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Find the smallest positive integer $k$ such that $k^3$ is divisible by 54 and $k^2$ is divisible by 72. | nvidia |
Consider a sequence of positive integers in which each term is the sum of the previous term and a fixed constant. If the sequence contains the number 2015, how many different values of the constant are possible such that the sequence also contains the number 2020? | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that the quantity $(a^2+b^2+c^2)^2-(a^4+b^4+c^4)$ is fixed and equals $k$, where $k>0$. Find the maximum possible value of $a^2b^2+b^2c^2+c^2a^2$. Express your answer in simplest form in terms of $k$. | nvidia |
Determine the value of x such that the equation $\sin(2x) = -\cos(x)$ is satisfied, given that x is an acute angle. | nvidia |
Find all real numbers $x$ such that \[|x^2-3x+2|<|x-1|.\] (Give your answer in interval notation.) | nvidia |
Given that the function $f(x)$ satisfies
\[f(ax) + f(bx) = f(cx)\]
for all real numbers $a$, $b$, and $c$ such that $a + b = c$ and $f(1) \neq 0.$ Find
\[\frac{f(3) + f(2)}{f(1)}.\] | nvidia |
The graph of the polynomial
$P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$
has five distinct $x$-intercepts, one of which is at $(0,0)$. Which of the following coefficients cannot be zero?
$\textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$ | nvidia |
A triangle is called balanced if the sum of the squares of its side lengths is equal to three times the product of its side lengths. A balanced triangle also must have integer side lengths. What is the sum of all the different possible perimeters of balanced triangles? | nvidia |
Find the number of positive integer solutions $(x,y)$ to the equation $2^x \cdot 3^y + 5^x \cdot 7^y = 10^3$. | nvidia |
What is the smallest positive value of $n$ such that the decimal representation of $n^2$ contains exactly three distinct digits, with no zeros? | nvidia |
Find the smallest positive integer whose square has the property that the sum of its digits is divisible by 13. | nvidia |
Compute the value of $x$ such that $\sqrt{2^x\cdot 3^{2x}} = 2^{x-1}\cdot 3^{x+1}$. | nvidia |
Given the function $f(x)=x^4-7x^3+6x^2-11x+1$, the value of $f(2)$ is 17 less than the sum of all the $x$-coordinates of the points where the graph of $y=f(x)$ intersects the line $y=x$. Find the sum of the squares of all the $x$-coordinates of these points. | nvidia |
In the equation
$$2^x + 3^x + 5^x = a^x$$
determine the value of $a$ such that the equation has exactly two distinct solutions. | nvidia |
In a triangle $ABC$, the side lengths $BC$ and $CA$ are both integers. A circle with center $O$ is tangent to side $AB$ and to the extensions of sides $BC$ and $CA$ beyond $B$ and $C$, respectively. Given that the circle's radius is $\frac{7}{4}$ and that $BC=8$, find the smallest possible perimeter of the triangle. | nvidia |
Suppose that $a$ is a nonzero constant for which the equation $ax^2+2x+a=0$ has two real solutions, $x_1$ and $x_2$, such that their reciprocal values have a difference of 2. Find $a$. | nvidia |
Find the smallest positive integer $k$ such that there exist exactly $k$ positive integers $n$ for which $2n$, $3n$, and $5n$ are all three-digit numbers. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x + y = 2.$ Find the maximum value of
\[\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}}.\] | nvidia |
Find the y-coordinate of the point on the y-axis that is equidistant from points C(0, -5) and D(3, 0). | nvidia |
The square of an integer is 182 greater than the integer itself. What is the sum of all integers for which this is true? | nvidia |
What is the largest positive integer that is not the sum of a positive integral multiple of 24 and a positive odd composite integer? | nvidia |
Find the maximum possible value of $x + y + z$ for positive integers $x$, $y$, and $z$ such that the sum of their reciprocals is equal to $\frac{3}{2004}$. | nvidia |
Find the number of non-similar rectangles with an area of 24 square units, where all sides have integer lengths. | nvidia |
Find the sum of the squares of the digits of the smallest positive integer that is divisible by the sum of its digits. | nvidia |
Find the smallest positive integer that is divisible by each of the numbers in the list $$12, 18, 24, 30, 36.$$ | nvidia |
Let $n$ be the smallest positive integer such that there exist positive relatively prime integers $a$ and $b$ such that $ab+a+b=n$. Prove that $n$ is prime and determine the number of distinct pairs of integers $(a,b)$ that satisfy the equation. | 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 |
Let there be three functions defined for all real numbers $x$:
$f(x) = x^2-2,$
$g(x) = x^2+2,$
$h(x) = x^2.$
These three functions satisfy $f(x)-h(x)=-2$ and $g(x)-h(x)=2.$
Is it possible to have three functions defined for all real numbers $x$ and satisfy the three conditions below?
(1) For all real numbers $x$, $... | nvidia |
Given a function $g: \mathbb{R} \to \mathbb{R}$ that satisfies the functional equation
\[g(x)g(y) = g(x + y) + g(xy)\]
for all real numbers $x$ and $y$, find the number of possible values of $g(2)$ and the sum of their squares. | nvidia |
What is the smallest positive integer $n$ such that $435n \equiv 917n \pmod{16}?$ | nvidia |
Consider the set of all integers from 1 to 100 that are not divisible by 3. What is the probability that a randomly chosen number from this set is also not divisible by 7? | nvidia |
Find the smallest positive $a$ such that $a$ is a multiple of $4$ and $a$ is a multiple of $14.$ | nvidia |
Find the smallest positive integer with exactly 12 positive integer divisors. | nvidia |
Find the maximum value of the expression
\[\begin{vmatrix} 1 & x & x^2 \\ x & x^2 & 1 \\ x^2 & 1 & x \end{vmatrix}\]
as $x$ ranges over all real numbers. | nvidia |
Six fair, six-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal? | nvidia |
Consider the sequence defined by $a_n = \frac{n!}{(n+2)!}$ for $n \geq 1$. Find the sum of the first $k$ terms of this sequence, where $k$ is a positive integer. | nvidia |
Consider the set of all positive integers less than or equal to $24$ that are relatively prime to $24$. Find the number of elements in this set. | nvidia |
Compute the sum of all terms of the sequence $\frac{16}{2^n}+\frac{12}{4^n}+\frac{9}{16^n}$, where $n$ takes all positive integer values. | nvidia |
Find the sum of all two-digit positive integers $n$ such that $\sqrt[3]{n}$ is an integer. | nvidia |
Compute the value of $x$ such that $\left(\sqrt{x^3+1}\right)^2-x^3=2$. | nvidia |
Consider the graph of $y=f(x)$, where $f(x)=\frac{x}{x^2+2}$. Find $k$ such that the graph of $y=f(x)+k$ contains the point $(6, -3)$. | nvidia |
Let $P(x) = x^3 + ax + 1$ be a polynomial with integer coefficients. Find the smallest positive integer $n$ such that $P(x)$ divides $x^n - 1$. | nvidia |
In a triangle, the length of the hypotenuse is $5\sqrt{2}$ units and one of the legs is $\sqrt{10}$ units. Find the area of the triangle. | nvidia |
What is the greatest value that can be obtained by multiplying three positive real numbers whose sum is 135? | nvidia |
Let $a_n = f(n)$ for $n \geq 0$, where $f(x)$ is a real-valued function such that $f(0)=1$, $f(1)=2$, and $f(x)=f(x-1)\cdot f(x-2)^2$ for $x\geq 2$. Determine the smallest positive integer $k$ such that $f(x)$ has no pole at $x=k+\frac{1}{2}$. | nvidia |
Find the 100th number in the sequence of positive integers that are divisible by 3 but not by 7. | nvidia |
Find the sum of all positive integers less than 100 that leave a remainder of 2 when divided by 5. | nvidia |
Count the number of positive integers smaller than $1{,}000{,}000$ that are powers of $3$ but not powers of $9$. You may find it useful to consider that $3^6=729$. | nvidia |
What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions? | nvidia |
Compute the value of $\cos(3\theta)$, given that $\theta$ is an acute angle such that $\tan(\theta) = \frac{3}{4}$. | nvidia |
The roots of the cubic equation
$$x^3-21x^2+35x-25=0$$
are all integers. If the least possible sum of the squares of the roots is $903$, find the value of the largest possible root. | nvidia |
Let $\alpha$ and $\beta$ be two distinct real numbers such that $\sqrt{2x+\alpha}+\sqrt{2x+\beta}=4x$. Determine the smallest possible value of $\beta$. | nvidia |
Consider the arithmetic sequence $a_n = 2n + 1$ and the geometric sequence $b_n = 3\cdot 2^n$. Find the value of $n$ for which the sum of the first $n$ terms of the arithmetic sequence is equal to the sum of the first $n$ terms of the geometric sequence. | nvidia |
Let $x$ and $y$ be positive real numbers such that $xy = 1.$ Find the maximum value of $\frac{x}{1 + x^2} + \frac{y}{1 + y^2}$. | nvidia |
The area of the smallest square that will contain a circle of radius 4 is what? | nvidia |
Find the number of 3x3 matrices whose entries are distinct integers from 1 to 9, such that the sum of the numbers in any row, column, or diagonal is a perfect square. | nvidia |
Find the smallest positive multiple of 6 that is not a "fortunate number", where a fortunate number is a positive integer that is divisible by the product of its digits. | nvidia |
In a regular hexagon, all the diagonals are drawn. Then around each diagonal, a circle is drawn such that its diameter is equal to the diagonal and the circles are not intersected pairwise. Determine the ratio of the area of all the circles to the area of the hexagon. | nvidia |
Let $S$ be a set of $n$ distinct positive integers such that no pair of numbers in $S$ has a common divisor other than 1. If a number is selected at random from the set of all possible products of two distinct elements of $S$, what is the probability that the selected number is not a perfect square? | nvidia |
Let $f(x) = x^3 + ax^2 + bx + c$ be a cubic function with real coefficients, where $a,$ $b,$ and $c$ are integers. Given that $f(-4) = 0,$ $f(-3) = -72,$ and $f(-1) = 84,$ find the sum of all integers $x$ such that $f(x) < 0.$ | nvidia |
How many four-digit numbers greater than 2999 can be formed such that the product of the middle two digits exceeds 5? | nvidia |
Let $x$ and $y$ be positive real numbers such that $x + y = 10$ and $xy = 4.$ Find \[\frac{x^2}{y} + \frac{y^2}{x}.\] | nvidia |
Given a circle with center at (3,-8) and passing through the point (-2,4), find the equation of the circle. | nvidia |
A circle is inscribed in a regular hexagon. If the area of the circle is $9\pi$, find the area of the hexagon. | nvidia |
Let $g(n)$ be the product of all the positive divisors of the integer $n$. Determine the smallest positive integer $n$ such that $g(n) = n^{\sqrt{n}}$. | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(xy) f(x/y) = (f(x) - f(y))^2 + 4xy f(y)\]for all non-zero real numbers $x$ and $y$. | nvidia |
Find the value of $a$ such that the infinite expression
$$\sqrt{a+\sqrt{a+\sqrt{a+\cdots}}}$$
is equal to $a+1$. | nvidia |
Let $a_1 , a_2 , \dots$ be a sequence for which $a_1=2$ , $a_2=3$, and $a_n=\frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$. What is $a_{2006}$? | nvidia |
In a convex pentagon, the lengths of the sides form a decreasing geometric sequence. How many such sequences are possible if the longest side is at most 30 cm and all side lengths are positive integers less than 15 cm? | nvidia |
Find the largest integer $n$ such that $n^2 + 3n - 5 < 0$. | nvidia |
Given that $\triangle PQR$ is a triangle where $P$ is at the origin, $Q$ is at $(2,0)$, and $R$ is at $(1,1)$, the Fermat point $F$ of $\triangle PQR$ has the property that $\angle PFR=\angle PQR$ and $\angle PRF=\angle PQF$. If $F$ is at the point $(x,y)$, where $x$ and $y$ are positive integers less than 4, what is $... | nvidia |
The sum of the squares of the distances from a point to the vertices of a square is equal to 200. The sum of the squares of the distances from the same point to the centers of the sides of the square is equal to 180. Find the area of the square. | nvidia |
Find all values of $t$ that satisfy the equation
$$e^t - e^{-t} = 2.$$ | nvidia |
Let $n$ be a positive integer such that the sum of the squares of $n$ and the next consecutive integer is $365.$ What is the value of $n$? | nvidia |
Find the range of the function $y=\arcsin(\log_{10}(x^2+1))$ for all real numbers $x$. | nvidia |
Find the base $b$ for which the number $225_b$ is a perfect square and can be expressed as the sum of two consecutive integers in base $b$. | nvidia |
Determine x such that $\sqrt[x]{2}+\sqrt[x]{8}=3$. | nvidia |
Determine the units digit of the sum of the squares of all the prime numbers between 20 and 120. | nvidia |
Let $f(x)$ be a function such that $f(0) = 1$ and
\[f(xy) = f \left( \frac{x^2 + y^2}{2} \right) + (x - y)^2\]for all real numbers $x$ and $y.$ Find $f(x).$ | nvidia |
Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2=1$ and $x+y+z=\sqrt{2}$. Let $n=\max\{xy,yz,xz\}$. Find the smallest possible value of $n$. | nvidia |
Simplify the expression $\sqrt[3]{\frac{27x^6}{64y^9}}$ where $x$ and $y$ are real numbers. | nvidia |
Find the sum of all numbers $x$ satisfying $x+25/x = 10.$ | nvidia |
Let $\{a_k\}$ be a sequence of integers such that $a_1=1$ and $a_{m+n}=a_m+a_n+mn,$ for all positive integers $m$ and $n.$ Find $a_{12}.$ | nvidia |
Compute the value of $x$ such that
$\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\cdots\right)\left(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots\right)=1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\cdots$. | nvidia |
Let a and b be positive real numbers such that a+b=2 and a^3 + b^3=56.
Find ab. | nvidia |
The product of two consecutive integers is 120 more than the sum of the two integers. What is the difference between the two integers? | nvidia |
Find the units digit in the sum of all natural numbers from 1 to 99, inclusive. | nvidia |
What is the sum of all the two-digit positive integers that are divisible by 7? | nvidia |
Find the smallest positive integer n such that $\dbinom{n}{5}$ is divisible by 11. | nvidia |
Consider a set of geometric objects - one circle, one parabola, and one straight line. Let's say that the parabola is $y=x^2$, the circle is $x^2+y^2=4$, and the line is $y=-x+4$. Find the sum of the coordinates of the points where the circle and the line intersect. | nvidia |
Let $a, b, c$ be positive real numbers such that $a + b + c = 1.$ Find the set of all possible values of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$ | nvidia |
How many positive integers less than 100 have an odd number of positive divisors? | nvidia |
In the unit cube, all the coordinates of the vertices are either 0 or 1.
Determine the number of equilateral triangles that can be formed using those vertices as the vertices of the triangles. | nvidia |
The function $f(x)$ satisfies
\[f(2^x) + xf(2^{-x}) = 1\]for all real numbers $x.$ Find $f(2).$ | nvidia |
Determine the largest possible value of $m$ such that the quadratic expression $3x^2 + mx + 72$ can be factored into the product of two linear factors with integer coefficients. | nvidia |
In the sequence of consecutive integers
$$\dots, a-4, a-3, a-2, a-1, a, a+1, a+2, a+3, a+4, \dots$$
all members whose index is a multiple of four are removed and the remaining integers are arranged in the sequence
$$c_1, c_2, c_3, c_4, c_5, \dots$$
Find the largest integer $n$ for which there exist integer values of $... | nvidia |
The sum of the squares of the roots of the equation $2x^2+5x-3=0$ is $p$, and the product of the squares of the roots is $q$. Find the value of $p+q$. | nvidia |
In the 2019 Math Competition, Team USA won 23 out of a possible 120 medals. Which one of the following is closest to the fraction of medals that they won?
$$
\frac{1}{5} \qquad \frac{1}{6} \qquad \frac{1}{7} \qquad \frac{1}{8} \qquad \frac{1}{9}
$$ | nvidia |
Determine the values of $a$ for which the function
$$f(x) = \frac{x^2+2ax+1}{x^2-2ax+2a}$$
is not defined, where $x$ is a real number. | nvidia |
Given that the point $(4,3)$ is on the graph of $y=g(x)$, there is one point that must be on the graph of $3y=\frac{g(3x)}3+1$. What is the product of coordinates of that point? | nvidia |
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