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Let $x,$ $y,$ $z$ be positive real numbers such that $xyz=1$. Determine the minimum possible value of
\[x^2y+y^2z+z^2x.\] | nvidia |
Let $a_1, a_2, \dots, a_n$ be a sequence of integers such that $a_i = 0$ or $a_i = 1$ for each $i$. Find the maximum possible value of the sum
$$\sum_{1 \le i < j < k \le n} a_i a_j a_k$$
subject to the condition that the sum
$$\sum_{1 \le i < j \le n} a_i a_j$$
is equal to a given positive integer $m$. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 1.$ Find the minimum value of $x^2 y^2 + y^2 z^2 + z^2 x^2.$ | nvidia |
Let $a_1, a_2, \ldots, a_8$ be real numbers such that $\sum_{i=1}^8 a_i = 0$ and $\sum_{i=1}^8 a_i^2 = 1.$ Find the minimum value of
\[\sum_{i=1}^8 a_i^4.\] | nvidia |
The sum of the digits of the number $3^{2007}$ is congruent to $n$ modulo 9. Determine $n$. | nvidia |
The area of a rectangle is inversely proportional to the perimeter of a square that has the same length as one of the sides of the rectangle. When the area of the rectangle is 30 square units, the perimeter of the square is 48 units. Find the area of the rectangle when the perimeter of the square is 120 units. | nvidia |
A convex polyhedron has six faces, each being a regular hexagon. Determine the number of edges of the polyhedron if its vertices can be colored with three colors such that any face has all three colors. | nvidia |
A certain function $f$ has the property that $f(f(x))=x$ for all real numbers $x$, and $f(3x)=3f(x)$ for all $x\geq 0$. If $f(x)=\frac{x^3}{2}$ when $0\leq x\leq 1$, find $f(100)$. | nvidia |
What is the largest positive three-digit number divisible by 11 which has all distinct digits and none of the digits is equal to 9? | nvidia |
Consider the polynomial function:
$Q(x) = x^4 + px^3 + qx^2 + rx + s$
The graph of this polynomial has four distinct $x$-intercepts, one of which is at $(1,0)$. Which of the following coefficients cannot be zero?
$\textbf{(A)}\ p \qquad \textbf{(B)}\ q \qquad \textbf{(C)}\ r \qquad \textbf{(D)}\ s$ | nvidia |
The positive numbers $a$, $b$, $c$, $d$ satisfy the equality
$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=1.$
Prove that $\max{(a+b)(c+d)}=4.$ | nvidia |
The sum of the digits of a three-digit integer is 12, and the product of its hundreds and tens digits is 15. How many such integers are there? | nvidia |
How many positive whole numbers have square roots that are between $5$ and $15$, inclusive? | nvidia |
Find the sum of all integers between 20 and 100 that are divisible by 6, given that these numbers form an arithmetic sequence with a common difference of 6. | nvidia |
Find the sum of all even divisors of 240. | nvidia |
Find the value of $x$ that maximizes the area of a rectangle with a fixed perimeter of 24 units, where the length and width of the rectangle are $x$ and $x+2$, respectively. | nvidia |
Find the maximum value of
\[x^2 - 2xy + 3y^2\]over all real numbers $x$ and $y.$ | nvidia |
Find the smallest positive integer that has exactly eight distinct positive factors and is a product of two prime numbers. | nvidia |
Let $f(x)$ be a function which satisfies $f(f(x)) = f(-x),$ and the value of $f(10)$ is known to be $5.$ If the function $f(x)$ is defined only on the integer values of $x$, and for each integer $a$ from $0$ to $9,$ the sum of the two values of $f(a)$ and $f(-a)$ is always the same integer $k$, find all possible values... | nvidia |
Let the value of the polynomial $P(x)=ax^4+bx^3+cx^2+dx+f$ at some $x=t$ be equal to 2. Determine all coefficients of the polynomial if it is known that
$2P(x)=P(2x)$ for any value of $x$. | nvidia |
Find the number of ways to express $2004^{2004}$ as a product of two positive integers such that the sum of their divisors is a perfect square. | nvidia |
Given the points A(1, 2) and B(4, 3), find the equation of the line that is the perpendicular bisector of the line segment AB, given that it also passes through the point C(2, 2). | nvidia |
The sum of two positive numbers is $27$. The sum of their squares is $251$. What is the difference between the squares of the numbers? | nvidia |
Let $f(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point. How many positive integer divisors does $f(2025)$ have? | nvidia |
Find all real numbers $m$ for which there exists a nonzero, 2-dimensional vector $\mathbf{w}$ such that
\[\begin{pmatrix} 3 & 5 \\ 1 & 3 \end{pmatrix} \mathbf{w} = m \mathbf{w}.\]Enter all the solutions, separated by commas. | nvidia |
In the following addition problem, the sum of the digits of each number equals 12, but the bases of the numbers are unknown.
$$_\square 4\square 6 + _\square \square 5\square = 47\square.$$
Determine the bases of the numbers and complete the addition problem. | nvidia |
Find the positive integer $n$ such that when $\frac{3}{5}$ is expressed as a decimal, the $n$th digit after the decimal point is the $1$st digit after the decimal point. | nvidia |
Find the smallest positive integer $n$ such that $n!$ is divisible by the fourth power of every prime less than or equal to $n$. | nvidia |
The roots $x_1$, $x_2$, $x_3$, and $x_4$ of the equation
$$\sin^2(x)+\sin^2(2x)+\sin^2(3x)+\sin^2(4x)=2$$
are such that
$x_1^2+x_2^2+x_3^2+x_4^2=10$.
Determine $a$ such that
$\sin^2(ax)+\sin^2(2ax)+\sin^2(3ax)+\sin^2(4ax)=2$ also has four real roots. | nvidia |
Find the number of four-digit numbers such that the sum of the first and last digits is equal to the sum of the second and third digits, and the sum of the first and second digits is equal to the sum of the third and last digits. | nvidia |
Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$. The largest integer which divides all possible numbers of the form $m^2-n^2$ is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$
| nvidia |
Find the smallest positive integer $x$ such that $2x \equiv 3 \pmod{13}$. | nvidia |
Let $x_1$, $x_2$ and $x_3$ be different prime numbers. Determine the number of all different natural numbers of the form $x_1^{k_1}x_2^{k_2}x_3^{k_3}$ ($k_1$, $k_2$, $k_3 \in N$) having exactly 12 different positive integer factors. | nvidia |
The product $N$ of three positive integers is $6$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$.
| nvidia |
Determine the value of $a$ such that $(3a+2)^2-(2a-1)^2=5a^2+14a-7$. | nvidia |
Find the value of $y$ such that $\sqrt{y-3} = 7$. | nvidia |
The domain of the function $f(x) = \arcsin(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the the smallest possible value of $m+n.$ | nvidia |
Find the number of positive integers $n$ for which the product of the first $n$ positive integers (i.e., $n!$) is divisible by the sum of the squares of the first $n$ positive integers. | nvidia |
If the product of the non-real roots of the polynomial $$P(x)=x^4-3x^3+5x^2-27x-36$$ is equal to $p$, find $|p|$.
(Please note that P(x) has no real roots as we already know that it has two pure imaginary solutions.) | nvidia |
Determine the value of $a$ such that the graphs of $y=a^x$ and $y=\log_a x$ intersect at the point $(2,4)$. | nvidia |
A bag contains 12 marbles, each with a different number from 1 to 12. If two marbles are drawn at random without replacement, what is the probability that the sum of the numbers on the two marbles is a perfect square? | nvidia |
Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.
| nvidia |
How many four-digit numbers are there such that the sum of the first two digits is equal to the sum of the last two digits? | nvidia |
Find the greatest value of $c$ such that $-c^2+9c-12 \ge 0$. | nvidia |
The sum of the squares of two consecutive positive integers is 365. What is the sum of the two integers? | nvidia |
A square matrix $A$ with integer entries is such that $A^3=A$. Find the smallest possible value of $\det(A)$. | nvidia |
Solve the system of equations:
$$x^2 + 2y^2 = 10$$
$$2x^2 + y^2 = 13$$
Find the ordered pair of real numbers $(x,y)$ that satisfy both equations. | nvidia |
Let $f(x)$ be a polynomial such that $f(0) = 1$ and for every integer $n$, $f(n + 1) = 2^{n + 1} - 3f(n)$. Find the smallest positive integer $k$ such that $f(k)$ is a multiple of $7$. | nvidia |
Find the sum of the digits of the base-3 expression for the smallest number that is divisible by 7 and has the same number of digits in base-3 as the number 222 has in base-10. | nvidia |
There exists a positive real number $x$ such that $\sin(\arccos(x)) = \frac{x}{\sqrt{2}}$. Find the value of $x^2$. | nvidia |
Find the product of all values of $y$ such that $\log_3{(y^2+2y-5)} = 2\log_3{(y-1)}$. | nvidia |
A certain sequence is constructed so that the sum of every pair of adjacent terms is equal to the sum of every other pair of terms separated by two places. The first three terms of the sequence are $1$, $3$, and $2$. What is the sum of the first $14$ terms of the sequence? | nvidia |
Given that the polynomial $x^2-5x+t$ has only positive integer roots, find the average of all distinct possible values of $t$. | nvidia |
A rectangular prism has a rectangular hole. The sides of the prism are $s$ and $3s$ and the sides of the hole are $s$ and $2s$. If the depth of the hole is half that of the prism, find the volume of the solid part of the prism in terms of $s$. | nvidia |
In a triangle $ABC$, the measure of $\angle A$ is $2x$ degrees, the measure of $\angle B$ is $5x$ degrees, and the measure of $\angle C$ is $3x$ degrees. Find the value of $x$ such that the sum of the measures of the exterior angles of triangle $ABC$ is $180^\circ$ greater than the sum of the measures of the interior a... | nvidia |
In the triangle $ABC$, the midpoint of the side $BC$ is $D$, and the foot of the perpendicular from $D$ to $AC$ is $E$. If $AD$ intersects $BE$ at $F$, and the area of the triangle $ABC$ is 16, find the area of the triangle $DEF$. | nvidia |
Find the sum of all the integers that are divisible by 3 and are between -12.1 and 3.3. | nvidia |
Given that the polynomial $x^2 - 7x + u$ has only negative integer roots, find the sum of all distinct possible values of $u$. | nvidia |
A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 3-primable positive integers are there that are less than 1000? | nvidia |
Let $x,$ $y,$ $z$ be positive real numbers such that $xyz = 8.$ Find the minimum value of $x + 2y + 4z.$ | nvidia |
Let $a_0$ and $b_0$ be real numbers such that $\frac{a_0}{b_0} > 1$, and define
\[a_{n + 1} = \frac{a_n^2 + b_n^2}{2b_n} \quad \text{and} \quad b_{n + 1} = \frac{a_n^2 - b_n^2}{2a_n}\]for all $n \ge 0.$ Show that the limit of the sequence $b_n$ is zero, and determine the rate at which it converges to zero. | nvidia |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/29 of the original integer. | nvidia |
Determine $n$ such that the sum of the digits of the $n$-digit number $2^{n-1}+2^{n-2}+...+2^2+2^1+2^0$ is divisible by 3. | nvidia |
Determine all values of $a$ such that the equation
$$\sum_{n=1}^{\infty} \frac{a^n}{x - n} = x + a$$
has a solution $x \in \mathbb{R}$. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $x + y + z = 1.$ Find set of all possible values of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}.$ | nvidia |
Let's say that $\omega$ is a constant for which the equation $x^2+20x+\omega=0$ has two solutions and the product of the two solutions is 5. Determine $\omega$. | nvidia |
Let $P$, $O$, and $W$ be distinct positive integers such that the product $P\cdot O\cdot W=3003$. What is the largest possible value of the sum $P+O+W$? | nvidia |
Given that $a$ and $b$ are integers, if the function $f(x) = ax^2 + bx + 9$ satisfies the condition that $\frac{f(3x)}{f(x)}$ is a constant for all real numbers $x$, find the product of $a$ and $b$. | nvidia |
What is the smallest positive integer that is a perfect square and is divisible by each of the first six positive integers that are perfect squares? | nvidia |
The sum of the infinite geometric series
$$1 + \frac{1 + a}{3} + \frac{1 + 2a}{3^2} + \frac{1 + 3a}{3^3} + \dotsb$$
is equal to 5. Find the value of $a$ such that the sum of the reciprocals of the terms of the series is equal to 3. | nvidia |
Problem:
A hyperbola with its transverse axis along the x-axis has its center at the point (2,3). The equation of the asymptotes are $y=2x$ and $y=-2x$. The distance between the vertices of the hyperbola is 6. Find the distance between the foci. | nvidia |
Given that $531n \equiv 1067n \pmod{24}$, find the smallest positive integer $n$ such that $531n \equiv 25n \pmod{23}$. | nvidia |
Let $x$ be a positive real number such that $\log_{3}{x} + \log_{5}{x} = 3$. Find the number of possible values of $x$. | nvidia |
Find the number of five-digit integers that do not contain the digit 7 and have digits arranged such that all occurrences of the digit 9 appear after all occurrences of the digit 2. | nvidia |
A polynomial of the form $x^2 + ax + b$ can be factored as the product of two linear factors with integer coefficients if and only if there exist two integers $m$ and $n$ such that $m \times n = b$ and $m + n = a$.
Find the number of positive integers $n$, $1 \le n \le 1000$, for which there exist two integers $m$ and... | nvidia |
Find all positive integers $n$ such that
$$\cos\left(\frac{n\pi}{7}\right) + \cos\left(\frac{3n\pi}{7}\right) + \cos\left(\frac{5n\pi}{7}\right) = 0.$$ | nvidia |
A function f is defined on the set of positive integers and takes values in the same set. The following facts are known about this function:
* f is an invertible function.
* For all positive integers x and y, f(x+y)=f(x)+f(y).
* The value of f(1) is 3.
Determine f(8). | nvidia |
Find all real numbers $x$ such that $x^2 - \lfloor x \rfloor^2 = 2.25$. | nvidia |
Find the slope of the line that passes through the point (2,3) and is perpendicular to the line that passes through the points (1,1) and (4,7). | nvidia |
There are 3 complex numbers $a+bi$, $c+di$, and $e+fi$. If $b=1$, $e=-a-c$, and the sum of the numbers is $-i$, find $d+f$. | nvidia |
Let $ABCD$ be a square of side length 1, and let $E$ and $F$ be two points chosen randomly inside the square. What is the probability that the area of $\triangle EDF$ is less than the area of $\triangle EBF$? | nvidia |
A quadratic equation has two real roots, and the coefficients are all positive integers. The roots can be written in the form $x = \frac{m \pm \sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers with a greatest common divisor of 1. The value of $n$ is the product of two distinct prime numbers. What is the larg... | nvidia |
How many unordered pairs of distinct positive integers have a product of 360 and a difference that is a perfect square? | nvidia |
A convex polygon with 20 sides has 10 vertices labeled with the numbers 1 through 10 in clockwise order. A diagonal is a line segment that connects two non-adjacent vertices. How many diagonals can be drawn such that the sum of the labels of the two vertices it connects is a multiple of 3? | nvidia |
Find the number of divisors of $2007$ that are divisible by $3$ but not by $9$, given that its prime factorization is $3^2 \cdot 13 \cdot 17$. | nvidia |
What is the smallest positive integer $n$ such that the inequality
$$\left|\frac{x}{2}+3\right| \leq \left|2x-1\right|$$
has exactly $n$ integer solutions? | nvidia |
Let $h(x)$ be a function defined on an open interval $(a,b)$. Suppose that there exists a polynomial $p(x)$ of degree $n$ such that the function $F(x)=h(x)+p(x)$ satisfies the following condition: for any four distinct points $x_1,x_2,x_3,x_4\in (a,b)$ the points $(x_1,F(x_1))$, $(x_2,F(x_2))$, $(x_3,F(x_3))$ and $(x_4... | nvidia |
Consider the set of all functions $f(x)$ defined for all real numbers $x$ and satisfying the following conditions:
1. $f(-x) = f(x)$ for all $x$
2. $f(x + k) = f(x)$ for all $x$, where $k$ is a fixed real number
A given function $f(x)$ has exactly $n$ roots on the interval $[-10, 10]$. Find the smallest possible valu... | nvidia |
What is the sum of the digits of the largest number that is a product of three distinct prime numbers, each less than 20? | nvidia |
Given a circle with equation $(x+2)^2+(y-3)^2=8^2$, find the point on the circle that is furthest from the line $y=x-5$. | nvidia |
In the equation $x^2 + mx + n = 0$, determine $m$ such that the sum of the roots is equal to $10$. | nvidia |
Find all real solutions to the equation
$$\frac{x-1}{x^2+x+1} + \frac{x+1}{x^2-x+1} = \frac{x^2+1}{x^2-1}$$
that satisfy the inequality $x^2-1>0$. What is the product of all such solutions? | 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 largest perfect square that is a sum of two consecutive cubes. | nvidia |
Find a nonzero $p$ such that $px^2-12x+4=0$ has only one solution. | nvidia |
Let $a$ and $b$ be positive integers such that
\[\frac{1}{a} + \frac{1}{b} = \frac{1}{\gcd(a,b)}.\]Find the maximum value of $\frac{a}{b}.$ | nvidia |
Let $a$ and $b$ be two positive integers such that $\lcm(a,b)=24$.
Determine the largest possible value of $\lcm(5a,7b)$.
(Note: $\lcm$ stands for least common multiple) | nvidia |
Let $a$ and $b$ be positive integers such that $a^2-b^2=3(a+b)$. Find the maximum possible value of $\gcd(a,b)$. | nvidia |
A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the square to the area of the circle? | nvidia |
Consider a regular hexagon with side length 1, whose vertices are labeled A, B, C, D, E, and F. Three points are randomly chosen from the vertices of the hexagon. Let b be the smallest possible number such that it is always possible to select one pair of points from these three such that the distance between them is eq... | nvidia |
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