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Two circles have radii that differ by 4 units. The area of the larger circle is 4 times the area of the smaller circle. Find the radius of the smaller circle. | nvidia |
Find the number of nonnegative integers less than $100$ that are not multiples of $2$, $3$, or $5$. | nvidia |
In an ellipse centered at the origin, the point $(5,4)$ lies on the ellipse and its tangent line makes a $30^\circ$ angle with the positive x-axis. Find the length of the semi-major axis of the ellipse. | nvidia |
Let $A$ be the set of positive integers $n$ such that $f(f(n))=2n$ and $f(2n+1)=2n+3$ for all positive integers $n.$ Let $B$ be the set of positive integers $n$ such that $f(f(n))=3n$ and $f(3n+1)=3n+3$ for all positive integers $n.$ Find the number of elements in the set $A\cap B$. | nvidia |
Determine the value of $x$ that satisfies $\sqrt[5]{x\sqrt{x^3}}=3$. | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-2ax+a^2=0$$
are such that
$x_1^2+x_2^2=0.75$.
Determine $a$ | nvidia |
How many positive integer values of $n$ satisfy the inequality $2^n < 100 < 2^{n+1}$? | nvidia |
Find the value of $x$ such that $\frac{x}{3} + \frac{2}{5} = \frac{3}{2}$. | nvidia |
Given that the absolute value of the difference between the cube of a number and its reciprocal is equal to 23, determine the sum of all possible values of the number. | nvidia |
Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\tan x + \sec x = \sqrt{3}.$ Enter all the solutions, separated by commas. | nvidia |
The seven points $A$, $B$, $C$, $D$, $E$, $F$ and $G$ are evenly spaced around a circle, in that order. Each of the points is assigned a number such that the sum of the numbers assigned to any three adjacent points is 13. What is the number assigned to point $D$? | nvidia |
Find the smallest positive integer $n$ such that $3n+7$ leaves a remainder of 5 when divided by 11. | nvidia |
Find the product of all values of $y$ that satisfy the equation $y^2 + 2y = 15$. | nvidia |
Find the absolute value of the sum of the solutions of $x^2-7x+3=0$. | nvidia |
Find the smallest positive integer $b$ such that the digits $0, 1, 2, 3, 4, 5$ can be arranged in a $2 \times 3$ rectangle so that each row and column sums to the same number in base $b$. | nvidia |
In an equilateral triangle with side lengths of 8 units each, find the sum of the squares of the lengths of the altitudes. | nvidia |
Parallelogram $ABCD$ has vertices $A$ and $B$ at $(2,0)$ and $(3,5)$, respectively. Find the length of side $CD$ if the area of the parallelogram is $30$. | nvidia |
Find the sum of all positive integers with three digits that are multiples of 7. | nvidia |
A circle with center $(1, 0)$ and radius $2$ intersects the line $y=x-1$ at two points $P$ and $Q$. Determine the slope of the line that passes through the midpoint of segment $\overline{PQ}$ and the point $(3, 4)$. | nvidia |
Find the smallest positive integer that is both an integer power of 13 and has exactly three distinct digits. | nvidia |
Quadrilateral $ABCD$ has sides $AB=10,BC=13,CD=8$, and $DA=11$. Lines $\ell_A,\ell_B,\ell_C$, and $\ell_D$ are drawn parallel to $\overline{BC},\overline{AD},\overline{AB}$, and $\overline{CD}$, respectively, such that the intersections of $\ell_A,\ell_B,\ell_C$, and $\ell_D$ with the interior of $ABCD$ are segments of... | nvidia |
What are the last two digits in the product of the sum of squares of the first 50 positive integers and the sum of cubes of the first 50 positive integers? | nvidia |
Find the sum of all positive divisors of 72 that are also multiples of 4. | nvidia |
The roots of the equation $x^3-6x^2+5x-1=0$ are $p$, $q$, and $r$. Given that $p+q+r=6$ and $pq+pr+qr=5$, find the value of $p^2+q^2+r^2$. | nvidia |
Let $f(x)$ be a polynomial of degree $2$ satisfying
\[f(x+1) = 3f(x)-f(x-1)\]for all real numbers $x$ and $f(0)=1$, $f(1)=10$.
Find $f(2)$. | nvidia |
Find the sum of all positive integers $a$ for which the equation $x^2+(2a-1)x+3a=0$ has integral roots. | nvidia |
A triangle is formed with one vertex at the vertex of the parabola $y=x^2-1$ and the other two vertices at the intersections of the line $y=r$ and the parabola. If the area of the triangle is between $8$ and $64$ inclusive, find all possible values of $r$. Express your answer in interval notation. | nvidia |
Find the value of $x$ such that the following equation in base $x$ is true:
$$\begin{array}{c@{}c@{}c@{}c@{}c@{}c} &&1&4&6&8_x\\ &\times&7&5&1&2_x\\ \cline{2-6} &1&0&6&8&0_x.\\ \end{array}$$ | nvidia |
Find the smallest two-digit prime number such that reversing the digits of the number produces a composite number. | nvidia |
Let $S$ be a set of 9 non-zero complex numbers such that the sum of any 3 distinct numbers in $S$ is a real number. Find the smallest possible value of the product of the absolute values of the elements of $S$. | nvidia |
What is the smallest positive integer $n$ such that the number $11n$ is divisible by 5 and $n$ is divisible by 3? | nvidia |
Let $a,b,c$ be the roots of the equation
$$x^3-px^2+qx+r=0$$
where $a$ and $b$ are integers, $c$ is a real number and $a<b<c$. If $|c|=|a-b|$ find the maximum possible value of $|ab|$. | nvidia |
Let $a_{n+2}=\frac{a_n+1}{1+a_{n+1}}$ for $n\geq 1$, where the terms of the sequence $(a_i)$ are positive integers. Determine the number of possible values of $a_1$ such that $a_1=a_3$. | nvidia |
A circular arrangement of 12 points is such that every pair of adjacent points is connected by a line segment of a different integer length, and these lengths are relatively prime to each other. Determine the smallest possible value of the sum of these lengths. | nvidia |
Find the number of non-similar quadrilaterals with distinct positive integer side lengths in geometric progression. | nvidia |
Find the value of $ab$ if $a$ and $b$ are positive integers such that $\frac{a}{b} + \frac{b}{a} = 6$ and $a^2 + b^2 = 56$. | nvidia |
For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \le 50$.
(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions? | nvidia |
Find the sum of all positive integers $n$ such that $n^3 + 3n^2 - 2n + 5$ is a perfect cube. Determine the remainder when this sum is divided by 2023. | nvidia |
Let $a$ and $b$ be real numbers such that $(a+bi)^2=3-4i$. Determine $a^4+b^4$. | nvidia |
Find the value of $x$ such that the graph of $y = \frac{x+2}{x-1}$ intersects the graph of $y = \frac{x-3}{x+4}$ at a single point. | nvidia |
The sum of the squares of two consecutive positive even integers is 488. What is their product? | nvidia |
Let $f(x)$ be a function that satisfies $f(f(x)) = x + f(x)$ for all real numbers $x.$ Given that $f(0) = 3,$ find $f(f(0))$. | nvidia |
The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum? | nvidia |
A rectangle has a perimeter of 24 cm. How many different rectangles with integer side lengths can be formed such that the area of the rectangle is a perfect square? | nvidia |
Let f(x) be a function that satisfies the equation f(2x) = 4f(x) + 1 for all real numbers x. Given that f(1) = 3, compute f(2). | nvidia |
Find the smallest positive integer value of x such that the sum of the squares of the digits of x is a multiple of 10 and the product of the digits is not divisible by 4. | nvidia |
Determine the minimum number of real roots that the polynomial $f(x)$ of degree 2015 with complex coefficients can have, if its nonreal roots come in 1008 pairs $a_i$ and $\bar{b}_i$ ($i=1,2,\dots,1008$), and the following conditions are satisfied:
$a_1=-b_1$,
$a_2=3b_2$,
$a_3=-5b_3$,
$a_4=7b_4$,
$\dots$,
$a_{1008}=-20... | nvidia |
Given the positive real numbers $x,$ $y,$ and $m,$ and the equation
\[4=m^2\left(\dfrac{x^3}{y^3}+\dfrac{y^3}{x^3}\right)+m\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right),\]
find the minimum possible value of $m.$ | nvidia |
How many three-digit positive integers have exactly two digits that are the same, and the sum of these two digits is equal to 5? | nvidia |
Let $f(n)$ be the probability that two randomly selected cards from a deck of $n$ cards have numbers whose difference is a multiple of 3. If $n \geq 3$, find the value of $n$ such that $f(n)$ is maximized. | nvidia |
Consider the equation $2x + 3y = 5xy = 11$, where $x$ and $y$ are real numbers. Given that the solutions for $x$ are in the form $x = \frac{p \pm q\sqrt{r}}{s}$, where $p$, $q$, $r$, and $s$ are positive integers and the expression is completely simplified, find the value of $p + q + r + s$. | nvidia |
Given a cubic polynomial $ax^3+bx^2+cx+d$ with rational coefficients and three real roots $r_1, r_2,$ and $r_3$, where $r_1 <r_2 <r_3$, suppose that for any two roots of the polynomial, their difference divided by their sum is also a root. Find the largest possible value of $\frac{b}{a}$. | nvidia |
Given that $x$, $y$, and $z$ are real numbers such that $\tan (x) + \tan (y) + \tan (z) = \tan (x) \tan (y) \tan (z)$, find the sum of all possible values of $\sin^2(x+y+z)$. | nvidia |
Suppose that $a,b,c,d,e$ are integers such that $a+b+c+d+e=0$. Let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $M$? | nvidia |
Let $S$ be the set of all positive integers $n$ such that $n!$ is divisible by $100$. What is the largest divisor of $100$ that divides $\prod_{n\in S} (n!)$? | nvidia |
Consider the linear transformation $\begin{pmatrix} x \\ y \end{pmatrix} \rightarrow \begin{pmatrix} x + 2y \\ 3x + 4y \end{pmatrix}.$ Determine the smallest possible value of $m + n,$ where $m$ and $n$ are positive integers such that the image of the point $(m, n)$ is on the $x$-axis. | nvidia |
In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is 56. Find the area of the polygon.
[asy]
unitsize(0.5 cm);
draw((3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(6,2)--(6,3)--(7,3)--(7,4)--(6,4)--(6,5)--(5,5)--(5,6)--(4,6)--(4,7)--(3,7)... | nvidia |
The sum of two consecutive positive integers is 2551. What is the product of the two integers? | nvidia |
Given that the average of three consecutive integers is less than 25, what is the largest possible integer value among them such that their sum does not exceed 70? | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x) g(y) + g(xy) = 2xy\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(3),$ and let $t$ be the sum of all possible values of $g(3).$ Find $m \times t.$ | nvidia |
What is the largest integer $n$ such that $\sqrt{2n+1}$ is an integer? | nvidia |
The points $A$, $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$. It is given that $AB=13$, $BC=14$, $CA=15$, and that the distance from $O$ to $\triangle ABC$ is $\frac{m\sqrt{n}}k$, where $m$, $n$, and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by t... | nvidia |
Determine whether there exist two real-valued functions $f(x)$ and $g(x),$ defined on the set of positive integers, such that
$f(n)g(m)=m+n$ and $f(n)+g(m)=mn$
for every pair of positive integers $m,n$. | nvidia |
Let $a$ be a real number, $a > 0,$ and let $x$ and $y$ be positive real numbers such that
\[x + y + \frac{1}{x} + \frac{1}{y} = a.\]Find $xy.$ | nvidia |
Given that the roots of $x^3+px^2+qx+r=0$ are $m$, $n$, and $k$, and the roots of $x^3+dx^2+ex+f=0$ are $\frac{1}{m}$, $\frac{1}{n}$, and $\frac{1}{k}$, find the value of $f$. | nvidia |
Let $p^a$ be the greatest power of a prime number $p$ that divides $n!$, where $n$ is a positive integer. Evaluate the following expression: $$\prod_{p\leq n}\left(1+\frac{1}{p}\right)^a$$ | nvidia |
Let $X$ be a set of all integers from 5 to 22, inclusive.
Let $A$ be a set of all odd numbers from set $X$.
Let $B$ be a set of all prime numbers from set $X$.
Let $C$ be a set of all even numbers from set $X$.
Find the probability that an integer randomly selected from set $X$ will be either a prime or a composite odd... | nvidia |
Find the minimum value of the sum
\[\sum_{k=1}^n \sqrt{k^2+b_k^2},\]where $b_1,b_2,\ldots,b_n$ are positive real numbers whose sum is $2019$ and $n$ is a positive integer such that $2n+1$ is a perfect square. | nvidia |
Let $a$ and $b$ be real numbers. If the polynomial $x^2 + ax + b$ has exactly one real root and $a = 2b - 1$, find the value of the sum of all possible values of $b$. | nvidia |
Given a set of positive integers, the operation $\ast$ is defined as $a \ast b = a + \frac{a}{b}$. If a list of positive integers is such that for any pair of distinct integers, $a$ and $b$, in the list, the number $a \ast b$ is equal to the product of all numbers in the list, what is the smallest possible list that ca... | nvidia |
Determine the number of positive integers $a$ less than $12$ such that the congruence $ax\equiv 1\pmod{12}$ has a solution in $x$. | nvidia |
What is the smallest positive integer n such that the total number of divisors of n^2 equals the product of n and the number of its perfect square divisors? | nvidia |
Find the value of $k$ if the graph of the equation $y=k\sqrt{x^2+6x+9}$ intersects the circle $x^2+y^2=36$ at exactly one point. | nvidia |
If $p$ and $q$ are positive integers such that $p^2+7pq+q^2=705$ then what is $q$? | nvidia |
How many positive integers less than 1000 are congruent to 6 (mod 11)?
Write another problem inspired by this one:
A positive integer $n$ is said to be "lucky" if it can be represented as a sum of three consecutive integers. How many positive integers less than 1000 are "lucky"? | nvidia |
Determine the complex number $w$ satisfying the equation $3w-2i\bar{w}=11-4i$. Note that $\bar{w}$ denotes the conjugate of $w$. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) - y) = f(x) + f(f(y) - f(-x)) + x\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$ | nvidia |
Given that $a^x \cdot a^{-x} = 1$ for any real number $a$ and $x$, find the value of $b$ such that $b^2 \cdot b^{-3} = \frac{1}{b}$. | nvidia |
Let $ a$, $ b$, $ c$ be nonzero real numbers such that $ a+b+c=0$ and $ a^3+b^3+c^3=a^5+b^5+c^5$. Find the value of
$ a^2+b^2+c^2$. | nvidia |
Find the number of positive integers $x$ that satisfy the inequality $\frac{1}{x}+\frac{1}{x+1}+\frac{1}{x+2} < \frac{1}{3}$. | nvidia |
Find the sum of all positive integer values of $k$ such that $kx^2-16x+2=0$ has two real roots whose ratio is less than 2. | nvidia |
The function $g(x)$ satisfies
\[g(x+y) = g(x) \cdot \frac{g(y)}{y}\]for all positive real numbers $x$ and $y.$ If $g(4) = 12,$ find $g(8).$ | nvidia |
Let $M$ be the product of the first $50$ positive even integers. Find the largest integer $n$ such that $M$ is divisible by $2^n$. | nvidia |
Let $A = \{1, 2, \dots, 100\}$, and let $S$ be the set of all positive integers $n$ such that there exists an element $a \in A$ such that $n$ is the remainder when $a^4$ is divided by 12. Compute the number of elements in $S$. | nvidia |
Let $P$ be a point inside a regular hexagon with side length $1$. Determine the largest possible value of the sum of the distances from $P$ to all vertices of the hexagon. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 1.$ Find the minimum value of
\[\frac{x}{y} + \frac{y}{z} + \frac{z}{x}.\] | nvidia |
The graph of the parabola $y=ax^2+bx+c$ passes through the points $(1,6)$, $(3,24)$ and $(5,42)$. Find the area of the triangle formed by these three points. | nvidia |
If $a$, $b$, $c$, and $d$ are positive integers such that $\frac{2^a \cdot 3^b}{2^c \cdot 3^d} = 1$, what is the smallest possible value of $a+b+c+d$? | nvidia |
Find the smallest positive integer n such that n^2 is a perfect square and n contains the digit 7. | 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 |
In the equation
$$2x^2-3x+1=0$$
determine $x$ such that the sum of the roots be equal to unity. | nvidia |
Find the sum of all the positive integer divisors of $12^2$. | nvidia |
The equation $y=\dfrac{x^2+1}{x^2-x+1}$ is a rational function with a numerator and denominator that are both quadratic expressions. Determine the values of $x$ that would make the denominator equal to zero, and express the domain of the function in interval notation. | nvidia |
Let $a$, $b$, $c$ be positive integers such that $\frac{a}{b}=\frac{b}{c}=\frac{c}{a+k}$, where $k$ is a positive integer. Find the smallest possible value of $k$. | nvidia |
Let $a$ and $b$ be the roots of the equation $x^2-3x-2=0$. Find the sum of the squares of all possible values of $c$ such that $|c-a|+|c-b|=3$. | nvidia |
Let $x$ be a positive real number such that $x^{2n}=n$ for some positive integer $n$. Compute the smallest possible value of $x+\frac{1}{x}$. | nvidia |
Let $a$ and $b$ be two positive integers such that $5^a$ is the largest power of $5$ that divides $8!$ and $3^b$ is the largest power of $3$ that divides $9!$. Find the value of $a+b$. | nvidia |
For how many positive integers $n$ is $2^n$ a perfect square? | nvidia |
The sum of two positive integers $a$ and $b$ is 1500. What is the largest possible value of $\gcd(a,b)$? | nvidia |
Given that $x,$ $y,$ $z$ are non-negative integers such that $x+y+z=20,$ find the maximum value of the expression $\sqrt{x-1}+\sqrt[3]{y-2}+\sqrt[4]{z-3}$ subject to the conditions that $x \geq 1,$ $y \geq 2,$ and $z \geq 3.$ | nvidia |
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