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The positive numbers $a$, $b$, and $c$ satisfy
$$a+b+c=3, \quad \frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}=3, \quad \text{and} \quad \frac{1}{a^2b^2}+\frac{1}{b^2c^2}+\frac{1}{c^2a^2}=4.$$
Find the value of $abc$. | nvidia |
A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is
$\mathrm{(A) \frac{\sqrt{3}}{12} } \qquad \mathrm{(B) \frac{\sqrt{6}}{16} } \qquad \mathrm{(C) \frac{1}{6} } \qquad \mathrm{(D) \frac{\sqrt{2}}{8} } \qquad \mathrm{(... | nvidia |
Find the area of an equilateral triangle $ABC$ with a side length of 25 centimeters, given that it is inscribed in a circle and one of its vertices is at the center of the circle. | nvidia |
Find the smallest integer $n$ such that the decimal representation of $\frac{1}{2n+1}$ has a repeating block of 5 digits. | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x) f(y) + f(x+y) - f(xy) \le 0\]for all real numbers $x$ and $y$ that satisfy $x+y \neq 0$. | nvidia |
Let $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d$ be two distinct polynomials with real coefficients such that the $x$-coordinates of the points of intersection of the graphs $y=f(x)$ and $y=g(x)$ are the solutions to the equation
$$\frac{x}{x+1}=\cos(\ln(x+2)).$$Find the smallest possible value of $|b-d|$. | nvidia |
Find the smallest positive integer $y$ that satisfies $y + 6723 \equiv 1045 \pmod{15}$. | nvidia |
Find the product of all real values of $r$ for which $\frac{1}{2x}=\frac{r-x}{7}$ has exactly one real solution. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(1) = 2$ and
\[g(x + y) = (x + y) (g(x) + g(y))\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(3),$ and let $t$ be the product of all possible values of $g(3).$ Find $m + t.$ | nvidia |
The coefficients of a quadratic polynomial $P(x)$ are integers and its roots are $x_1$ and $x_2$. It is known that $x_1^2+x_2^2=2x_1x_2$. Find the smallest possible value of the product of the leading coefficient and the constant term of $P(x)$. | nvidia |
Let $b_1,$ $b_2,$ $\dots$ be a sequence of positive real numbers such that
\[b_n = b_{n - 1} + \frac{b_{n - 1}^2}{n^2}\]for all $n > 1.$ Find the largest possible value of $b_1$ such that the sequence converges. | nvidia |
A particular sequence of ten 0's and/or 1's is randomly generated. If the probability that there are no two consecutive 1's can be written in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers, find $m+n$. | 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 |
Determine all $z \in [-\pi, \pi]$ such that
$$\cos(x+z)+\cos(x-z)\geq 2\cos(x)\cos(z)$$
for every $x$ between $-\pi$ and $\pi$, inclusive. | nvidia |
The sum of the squares of two consecutive positive integers is 997. What is the sum of the two integers? | nvidia |
What is the smallest positive integer that can be expressed as the sum of two distinct prime numbers, where the sum of the two prime numbers is equal to the greatest common factor of 40 and 48? | nvidia |
Let $S$ be a set of binary strings of length $n$ such that no proper subset of $S$ satisfies the condition:
For every pair of strings $s,t \in S$, there exists a string $r \in S$ such that for all $i \in [1,n]$, the $i^{th}$ bit of $r$ is $1$ if and only if the $i^{th}$ bits of $s$ and $t$ are different.
What is the ... | nvidia |
Find the smallest positive integer $k$ such that $3k$ and $k+7$ leave the same remainder when divided by 11. | nvidia |
Find the smallest possible value of $n$ for which $2^n$ can be expressed as the difference of two positive integers whose product is a perfect square. | nvidia |
The sum of four consecutive odd positive integers is equal to three times their product. What is the sum of the four integers? | nvidia |
Find the smallest positive integer $m$ such that:
(i) $m^3$ can be expressed as the sum of two consecutive squares;
(ii) $3m - 17$ is a perfect cube.
Determine the value of $m$. | nvidia |
Find the polynomial $p(x),$ with real coefficients, such that $p(2) = 5$ and
\[p(x) p(y) = p(x) + p(y) + p(xy) - 2\]for all real numbers $x$ and $y.$ | nvidia |
Find the value of $\frac{x^6-8x^3+16}{x^3-4}$ when $x=2$, given that $x$ satisfies the equation $x^3-3x^2-4x+12=0$. | nvidia |
Find $n$ such that $2^5 \cdot 3^2 \cdot n = 8!$. | nvidia |
Compute the sum of all decimal numbers from 0.1 to 1.0, inclusive, rounded to one decimal place, such that each digit appears only once in the sum. | nvidia |
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression with first term $a$ and common difference $d$.
Find the largest value of $n$ such that the fraction $\frac{S_n}{S_{n+1}}$ is less than or equal to $\frac{5}{6}$. | nvidia |
Find the coordinates of the point on the line passing through the points $(3,7)$ and $(5,1)$, which is equidistant from the lines $x=2$ and $x=6$. | nvidia |
What is the smallest even number that is the product of two distinct prime numbers and has a prime factorization that can be written as a sum of two perfect squares? | nvidia |
Divide 80 into two positive integers such that the sum of their squares is minimized. | nvidia |
Find all real numbers $x$ such that $x^3-2x^2-5x+6>0$. Express your answer in interval notation. | nvidia |
Find all the real solutions to the equation $a^3(b-c)^2+b^3(c-a)^2+c^3(a-b)^2=0.$ | nvidia |
In a convex pentagon, the sum of the interior angles whose measures are prime numbers is 756 degrees. Determine the number of possible configurations of such pentagons. | nvidia |
Two circles are drawn on a coordinate plane, one with center at $(a,b)$ and radius $r$, and the other with center at $(c,d)$ and radius $s$. The circles are externally tangent to each other at a point in the second quadrant. A line is drawn that is tangent to both circles and intersects the line segment connecting the ... | nvidia |
Find all complex numbers $z$ such that $z^3 = 128 + 128i$. Enter all complex numbers, separated by commas. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(x+y) + (x-y)^2\]for all $x,$ $y \in \mathbb{R}.$
Determine the number of possible values of $f(1)$ and find the sum of all possible values of $f(1)$. | nvidia |
Find the least number that has exactly 6 positive divisors and is divisible by 7. | nvidia |
If $0.8= \frac{y}{186+x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers. | nvidia |
Let $g:\mathbb{R}\to\mathbb{R}$ be a function such that $g(x)g(y)=xg(y)+yg(x)-g(xy)$ for all real numbers $x$ and $y.$ Find the number of solutions $a$ to the equation $g(a)=0.$ | nvidia |
Parabola $y^2 = 4ax$ contains a chord $\overline{AA'}$, which may be described by a line parallel to the directrix.
Prove that the area enclosed by the arc and chord $\overline{AA'}$ of the parabola bounded by these two points has an area equal to $4/3$ times the area of the triangle bounded by this chord and the direc... | nvidia |
Let $x$ and $y$ be positive real numbers such that $2x + 3y = 5$. Find the minimum value of
\[\frac{2}{x} + \frac{3}{y}.\] | nvidia |
For each nonnegative integer n, let Sn denote the sum of the digits of n. For positive integers a,b,c, let Δ(a,b,c)=SaSbScScSbSa, where, for example, we have that S444=(4+4+4)+(4+4+4)+(4+4+4)=36. Now, assume that for every positive integer n, we have that Δ(n,nn,n(n+1)). Find the maximum value of Δ(a,b,c), for posit... | nvidia |
Find the minimum value of
\[x^2 + xy + y^2\]over all real numbers $x$ and $y.$ | nvidia |
A circle with center $(0, 0)$ and radius $r$ is given. There are real numbers $a$ and $b$ such that the line with equation $ax+by=c$ is tangent to the circle if and only if $a < c < b$. What is $a+b$? | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(f(x) + y) = x + y^2\]for all real numbers $x$ and $y.$ | nvidia |
Determine the number of integers divisible by 5 in the set of the first 24 positive integers that are not divisible by 3. | nvidia |
The product of two positive integers is 50 and their difference is 12. What is the value of the positive difference of the cubes of the integers? | nvidia |
Let $S(n, b)$ be the sum of the digits of the $n$th power of the base-$b$ number $11$. Find the smallest $n$ such that $S(n, 6) = S(n, 8)$. | nvidia |
Consider a sequence of positive integers defined by $a_1=1$, $a_2=2$, and $a_n=a_{n-1}+a_{\lfloor \log_2 n \rfloor}$ for $n \geq 3$. Find the greatest integer $k$ such that $a_k < 10^6$. | nvidia |
The pattern of Pascal's triangle is illustrated in the diagram shown. What is the fourth element in Row 15 of Pascal's triangle? $$
\begin{array}{ccccccccccccc}\vspace{0.1in}
\textrm{Row 0}: & \qquad & & & & & 1 & & & & & & \\ \vspace{0.1in}
\textrm{Row 1}: & \qquad & & & & 1 & & 1 & & & & &\\ \vspace{0.1in}
\textrm{R... | nvidia |
Let $d$ be a positive number such that when $109$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$. | nvidia |
Find the positive value of $n$ such that the equation $9x^2+nx+1=0$ has exactly one solution in $x$. | nvidia |
Find the largest integer $n$ satisfying the following conditions:
(i) $n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $2n + 79$ is a perfect square.
| nvidia |
Find the sum of the reciprocals of the positive divisors of 24. | nvidia |
The solid $T$ consists of the set of all points $(x,y,z)$ such that $x^2 + y^2 \le z$ and $x^2 + z^2 \le y.$ Find the surface area of $T.$ | nvidia |
The graphs of the equations $x^2+y^2=|x|+|y|$ and $y=\frac{1}{3}x+1$ intersect at several points. Determine the sum of the $x$-coordinates of all intersection points that lie in the second quadrant. | nvidia |
Find all real values of $b$ such that the polynomial $x^3 - 6x^2 + bx + 4$ has exactly one real root. | nvidia |
A polynomial $P(x)$ with integer coefficients is said to be $\textit{golden}$ if it has two distinct real roots and the sum of these roots is equal to twice their product. How many golden polynomials $P(x) = x^2 + px + q$ are there, where $p$ and $q$ are positive integers less than or equal to $2004$? | nvidia |
Determine the number of real solutions of the equation \[\sin (e^x) = xe^{-x}.\] | nvidia |
The three vertices of a triangle are located at $A(1,2,3)$, $B(4,5,6)$, and $C(7,8,9)$. The midpoints of the sides of the triangle form a parallelogram. Find the coordinates of the point where the diagonals of the parallelogram intersect. | nvidia |
Let $S$ denote the set of all positive integers $n$ such that $n$ and $2n+1$ have the same greatest common divisor. Find the sum of all possible values of the greatest common divisor of $n$ and $2n+1$ for all $n \in S$. | nvidia |
Given a set of positive real numbers $a_1, a_2, \ldots, a_n$, find the minimum value of the expression
\[\frac{\sum_{i=1}^{n} a_i \sum_{i=1}^{n} a_i^2}{\left( \sum_{i=1}^{n} a_i^2 \right)^2}.\] | nvidia |
A sequence of positive integers is called $n$-complementary if the sum of its digits is divisible by $n$ and each of its digits is either 1 or 0. How many 4-complementary positive integers are there that are less than 10000 and have at least two 1's? | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a+b+c=1$. Find the minimum value of
\[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+ \frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}.\] | nvidia |
Find the product of all positive integer values of $c$ such that $8x^2+15x+c=0$ has two real roots. | nvidia |
A diamond-shaped grid is formed by connecting the midpoints of the sides of a square, creating a series of triangles and smaller squares. Each small square can be filled with either a 0 or a 1. The numbers in the triangles are determined by the sum of the two squares above it. How many distinct arrangements of 0s and 1... | nvidia |
Evaluate the product \[ (n-1) \cdot n \cdot (n+1) \cdot (n+2) \cdot (n+3), \] where $n=2$. | nvidia |
Let $a,b,c$ be positive integers such that $(a \star b) \star c = a \star (b \star c)$.
Find the smallest possible value of $a+b+c$. | nvidia |
Count the number of positive integers $n$ that can be expressed as the difference of two perfect squares in exactly 3 ways. | nvidia |
Find the sum of all the positive integers that have exactly three factors and are less than 20. | nvidia |
Let $f(n)$ be the sum of the proper positive integer divisors of $n$. (Recall that a proper divisor of $n$ is a divisor other than $n$.) For how many values of $n$ does $n$ not divide $f(n)$, given that $3 \le n \le 75$? | nvidia |
Given that $x$, $y$, and $z$ are positive integers such that $x^2+y^2-z^2+2xy=12$ and $x+y+z=8$, determine the value of $xyz$. | nvidia |
Find the number of integers $n$ that satisfy
\[10 < n^2 < 99.\] | nvidia |
Three consecutive odd integers are such that the product of the first and third integers is 483. What is the value of the second integer? | nvidia |
Four distinct integers a, b, c, and d are such that each has exactly 4 positive integer factors. Find the number of factors of a^3 * b^2 * c^5 * d^7. | nvidia |
On triangle $ABC$, point $D$ lies on side $AB$ and point $E$ lies on side $BC$, so that $BD=DE=EC=20$. Find the area of the triangle $ABC$. | nvidia |
A 100-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the 100 vertices equals 2009. The midpoints of the sides of $P_1$ form a second 100-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third 100-gon, $P_3$. Find the sum of the $x$-coordinates of the vertices of $P_3$.
A 50... | nvidia |
Find all values of $x$ that satisfy the equation $|x-2|=3x-1$. Express your answers in simplest fractional form. | nvidia |
Find the set of all real numbers $x$ that satisfy the inequality $x^2-3x-4 \le 0$ and are more than $5$ units away from the roots of the equation $x^2-3x-4 = 0$.
Express your answer in interval notation. | nvidia |
Find the largest integer $n$ such that the difference between $2^n$ and $2^{n+1}$ is less than 240. | nvidia |
How many four-digit numbers have the property that the first two digits are multiples of 3, the last two digits are multiples of 5, and no digit is repeated? | nvidia |
Four circles each of radius $r$ are placed on a coordinate plane so that their centers are located on the vertices of a square with side length $2r$. Another circle is inscribed inside this square, and its center coincides with the center of the square. Find the area of the shaded region.
[asy]
size(200);
import graph... | nvidia |
Given that the four roots of the equation \[ax^4+bx^3+cx^2+dx+e=0\] are $1$, $-1$, $i$, and $-i$, determine the value of $\frac{d}{e}$. | nvidia |
Find the smallest positive integer $m$ such that $2x^2 + mx + 24$ can be expressed as the sum of three squares of linear factors with integer coefficients. | nvidia |
Find the smallest positive integer $a$ such that $a$ is divisible by $7$ and the sum of its digits is equal to $9$. | nvidia |
A triangular array of numbers is formed by starting with a single 1 at the top, and each subsequent row is obtained by adding adjacent pairs of numbers in the previous row. The sum of all the numbers in the row with 23 numbers is 8388608. What is the sum of all the numbers in the row with 42 numbers? | 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 |
Prove that $\left(\frac{a}{b}\right)^2 \cdot \left(\frac{a}{b}\right)^{-2}=1$ for any nonzero integers $a$ and $b$. | nvidia |
Find the number of divisors of 2100 that are not divisible by any perfect cube greater than 1. | nvidia |
The sum of the squares of three consecutive positive integers is equal to 365. What is the middle integer? | nvidia |
Let $a_n = \frac{1}{n^2}$ for all positive integers $n$. Let $X$ be the sum of all terms of the form $a_n$ where $n$ is not an even multiple of 5, and let $Y$ be the sum of all terms of the form $a_n$ where $n$ is an even multiple of 5.
Determine $\frac{X}{Y}$. | nvidia |
Let n>5 be an integer and let T={1,2,3,...,n}. Find the smallest value of n such that for every partition of T into three subsets, at least one of the subsets contains integers x, y, and z (not necessarily distinct) such that x+y=z. | nvidia |
In triangle $ABC$, $AB=10$, $BC=11$, and $AC=12$. Points $D$, $E$, and $F$ are on sides $BC$, $AC$, and $AB$, respectively, such that $BD:DC=BF:FC=AE:EC=1:2$. The perpendicular bisectors of segments $AD$, $BE$, and $CF$ intersect at a point $O$. Find the area of triangle $ABC$ that is inside the circle with center $O$ ... | nvidia |
The first 100 positive integers are written on 100 separate cards. The cards are then randomly divided into two groups, A and B, each containing 50 cards. The numbers on each card in group A are multiplied by 2, while the numbers on each card in group B are multiplied by 3. What is the positive difference between the s... | nvidia |
The sum of the squares of the first 25 positive odd integers is also the sum of five consecutive perfect squares. What is the largest of these five perfect squares? | nvidia |
Determine the number of ways to arrange the letters of the word PROOF such that the vowels (O and O) are never adjacent. | nvidia |
The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ intersects the circle $x^2+y^2=r^2$ at four points. Prove that if the distance between its vertices is equal to $2\sqrt{2}$, then the equation of the tangent line passing through the origin is $y=x$. | nvidia |
Find the largest integer k such that k^3 is less than 127. | nvidia |
A circular table has 15 seats, numbered from 1 to 15. A certain number of people are seated around the table, such that the sum of the seat numbers of every pair of people sitting next to each other is always a perfect square. What is the maximum number of people that can be seated around the table? | nvidia |
The sum of the lengths of all the edges of a cube is 60 cm. Find the number of cubic centimeters in the volume of the cube. | nvidia |
Find the largest $k$ such that for any unit vector $\mathbf{u},$ the equation
\[(\mathbf{u}\cdot\mathbf{a})^2+(\mathbf{u}\cdot\mathbf{b})^2+(\mathbf{u}\cdot\mathbf{c})^2=k\]
holds, where $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are any vectors satisfying $\mathbf{a}\cdot\mathbf{a}+\mathbf{b}\cdot\mathbf{b}+\mathb... | nvidia |
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