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What is the largest integer $n$ such that $n^3$ divides $10!$? | nvidia |
Find the value of $x$ such that $\log_{\sqrt{x}}(3x)=4$. | nvidia |
Determine the value of $k$ such that the line $y=3x-2$ is tangent to the circle $(x-k)^2+y^2=3$. | nvidia |
Find the equation of the hyperbola that passes through the points $(\pm\sqrt{2}, 0)$ and has foci at $(\pm2, 0)$. | nvidia |
If $x$ and $y$ are real numbers such that $x^2 + y^2 = 16$, what is the maximum possible value of $x + y$? | nvidia |
Let $f(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. Given that $n$ is a multiple of 15, what is the smallest value of $n$ such that $f(n) > 15$? | nvidia |
Find the number of positive integers less than 100 that are not divisible by 2, 3, or 5. | nvidia |
Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 2xy.$ Simplify
\[\frac{x^2}{y^2} + \frac{y^2}{x^2} - \frac{2}{xy}.\] | nvidia |
Evaluate the product \[ (n-2) \cdot (n-1) \cdot n \cdot (n+1) \cdot (n+2), \] where $n=3$. | nvidia |
A mathematician picks a random real number $x$ in the interval $(0,1)$.
What is the probability that the quadratic equation
$xz^2+\sqrt x z+1=0$
has two distinct real roots $z$? | nvidia |
Let $*$ be a binary operation on positive integers such that for every $n,m$, $n*m$ is also a positive integer. The operation $*$ satisfies the following properties:
(i) For every $n$, $1*n=n$.
(ii) For every $m,n$, $m*(n+1)=m*n+m$.
(iii) For every $m,n$, $(m+1)*n=m*n+n$.
Calculate $6*3$. | nvidia |
Given a positive integer $n$, let $S_n$ be the set of all permutations of the set $\{1,2,\dots,n\}$. Define the function $f_n : S_n \to \mathbb{R}$ by
\[f_n(\pi) = \sum_{k=1}^n \frac{\pi(k) + k}{2}.\]
Determine the maximum value of $f_n(\pi)$ over all permutations $\pi \in S_n$, and find all permutations that attain th... | nvidia |
The average of two 2-digit positive integers is equal to the decimal number obtained by writing one of the two-digit integers before the decimal point and the other two-digit integer after the decimal point. What is the smaller of the two integers? | nvidia |
Find all the solutions to the equation
$$\frac{x+y}{1+xy}+\frac{2xy}{x+y}=\frac{5}{2},$$
where $x$ and $y$ are positive integers. | nvidia |
A polynomial $p(x)$ of degree 4 satisfies that the sum of the absolute values of the coefficients of $p(x^n)$ is not equal to $2^4$ for all positive integers $n$. Find the largest possible number of distinct roots that $p(x)$ can have. | nvidia |
A sequence of positive integers $a_1, a_2, \ldots$ has the property that for every $k$, the number of divisors of $a_k$ is equal to $a_{k+1}$. If $a_1 = 2$, find $a_3$. | nvidia |
Let $n$ be a positive integer, and let $b_0, b_1, \dots, b_n$ be a sequence of real numbers such that $b_0 = 1$, $b_1 = 2$, $b_n=0$, and $$ b_{k+1} = b_{k-1} + \frac{2}{k \cdot b_k} $$for $k = 1, 2, \dots, n-1$. Find the sum of all the terms in the sequence. | nvidia |
Four integers are written on four cards, one number on each card. When the cards are placed into two rows of two cards each, it turns out that the products of the two numbers in each row are equal. However, if the cards are placed into two columns of two cards each, then the sums of the two numbers in each column are n... | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $abc = 27.$ Find the minimum value of $a + 3b + 9c.$ | nvidia |
Find the minimum positive difference between two composite numbers whose sum is 95. | nvidia |
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 312^\circ.$ | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 1.$ Find the maximum value of $a^2 b^2 c^2.$ | nvidia |
Compute the smallest positive integer $n$ such that
\[\sum_{k = 0}^n \log_2 \left( 1 + \frac{1}{2^{2^k}} \right) \ge 1 + \log_2 \frac{2014}{2015}.\] | nvidia |
Let $x$ and $y$ be real numbers such that $3x + 2y \le 7$ and $2x + 4y \le 8.$ Find the largest possible value of $x + y.$ | nvidia |
Find all values of $x$ such that $\arctan x < \arccot x$. | nvidia |
Find the equation of the line that passes through the point (2,4) and is perpendicular to the line through the points (1,1) and (3,5). | nvidia |
Find the sum of the smallest and largest integers between 10 and 200 (inclusive) whose number of positive divisors is a perfect square. | nvidia |
Find the smallest positive integer $n$ such that the sum of the digits of $n$ is a multiple of 11. | nvidia |
The sum of three different numbers is 67. The two larger numbers differ by 7 and the two smaller numbers differ by 3. What is the value of the largest number? | nvidia |
A sphere intersects the x-axis at two points: (a,0,0) and (-a,0,0). It also intersects the z-axis at the point (0,0,b). Determine the radius of the circle formed by the intersection of the sphere and the xy-plane. | nvidia |
Let $a$ be the sum of the coefficients of the polynomial $2x^2-5x+1$, $b$ the sum of the coefficients of the polynomial $-x^2+2x-3$, and $c$ the sum of the coefficients of the polynomial $x^2+x+1$. Determine $a-b+c$. | nvidia |
Find the value of $x$ in the expression $\frac{x}{10} \cdot \frac{25}{3} \cdot \frac{6}{5}$, given that the result is equal to $\frac{3}{2}$. | nvidia |
Find the number of three-digit numbers that are divisible by 4 and have exactly one repeated digit. | nvidia |
A polygon is circumscribed about a circle and its sides are tangent to the circle. Given that the area of the polygon is 36, and the radius of the circle is 2, determine the perimeter of the polygon. | nvidia |
Problem:
The number $2001$ can be expressed as the sum of two positive integers $A$ and $B$ in several ways. What is the maximum possible value of the product $A\cdot B$? | nvidia |
Let $a$, $b$ and $c$ be the side lengths of a triangle with a non-zero area.
Find the largest positive integer $m$ such that there exists a unique set of integers $a$, $b$, $c$ satisfying $m=a+b+c$. | nvidia |
Find the smallest positive integer $m$ such that $2m \equiv 432 \pmod{27}.$ | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=7$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | nvidia |
The sum of three different numbers is 75. The two larger numbers differ by 5 and the two smaller numbers differ by 2. What is the value of the largest number? | nvidia |
A function $g$ is defined for all positive integers $n$ and satisfies the following conditions:
* $g(1) = 2$
* For any positive integer $n$, $g(n+1) = 2g(n) - 1$
* For any two distinct positive integers $m$ and $n$, the ratio $\frac{g(m)}{g(n)}$ is not an integer.
What is the value of $g(5)$? | nvidia |
Find the smallest integer $n$ greater than 10 such that the product of all prime numbers less than or equal to $n$ is greater than 1000. | nvidia |
A right triangle has one side that is 8 inches long and the other side that is 15 inches long. Find the ratio of the length of the hypotenuse to the sum of the lengths of the other two sides. Express your answer in the form $a:b$. | nvidia |
Let $A$ be a set of $n$ positive integers such that for every subset $B$ of $A$, there exist two elements $x$ and $y$ in $B$ such that $x+y$ is also in $A$. Find the smallest possible value of the sum of all elements in $A$. | nvidia |
A positive integer is divisible by $5$ if and only if its units digit is $0$ or $5$. Furthermore, a positive integer is divisible by $3$ if and only if the sum of its digits is divisible by $3$. What is the smallest positive integer that is divisible by both $3$ and $5$? | nvidia |
In a regular octahedron, the midpoints of the twelve edges are the vertices of a smaller octahedron. The ratio of the surface area of the smaller octahedron to that of the larger is $a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. | nvidia |
Find the sum of all positive integers $n$ such that $2.5n-6.3<8.2$. | nvidia |
Find the sum of all distinct positive integers whose product is 120 and that have no prime factors other than 2, 3, or 5. | nvidia |
The roots $x_1$, $x_2$, and $x_3$ of the equation $x^3+2x^2-x-3=0$ are such that $\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}=0$. Determine the value of $\frac{1}{x_1^2}+\frac{1}{x_2^2}+\frac{1}{x_3^2}$. | nvidia |
What is the smallest positive integer $n$ such that $3^n \equiv 1 \pmod{10}$? | nvidia |
Consider the sequence $a_n$ such that $a_0=5$ and $a_{n+1}=f(a_n)$, where $f(x)$ is a function that returns the sum of the digits of $x$. Find the smallest positive integer $n$ such that $a_n=5$. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 4.$ Find the maximum value of
\[3xy \sqrt{3} + 2yz + 3zx.\] | nvidia |
Let p>3 be a prime number. Prove that there are exactly p-1 integer solutions of the equation $(n-p)p^n+n^p=0$. | nvidia |
Find the value of $x$ such that $\sqrt{x+7} + \sqrt{x-2} = 11$. | nvidia |
Find the maximum value of $\frac{y + x}{x + y^2 + 1}$ over all real numbers $x$ and $y.$ | nvidia |
A fair six-sided die is rolled three times. If the probability that the sum of the three rolls is greater than 14 can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $a+b$. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of
\[\frac{1}{a} + \frac{1}{b}.\] | nvidia |
What is the smallest positive integer $n$ such that $n!$ is divisible by $2323$? | nvidia |
Find the value of $x$ such that $\cos (2x) = \frac{1}{2}$. | nvidia |
A palindrome is a number that reads the same forwards and backwards, such as 3003. How many positive four-digit integers are palindromes? | nvidia |
Find all natural numbers $n$ such that $n$, $n^2+1$ and $n^2+2$ are all prime. | nvidia |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}^n = \mathbf{I}.\] | nvidia |
Let $f(x)$ be a function such that $f(f(x)) = 2$ for all real numbers $x$. Find the value of $f(2)$ if $f(2) \neq 2$. | nvidia |
How many positive four-digit integers have the property that their digits are in increasing order, with no repeated digits? | nvidia |
Let $a,$ $b,$ $c,$ $d$ be positive integers such that $\gcd(a,b)=\gcd(c,d)=1$ and $ab=cd.$ If $x$ is a positive integer such that $$a\cdot x\equiv c\pmod{b}$$and $$b\cdot x\equiv d\pmod{a},$$find the remainder when $x^2$ is divided by $ab$. | nvidia |
Let $p$ be a prime number greater than $7$. For any integer $k$, let $\phi_k(n)$ denote the remainder when $n$ is divided by $k$. An integer $a$ is called a $\textit{modified primitive root modulo $p$}$ if for every $k$ with $2\le k \le p-1$, there exists some integer $n$ for which $p|\phi_k(a^n)$. Find the smallest in... | nvidia |
Find the largest positive integer $b$ such that $x^3 - b^2$ is a perfect square for some integer $x.$ | nvidia |
Find the minimum value of
\[e^x\left(x^2+\frac{2}{e^x}\right),\]
where $x$ is a real number. | nvidia |
Find the length of the side of a square that has the same area as a rectangle with dimensions 0.4 meters and 0.6 meters. | nvidia |
Determine the value of $b$ such that the graph of the function
$$y=\frac{\sin^2 x+b}{\cos x}$$
has no vertical asymptotes. | nvidia |
Find the number of distinct scalene triangles with integer side lengths that have a perimeter of 25. | nvidia |
How many nonnegative integers can be written in the form\[a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,\]where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$?
$\textbf{(A) } 512 \qquad \textbf{(B) } 729 \qquad \textbf{(C) } 1094 \qquad \textbf{(D) } 3281 \qquad \textbf{(E)... | nvidia |
If $m$ and $n$ are positive 3-digit integers such that $\gcd(m,n)=2$, what is the smallest possible value for $\mathop{\text{lcm}}[m,n]$? | nvidia |
Find the number of distinct ways to arrange four non-zero digits to form a number that is divisible by 5 and has a sum of digits equal to 12. | nvidia |
Find the smallest three-digit prime number such that reversing the digits of the number produces a palindrome. | nvidia |
Find the sum of all values of $x$ such that $2^{x^2-3x-2} = 4^{x - 4}$. | nvidia |
Find the number of non-negative integer solutions to the equation $x+y+z=10$ such that $x-y+z$ is a multiple of 3. | nvidia |
The quadratic expression $\frac{x^2-2x+2}{x^2-2x-4}$ has two real and distinct zeroes. Find the value of $K$ such that the expression can be written in the form $(x-a)^K(x-b)^{2-K}$ for some constants $a$ and $b$. | nvidia |
Consider a sequence of real numbers $x_1,$ $x_2,$ $\ldots$ that satisfies the equation
$$x_{n+1} = x_n + \sqrt{x_n - n^2}$$for all positive integers $n.$ Find the value of $x_{11}.$ | nvidia |
Find the value of $x$ such that $\left(\sqrt{x^2+1}\right)^3=x^3+2$. | nvidia |
Let $g$ be a function defined on the set of positive real numbers such that \[g(x)= \begin{cases} 4x & \mbox{if }x\ge 1 \\ \frac{1}{g(g(\frac{1}{x}))} & \mbox{if }0<x<1. \end{cases}\]Find $g(\frac{1}{8})$. | nvidia |
Let $S$ be a subset of $\{1,2,\ldots,1997\}$ containing more than 1000 elements such that no two members of $S$ differ by 1, 2, or 3. Prove that there exist two elements $x,y$ in $S$ such that $|x-y|=377$ or $|x-y|=390$. | nvidia |
Let $a$ and $b$ be positive integers such that $ab+4a+4b=37$ and $a>b$. Determine $a$. | nvidia |
Determine all integer values of $n$ such that the cube of the sum of the squares of two consecutive positive integers is less than 6000. | nvidia |
The sum of the proper divisors of a number is defined as the sum of all positive divisors of the number excluding the number itself. What is the smallest positive integer that has exactly five proper divisors? | nvidia |
Find the smallest positive integer $a$ such that $x^4 + a^2$ is not prime for any integer $x.$ | nvidia |
A square is divided into nine squares of the same area. The whole square is then divided into rectangles so that the area of each rectangle corresponds to one of the terms of the infinite series $\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots$. Determine the side lengths of the rectangles. | nvidia |
The number obtained by reversing the digits of a three-digit number is 495 more than the original number. Find the sum of the units digits of all such three-digit numbers. | nvidia |
Determine the value of $x$ such that the expression $\left(\dfrac{3x+5}{x-2}\right)^0$ is undefined. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(0) = 0$ and
\[g(xy) = xg(y) + yg(x)\]for all real numbers $x$ and $y.$
Determine the number of possible values of $g(1),$ and let $t$ be the sum of all possible values of $g(1).$ Find $2t.$ | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
Find the sum of all $x$ such that the expression $\frac{x^2+5x+6}{x^2-7x-18}$ is undefined. | nvidia |
A convex hexagon $ABCDEF$ is circumscribed about a circle centered at $O$ with a radius of 10. The segment $AO$ intersects the circle at $T$. The area of the hexagon is 2400 and the area of triangle $ATO$ is 15 times the area of triangle $OBF$. Find the length of side $CD$. | nvidia |
Find the number of perfect squares between 1 and 100 that are also the sum of two consecutive integers. | nvidia |
There are values $C$ and $D$ such that
\[\frac{Dx+15}{x^2+8x+12}=\frac{2}{x+2}+\frac{C}{x+6}.\]
Find $C+D$. | nvidia |
Find the sum of the proper divisors of 512, where a proper divisor is a divisor that is not equal to the number itself. Instead of finding the divisors directly, use the fact that the sum of the proper divisors of a number is equal to the sum of the divisors of its prime factorization. The prime factorization of 512 is... | nvidia |
Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$. | nvidia |
When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively.
When the sum of the three integers is divided by $12$, what is the remainder? | nvidia |
Given that $\frac{10!}{7!4!}=210$, find the smallest positive integer $m$ for which $\frac{n!}{(n-m)!m!}$ is not divisible by $n$, for some $n$. | nvidia |
Find the sum of the divisors of 24. | nvidia |
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