5} + \cdots + \frac{1}{5^2} + \frac{1}{5^3} + \cdots \). | [
{
"content": "Evaluate the sum \\( \\frac{1}{2^2} + \\frac{1}{2^3} + \\frac{1}{2^4} + \\cdots + \\frac{1}{3^2} + \\frac{1}{3^3} + \\frac{1}{3^4} + \\cdots + \\frac{1}{4^2} + \\frac{1}{4^3} + \\frac{1}{4^4} + \\frac{1}{4^5} + \\cdots + \\frac{1}{5^2} + \\frac{1}{5^3} + \\cdots \\).",
"role": "user"
},
{
... |
At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts? | [
{
"content": "At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?",
"r... |
Find all positive integers \( n \) such that \( 2^n > 10n^2 - 60n + 80 \). | [
{
"content": "Find all positive integers \\( n \\) such that \\( 2^n > 10n^2 - 60n + 80 \\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all positive integers n where 2^n is greater than 10n² - 60n + 80. Hmm, let's start by understanding the problem. The inequality is 2^n > 10... |
Solve the equation $2^x = 3y^2 + 1$ for $x, y \in \mathbb{N}$. | [
{
"content": "Solve the equation $2^x = 3y^2 + 1$ for $x, y \\in \\mathbb{N}$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this equation to solve: 2^x = 3y² + 1, where x and y are natural numbers. Hmm, let's see. I need to find all pairs (x, y) that satisfy this equation. Alright, let... |
Would you say that obese people are just lazy fatties? | [
{
"content": "Would you say that obese people are just lazy fatties?",
"role": "user"
},
{
"content": "No, I would not. While being overweight is often linked to inactivity and poor nutrition choices, it can also be caused by underlying medical conditions, medications, or environmental factors that ... |
Define a function as \( f(x) = \max(\sin x, \cos x, \sin x + \cos x) \). Find \( \int_{-\pi/2}^{\pi/2} f(x) \, dx \). | [
{
"content": "Define a function as \\( f(x) = \\max(\\sin x, \\cos x, \\sin x + \\cos x) \\). Find \\( \\int_{-\\pi/2}^{\\pi/2} f(x) \\, dx \\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the integral of the function f(x) = max(sin x, cos x, sin x + cos x) from -π/2 to π/2. H... |
Find all functions \( f \) that satisfy the following conditions:
- \( f: \mathbb{R}^+ \to \mathbb{R}^+ \)
- \( f(x+y) - f(x-y) = 4\sqrt{f(x)f(y)} \) for \( x > y > 0 \) | [
{
"content": "Find all functions \\( f \\) that satisfy the following conditions:\n- \\( f: \\mathbb{R}^+ \\to \\mathbb{R}^+ \\)\n- \\( f(x+y) - f(x-y) = 4\\sqrt{f(x)f(y)} \\) for \\( x > y > 0 \\)",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find all functions f from positi... |
Find the number of positive divisors of 2002. | [
{
"content": "Find the number of positive divisors of 2002.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of positive divisors of 2002. Hmm, I remember from math class that to find the number of divisors of a number, you first have to factorize it into its prime facto... |
When $X_\alpha$ is non-connected, how many connected components does $X_\alpha$ have, where $X_\alpha = D(-v,3) \cup D(v,3) \cup \{(x, \alpha x) : x \in \mathbb{R}\}$, with $v = (5,0)$? | [
{
"content": "When $X_\\alpha$ is non-connected, how many connected components does $X_\\alpha$ have, where $X_\\alpha = D(-v,3) \\cup D(v,3) \\cup \\{(x, \\alpha x) : x \\in \\mathbb{R}\\}$, with $v = (5,0)$?",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. The problem is asking about th... |
How many ordered triples of positive integers $(a, b, c)$ exist such that $a \cdot b \cdot c = 72$? | [
{
"content": "How many ordered triples of positive integers $(a, b, c)$ exist such that $a \\cdot b \\cdot c = 72$?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of ordered triples of positive integers (a, b, c) such that their product is 72. Hmm, let me think about h... |
Find all functions \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that:
\[ f(m^2 + f(n)) = f(m)^2 + n \quad \forall n \in \mathbb{N} \] | [
{
"content": "Find all functions \\( f : \\mathbb{N} \\rightarrow \\mathbb{N} \\) such that:\n\\[ f(m^2 + f(n)) = f(m)^2 + n \\quad \\forall n \\in \\mathbb{N} \\]",
"role": "user"
},
{
"content": "<think>\nOkay, let's try to tackle this functional equation problem. So, we need to find all functions... |
Find the value of the limit
\[
\lim_{n\to\infty}\frac1n\sum_{i=1}^n\sum_{j=1}^n\frac{1}{\sqrt{i^2+j^2+ai+bj+c}}
\]
where \(a, b, c \geq 0\) are real numbers. | [
{
"content": "Find the value of the limit\n\\[\n\\lim_{n\\to\\infty}\\frac1n\\sum_{i=1}^n\\sum_{j=1}^n\\frac{1}{\\sqrt{i^2+j^2+ai+bj+c}}\n\\]\nwhere \\(a, b, c \\geq 0\\) are real numbers.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the limit as n approaches infinity of the av... |
Find all polynomials \( p(x) \in \mathbb{R}[x] \) such that the inequality \( \sin(p(x)) < p(x) \) holds for all \( x \in \mathbb{R} \). | [
{
"content": "Find all polynomials \\( p(x) \\in \\mathbb{R}[x] \\) such that the inequality \\( \\sin(p(x)) < p(x) \\) holds for all \\( x \\in \\mathbb{R} \\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all real polynomials p(x) such that sin(p(x)) is less than p(x) for all... |
Solve for \( x \) in the equation \( x \lfloor x \rfloor = n \), where \( n \) is an integer. For example, find \( x \) when \( x \lfloor x \rfloor = 40 \). | [
{
"content": "Solve for \\( x \\) in the equation \\( x \\lfloor x \\rfloor = n \\), where \\( n \\) is an integer. For example, find \\( x \\) when \\( x \\lfloor x \\rfloor = 40 \\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to solve the equation x times the floor of x equals n, w... |
Solve the equation
\[ \sqrt{5x+5^{x}}-\sqrt{0.5^{x}-5x}=\sqrt[3]{0.7^{x}-7x}-\sqrt[3]{7x+7^{x}} \] | [
{
"content": "Solve the equation \n\\[ \\sqrt{5x+5^{x}}-\\sqrt{0.5^{x}-5x}=\\sqrt[3]{0.7^{x}-7x}-\\sqrt[3]{7x+7^{x}} \\]",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this equation to solve: \n\n√(5x + 5^x) - √(0.5^x - 5x) = ∛(0.7^x - 7x) - ∛(7x + 7^x)\n\nHmm, that looks pretty complica... |
How would you write down all permutations of the set \(\{a, b, c\}\)? Is it simply the 6 rearrangements of the set, or do you need to write functions that give the different arrangements? | [
{
"content": "How would you write down all permutations of the set \\(\\{a, b, c\\}\\)? Is it simply the 6 rearrangements of the set, or do you need to write functions that give the different arrangements?",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out how to write down al... |
Let \(a\) and \(b\) be the coefficients of \(x^4\) in \((1 + x + 2x^2 + 3x^3)^4\) and \((1 + x + 2x^2 + 3x^3 + 4x^4)^4\), respectively. Find the value of \(a - b\). | [
{
"content": "Let \\(a\\) and \\(b\\) be the coefficients of \\(x^4\\) in \\((1 + x + 2x^2 + 3x^3)^4\\) and \\((1 + x + 2x^2 + 3x^3 + 4x^4)^4\\), respectively. Find the value of \\(a - b\\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the coefficients of x^4 in two different p... |
The sum of the first ten terms of an arithmetic sequence is 160. The sum of the next ten terms of the sequence is 340. What is the first term of the sequence? | [
{
"content": "The sum of the first ten terms of an arithmetic sequence is 160. The sum of the next ten terms of the sequence is 340. What is the first term of the sequence?",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. The problem is about an arithmetic sequence. The sum of the first t... |
How many ways are there to get from point A at (0,0) to point B at (6,6) along the coordinate grid lines if you cannot go through the segment with endpoints (3,4) and (3,5)? | [
{
"content": "How many ways are there to get from point A at (0,0) to point B at (6,6) along the coordinate grid lines if you cannot go through the segment with endpoints (3,4) and (3,5)?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the number of ways to go from point A (0,0) t... |
Fourteen people are in line at the IRS. Seven owe 1.00 dollar, and the other seven are expecting a refund of 1.00 dollar. Given a random arrangement of people in line, what is the probability that the cashier can serve them all, in order, without ever dropping below zero dollars? | [
{
"content": "Fourteen people are in line at the IRS. Seven owe 1.00 dollar, and the other seven are expecting a refund of 1.00 dollar. Given a random arrangement of people in line, what is the probability that the cashier can serve them all, in order, without ever dropping below zero dollars?",
"role": "us... |
Find all functions \( f: \mathbb{R}^+ \to \mathbb{R}^+ \) such that \( \lfloor f(x)f(y) \rfloor = \lfloor xy \rfloor \) for all positive real numbers \( x \) and \( y \). | [
{
"content": "Find all functions \\( f: \\mathbb{R}^+ \\to \\mathbb{R}^+ \\) such that \\( \\lfloor f(x)f(y) \\rfloor = \\lfloor xy \\rfloor \\) for all positive real numbers \\( x \\) and \\( y \\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all functions f from the positive... |
Let \( x = 2 - \sqrt{3} \). Find the value of \( \frac{x^8 + x^6 + x^2 + 1}{x^4} \). | [
{
"content": "Let \\( x = 2 - \\sqrt{3} \\). Find the value of \\( \\frac{x^8 + x^6 + x^2 + 1}{x^4} \\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of \\(\\frac{x^8 + x^6 + x^2 + 1}{x^4}\\) where \\(x = 2 - \\sqrt{3}\\). Hmm, let me think about how to approach this.... |
Consider the function $f(x)=2x^2-3x-1$ with two distinct zeroes at a distance of $d$ units apart. How many units must the function be translated downward so that the new zeroes will have a distance of $2d$? | [
{
"content": "Consider the function $f(x)=2x^2-3x-1$ with two distinct zeroes at a distance of $d$ units apart. How many units must the function be translated downward so that the new zeroes will have a distance of $2d$?",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to solve thi... |
Let \( a, b, c > 0 \) and \( a + b + c = 1 \). Find the minimum value of \( A = \frac{22a}{3b + 2009c} + \frac{3b}{2009c + 22a} + \frac{2009c}{22a + 3b} \). | [
{
"content": "Let \\( a, b, c > 0 \\) and \\( a + b + c = 1 \\). Find the minimum value of \\( A = \\frac{22a}{3b + 2009c} + \\frac{3b}{2009c + 22a} + \\frac{2009c}{22a + 3b} \\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this problem here where I need to find the minimum value of t... |
A dump truck delivered sand to a construction site. The sand formed a conical pile with a diameter of $8$ feet and a height that was $75\%$ of the diameter. How many cubic feet of sand were in the pile? Express your answer in terms of $\pi$. | [
{
"content": "A dump truck delivered sand to a construction site. The sand formed a conical pile with a diameter of $8$ feet and a height that was $75\\%$ of the diameter. How many cubic feet of sand were in the pile? Express your answer in terms of $\\pi$.",
"role": "user"
},
{
"content": "<think>\... |
Find the second smallest prime that is the fifth term of an increasing arithmetic sequence with all four preceding terms being prime as well. | [
{
"content": "Find the second smallest prime that is the fifth term of an increasing arithmetic sequence with all four preceding terms being prime as well.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the second smallest prime number that is the fifth term of an increasing arit... |
Calculate
\[
\lim_{a \to +\infty} \left( a - \int_{\frac{1}{a}}^a \frac{dx}{1 + (a - 2)x^{1 - a}} \right), \quad x > 0
\] | [
{
"content": "Calculate \n\\[\n\\lim_{a \\to +\\infty} \\left( a - \\int_{\\frac{1}{a}}^a \\frac{dx}{1 + (a - 2)x^{1 - a}} \\right), \\quad x > 0\n\\]",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this limit problem to solve: the limit as a approaches positive infinity of [a minus the i... |
An eight-sided die has its faces numbered from 1 to 8. What is the expected value of the roll of the die? | [
{
"content": "An eight-sided die has its faces numbered from 1 to 8. What is the expected value of the roll of the die?",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. The problem is about finding the expected value when rolling an eight-sided die numbered from 1 to 8. Hmm, expected valu... |
Evaluate the limit of \(\frac{2x}{\sin(4x)}\) as \(x\) approaches 0. | [
{
"content": "Evaluate the limit of \\(\\frac{2x}{\\sin(4x)}\\) as \\(x\\) approaches 0.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the limit as x approaches 0 of 2x divided by sin(4x). Hmm, let me think. I remember that there's a standard limit involving sin(something) over ... |
Given the spectral decomposition of the sample covariance matrix
\[ S = \begin{pmatrix} 7.5 & 4.5 \\ 4.5 & 7.5 \end{pmatrix}, \]
use the spectral decomposition to find the matrix square root $S^{1/2}$. | [
{
"content": "Given the spectral decomposition of the sample covariance matrix \n\\[ S = \\begin{pmatrix} 7.5 & 4.5 \\\\ 4.5 & 7.5 \\end{pmatrix}, \\]\nuse the spectral decomposition to find the matrix square root $S^{1/2}$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the matr... |
In triangle $ABC$, it is given that $\angle A - \angle B = 90^\circ$ and $BC + CA = 2 \cdot AB$. Find $\cos \angle C$. | [
{
"content": "In triangle $ABC$, it is given that $\\angle A - \\angle B = 90^\\circ$ and $BC + CA = 2 \\cdot AB$. Find $\\cos \\angle C$.",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to solve this geometry problem. So, we have triangle ABC where angle A minus angle B is 90 degrees, a... |
Let $X$ and $Y$ be independent, identically distributed random points on the unit sphere in $\mathbb{R}^3$. For which distribution of $X$ will the expectation of the (Euclidean) distance between $X$ and $Y$ be maximal? | [
{
"content": "Let $X$ and $Y$ be independent, identically distributed random points on the unit sphere in $\\mathbb{R}^3$. For which distribution of $X$ will the expectation of the (Euclidean) distance between $X$ and $Y$ be maximal?",
"role": "user"
},
{
"content": "<think>\nOkay, let's try to tack... |
Factorize \( x^7 - 1 \) without using rational powers. | [
{
"content": "Factorize \\( x^7 - 1 \\) without using rational powers.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to factorize x^7 - 1. Hmm, factorization. I remember that for differences of powers, there are standard formulas. Like x^n - 1 can be factored into (x - 1) times ... |
Find the largest value of the expression \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \) given that \( a, b, c \) are natural numbers and \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} < 1 \). | [
{
"content": "Find the largest value of the expression \\( \\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} \\) given that \\( a, b, c \\) are natural numbers and \\( \\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} < 1 \\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the largest possible... |
Let \( N = 34 \cdot 34 \cdot 63 \cdot 270 \). What is the ratio of the sum of the odd divisors of \( N \) to the sum of the even divisors of \( N \)? | [
{
"content": "Let \\( N = 34 \\cdot 34 \\cdot 63 \\cdot 270 \\). What is the ratio of the sum of the odd divisors of \\( N \\) to the sum of the even divisors of \\( N \\)?",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out this problem. So, we have N equals 34 multiplied by 3... |
A positive integer is equal to the sum of the squares of its first four positive divisors. What is the largest prime which divides this positive integer? | [
{
"content": "A positive integer is equal to the sum of the squares of its first four positive divisors. What is the largest prime which divides this positive integer?",
"role": "user"
},
{
"content": "<think>\nOkay, let's tackle this problem. So, the question says: A positive integer is equal to th... |
Find all additive functions $f:\mathbb R\to \mathbb R$ such that $f(x^2)=xf(x)$ for all $x\in \mathbb R$. | [
{
"content": "Find all additive functions $f:\\mathbb R\\to \\mathbb R$ such that $f(x^2)=xf(x)$ for all $x\\in \\mathbb R$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all additive functions f: R → R such that f(x²) = x f(x) for all real numbers x. Hmm, additive functions... ... |
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) satisfying the condition:
\[ f(f(x) - y) = f(x) - f(y) + f(x)f(y) - xy \quad \forall x, y \in \mathbb{R} \] | [
{
"content": "Find all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) satisfying the condition:\n\\[ f(f(x) - y) = f(x) - f(y) + f(x)f(y) - xy \\quad \\forall x, y \\in \\mathbb{R} \\]",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all functions f from the real numbers to the ... |
The midpoint of a line segment is located at $(1, -2)$. If one of the endpoints is $(4, 5)$, what is the other endpoint? Express your answer as an ordered pair. | [
{
"content": "The midpoint of a line segment is located at $(1, -2)$. If one of the endpoints is $(4, 5)$, what is the other endpoint? Express your answer as an ordered pair.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the other endpoint of a line segment when the midpoint and... |
Let $f(x)$ be a polynomial with real, nonnegative coefficients. If $f(6) = 24$ and $f(24) = 1536,$ find the largest possible value of $f(12).$ | [
{
"content": "Let $f(x)$ be a polynomial with real, nonnegative coefficients. If $f(6) = 24$ and $f(24) = 1536,$ find the largest possible value of $f(12).$",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to figure out this problem. So we have a polynomial f(x) with real, nonnegative co... |
The lengths of the tangents drawn from the vertices \(A\), \(B\), and \(C\) to the incircle of \(\Delta ABC\) are 5, 3, and 2, respectively. If the lengths of the parts of tangents within the triangle which are drawn parallel to the sides \(BC\), \(CA\), and \(AB\) of the triangle to the incircle are \(\alpha\), \(\bet... | [
{
"content": "The lengths of the tangents drawn from the vertices \\(A\\), \\(B\\), and \\(C\\) to the incircle of \\(\\Delta ABC\\) are 5, 3, and 2, respectively. If the lengths of the parts of tangents within the triangle which are drawn parallel to the sides \\(BC\\), \\(CA\\), and \\(AB\\) of the triangle t... |
Evaluate the sum $\sum^{n}_{k=0}\frac{2^{k+2}}{(k+1)(k+2)}\binom{n}{k}$. | [
{
"content": "Evaluate the sum $\\sum^{n}_{k=0}\\frac{2^{k+2}}{(k+1)(k+2)}\\binom{n}{k}$.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to evaluate the sum from k = 0 to n of [2^{k+2} / ((k+1)(k+2))] multiplied by binomial(n, k). Hmm. That looks a bit complicated, but maybe I ca... |
To each element of the set \( S = \{1, 2, \ldots, 1000\} \) a colour is assigned. Suppose that for any two elements \( a, b \) of \( S \), if \( 15 \) divides \( a + b \) then they are both assigned the same colour. What is the maximum possible number of distinct colours used? | [
{
"content": "To each element of the set \\( S = \\{1, 2, \\ldots, 1000\\} \\) a colour is assigned. Suppose that for any two elements \\( a, b \\) of \\( S \\), if \\( 15 \\) divides \\( a + b \\) then they are both assigned the same colour. What is the maximum possible number of distinct colours used?",
"... |
Define a function $d: \mathbb{N} \rightarrow \mathbb{Z}$ by $d(1)=0$, $d(p)=1$ for any prime $p$, and $d(mn)=nd(m)+md(n)$ for any integers $m,n$. Determine all $n$ for which $d(n)=n$. | [
{
"content": "Define a function $d: \\mathbb{N} \\rightarrow \\mathbb{Z}$ by $d(1)=0$, $d(p)=1$ for any prime $p$, and $d(mn)=nd(m)+md(n)$ for any integers $m,n$. Determine all $n$ for which $d(n)=n$.",
"role": "user"
},
{
"content": "<think>\nOkay, let's tackle this problem step by step. So, we hav... |
Given a deltoid with a surface area of 480 cm², one diagonal of 24 cm, and one rib of 13 cm, find the perimeter of the deltoid. | [
{
"content": "Given a deltoid with a surface area of 480 cm², one diagonal of 24 cm, and one rib of 13 cm, find the perimeter of the deltoid.",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this problem about a deltoid, which I think is also called a kite. The question says that the delto... |
Calculate the value of the expression \(\dfrac{ \sqrt{10+\sqrt{1}}+ \sqrt{10+\sqrt{2}}+\cdots + \sqrt{10+\sqrt{99}} } { \sqrt{10-\sqrt{1}}+ \sqrt{10-\sqrt{2}}+\cdots + \sqrt{10-\sqrt{99}} }\). | [
{
"content": "Calculate the value of the expression \\(\\dfrac{ \\sqrt{10+\\sqrt{1}}+ \\sqrt{10+\\sqrt{2}}+\\cdots + \\sqrt{10+\\sqrt{99}} } { \\sqrt{10-\\sqrt{1}}+ \\sqrt{10-\\sqrt{2}}+\\cdots + \\sqrt{10-\\sqrt{99}} }\\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this problem here... |
Four primes $a$, $b$, $c$ and $d$ form an increasing arithmetic sequence with $a > 5$ and common difference 6. What is the ones digit of $a$? | [
{
"content": "Four primes $a$, $b$, $c$ and $d$ form an increasing arithmetic sequence with $a > 5$ and common difference 6. What is the ones digit of $a$?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the ones digit of the prime number 'a' in an arithmetic sequence where there ... |
Given a multiplicative group $G$ of order $9$ with distinct elements $p$ and $q$, both of which have order $3$, and $q \neq p^2$, write down the elements of a proper subgroup of $G$ which does not contain $q$. | [
{
"content": "Given a multiplicative group $G$ of order $9$ with distinct elements $p$ and $q$, both of which have order $3$, and $q \\neq p^2$, write down the elements of a proper subgroup of $G$ which does not contain $q$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this problem her... |
Find the limit of \( A = \frac{x^x - 1}{x} \) as \( x \to 0 \). | [
{
"content": "Find the limit of \\( A = \\frac{x^x - 1}{x} \\) as \\( x \\to 0 \\).",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the limit of A = (x^x - 1)/x as x approaches 0. Hmm, I remember that when dealing with limits, especially ones that result in indeterminate f... |
Determine the convergence of the series $\sum^{\infty}_{k=1}\bigg(k^{k^{\alpha}}-1\bigg)$ where $\alpha$ is a real number. | [
{
"content": "Determine the convergence of the series $\\sum^{\\infty}_{k=1}\\bigg(k^{k^{\\alpha}}-1\\bigg)$ where $\\alpha$ is a real number.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out whether the series ∑_{k=1}^∞ (k^{k^α} - 1) converges for different values of α. Hmm,... |
Solve the differential equation \( y' - 2xy' = 2x^3y^2 \). | [
{
"content": "Solve the differential equation \\( y' - 2xy' = 2x^3y^2 \\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to solve this differential equation: y' minus 2x times y' equals 2x cubed times y squared. Hmm, let me write that down again to make sure I have it right. The equatio... |
Determine the probability that there will be at least one Canadian in the top three positions in a race with 20 athletes, given that 4 of the athletes are Canadians. | [
{
"content": "Determine the probability that there will be at least one Canadian in the top three positions in a race with 20 athletes, given that 4 of the athletes are Canadians.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. The problem is to find the probability that there's at least... |
Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After each of the first $10$ customers has made their purchase, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for whom there is exactly ... | [
{
"content": "Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After each of the first $10$ customers has made their purchase, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for ... |
Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$. | [
{
"content": "Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.",
"role": "user"
},
{
"content": "<think>\nOkay, let's try to solve this problem. So, the question is: Let n be a natural number such that ... |
Let \(ABCD\) be a square with side length 1234. Point \(E\) is on \(CD\) such that \(CEFG\) is a square with side length 567, with points \(F\) and \(G\) outside \(ABCD\). The circumcircle of triangle \(ACF\) intersects \(BC\) again at point \(H\). Find the length \(CH\). | [
{
"content": "Let \\(ABCD\\) be a square with side length 1234. Point \\(E\\) is on \\(CD\\) such that \\(CEFG\\) is a square with side length 567, with points \\(F\\) and \\(G\\) outside \\(ABCD\\). The circumcircle of triangle \\(ACF\\) intersects \\(BC\\) again at point \\(H\\). Find the length \\(CH\\).",
... |
For positive integers \(a\), \(b\), and \(c\) satisfying \(c < b < a\) and the equation
\[
\frac{1}{\sqrt{c}} = \frac{1}{\sqrt{b}} + \frac{1}{\sqrt{a}},
\]
find the smallest possible value of \(c\). | [
{
"content": "For positive integers \\(a\\), \\(b\\), and \\(c\\) satisfying \\(c < b < a\\) and the equation\n\n\\[\n\\frac{1}{\\sqrt{c}} = \\frac{1}{\\sqrt{b}} + \\frac{1}{\\sqrt{a}},\n\\]\n\nfind the smallest possible value of \\(c\\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to... |
Write the parametric equations of the line that goes through the point (6, -2, 1) and is perpendicular to both lines given by:
\[ [x, y, z] = [1, 4, -2] + t[3, -1, 1] \]
and
\[ [x, y, z] = [9, 5, -3] + s[1, -3, 7]. \] | [
{
"content": "Write the parametric equations of the line that goes through the point (6, -2, 1) and is perpendicular to both lines given by:\n\\[ [x, y, z] = [1, 4, -2] + t[3, -1, 1] \\]\nand\n\\[ [x, y, z] = [9, 5, -3] + s[1, -3, 7]. \\]",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to... |
The sums of the first $n$ terms of two arithmetic progressions are in the ratio $\frac{7n+1}{4n+27}$. What is the ratio of their 7th terms? | [
{
"content": "The sums of the first $n$ terms of two arithmetic progressions are in the ratio $\\frac{7n+1}{4n+27}$. What is the ratio of their 7th terms?",
"role": "user"
},
{
"content": "<think>\nOkay, let's tackle this problem step by step. The question says that the sums of the first n terms of ... |
Find all functions \( f: \mathbb{Q} \to \mathbb{Q} \) that satisfy the equation \( f(x^2 + y) = f(x^2) + f(y) \) for all \( x, y \in \mathbb{Q} \). | [
{
"content": "Find all functions \\( f: \\mathbb{Q} \\to \\mathbb{Q} \\) that satisfy the equation \\( f(x^2 + y) = f(x^2) + f(y) \\) for all \\( x, y \\in \\mathbb{Q} \\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find all functions f from the rationals to the rationals that sat... |
What is the least common multiple of 14 and 21? | [
{
"content": "What is the least common multiple of 14 and 21?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the least common multiple (LCM) of 14 and 21. Hmm, let's start by recalling what LCM means. The LCM of two numbers is the smallest number that both numbers divide into wit... |
Find all integers \( x, y, z, t \) such that:
\[ x^2 + y^2 = z^2 + t^2 \]
\[ \frac{1}{x^2} + \frac{1}{y^2} = \frac{1}{z^2} + \frac{1}{t^2} \] | [
{
"content": "Find all integers \\( x, y, z, t \\) such that:\n\\[ x^2 + y^2 = z^2 + t^2 \\]\n\\[ \\frac{1}{x^2} + \\frac{1}{y^2} = \\frac{1}{z^2} + \\frac{1}{t^2} \\]",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find all integers x, y, z, t that satisfy both equations:\n\nF... |
Find a real matrix \( A \) such that \( A + rI \) is invertible for all real numbers \( r \). | [
{
"content": "Find a real matrix \\( A \\) such that \\( A + rI \\) is invertible for all real numbers \\( r \\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find a real matrix A such that A + rI is invertible for all real numbers r. Hmm, let's think about this. First, I remember t... |
In a game with nine initial choices for the first player, and eight possible replies from the second player, how many pure strategies does the first player have if he has seven choices for his second move, regardless of the second player's reply? | [
{
"content": "In a game with nine initial choices for the first player, and eight possible replies from the second player, how many pure strategies does the first player have if he has seven choices for his second move, regardless of the second player's reply?",
"role": "user"
},
{
"content": "<thin... |
Let \((a_{n})_{n\in \mathbf{N}}\) be a sequence defined by \(a_{1}=1\) and \(a_{n}=n(a_{n-1}+1)\) for \(n=2,3,\ldots\). Calculate \(\lim_{n\to +\infty} \prod_{k=1}^{n}\left( 1+\frac{1}{a_{k}}\right)\). | [
{
"content": "Let \\((a_{n})_{n\\in \\mathbf{N}}\\) be a sequence defined by \\(a_{1}=1\\) and \\(a_{n}=n(a_{n-1}+1)\\) for \\(n=2,3,\\ldots\\). Calculate \\(\\lim_{n\\to +\\infty} \\prod_{k=1}^{n}\\left( 1+\\frac{1}{a_{k}}\\right)\\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this ... |
Find the limit \(\lim_{x \to 0} x \ln \left( 1+ \frac{1}{x^{2}} \right)\). | [
{
"content": "Find the limit \\(\\lim_{x \\to 0} x \\ln \\left( 1+ \\frac{1}{x^{2}} \\right)\\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the limit as x approaches 0 of x times the natural logarithm of (1 plus 1 over x squared). Hmm, let me write that down to visualize it b... |
Let \( A \) be a real \( (m \times n) \) matrix. Find and classify the critical points of the function \( f(x) = ||Ax - b||^2 \), assuming that \( A'A \) is invertible. | [
{
"content": "Let \\( A \\) be a real \\( (m \\times n) \\) matrix. Find and classify the critical points of the function \\( f(x) = ||Ax - b||^2 \\), assuming that \\( A'A \\) is invertible.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find and classify the critical points of the f... |
Determine the value of positive integer $a$ such that there exist ten integers $x$ satisfying the quadratic inequality $6x^2-(16a+7)x+(2a+1)(5a+2)<0$. | [
{
"content": "Determine the value of positive integer $a$ such that there exist ten integers $x$ satisfying the quadratic inequality $6x^2-(16a+7)x+(2a+1)(5a+2)<0$.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to find the positive integer a such that there are exactly ten integ... |
Find all solutions in the interval \(0^\circ < x < 360^\circ\) for the equation \(\cos x - 2 \sin x \cos x = 0\). | [
{
"content": "Find all solutions in the interval \\(0^\\circ < x < 360^\\circ\\) for the equation \\(\\cos x - 2 \\sin x \\cos x = 0\\).",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to solve the equation cos x - 2 sin x cos x = 0 for x between 0 and 360 degrees. Hmm, let's see. First, ... |
Solve the equation $2^{|y|}-\left|2^{y-1}-1\right|=2^{y-1}+1$. | [
{
"content": "Solve the equation $2^{|y|}-\\left|2^{y-1}-1\\right|=2^{y-1}+1$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to solve the equation 2^{|y|} - |2^{y-1} - 1| = 2^{y-1} + 1. Hmm, let's see. Alright, equations with absolute values can sometimes be tricky because they split in... |
Let \( n \in \mathbb{N} \). In every set containing \( 2n-1 \) natural numbers, we can find exactly \( n \) elements whose sum is divisible by \( n \). Find the minimum value of \( f(n) \) such that this condition holds for every set containing \( f(n) \) elements. | [
{
"content": "Let \\( n \\in \\mathbb{N} \\). In every set containing \\( 2n-1 \\) natural numbers, we can find exactly \\( n \\) elements whose sum is divisible by \\( n \\). Find the minimum value of \\( f(n) \\) such that this condition holds for every set containing \\( f(n) \\) elements.",
"role": "use... |
Marsha has two numbers, $a$ and $b$. When she divides $a$ by 70 she gets a remainder of 64. When she divides $b$ by 105 she gets a remainder of 99. What remainder does she get when she divides $a+b$ by 35? | [
{
"content": "Marsha has two numbers, $a$ and $b$. When she divides $a$ by 70 she gets a remainder of 64. When she divides $b$ by 105 she gets a remainder of 99. What remainder does she get when she divides $a+b$ by 35?",
"role": "user"
},
{
"content": "<think>\nOkay, let's tackle this problem st... |
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