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What is the positive integer $M$ for which $$ 33^2 \times 77^2 = 11^2 \times M^2 \? $$ | nvidia |
Find the number of integer values of $n$ that satisfy the inequality $\sqrt{n^2+1}+\sqrt{n^2+2}<9$. | nvidia |
Find the minimum value of $\frac{1}{x} + \log x$, over all positive real numbers $x$. | nvidia |
Compute the number of ways to arrange 5 identical red balls and 5 identical blue balls in a row such that no two balls of the same color are adjacent. | nvidia |
Find the largest three-digit positive integer that is a sum of two consecutive perfect squares. | nvidia |
Find the largest positive integer that has exactly four distinct factors. | nvidia |
A convex polyhedron has a certain number of faces, each of which is either a triangle or a quadrilateral. The total number of edges in the polyhedron is 60, and the number of triangular faces is 5 more than the number of quadrilateral faces. Determine the number of possible values for the number of vertices in the poly... | nvidia |
Let $f$ be a non-constant polynomial such that
\[f(x - 1) + f(x) + f(x + 1) = \frac{[f(x)]^2}{2013x}\]for all nonzero real numbers $x.$ Find the sum of all possible values of $f(1).$ | nvidia |
How many three-digit positive integers exist, all of whose digits are 2's and/or 5's? | 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 |
What is the positive integer $N$ for which $$ 22^2 \times 55^2 = 10^2 \times N^2 \ ? $$ | nvidia |
A circle is inscribed inside a right triangle. The radius of the circle is 6 cm, and the lengths of the two legs of the triangle are in the ratio 3:4. Find the area of the triangle. | nvidia |
Find the positive real number(s) $y$ such that $\frac{1}{3}\left( 2y^2+5\right) = \left( y^2-30y-15\right)\left( y^2+20y+10\right)$. | nvidia |
Find the largest three-digit integer that is divisible by each of its distinct, non-zero digits. | nvidia |
In a 3x3 grid, nine different positive integers are arranged such that the sum of the numbers in each row, column, and diagonal is the same. What is the smallest possible value of this common sum? | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 1.$ Find the minimum value of
\[(x + 2y)(y + 2z)(xz + 1).\] | nvidia |
Consider a set of 100 consecutive integers, $\{n, n+1, n+2, \ldots, n+99\}$. After removing all the integers that are divisible by 3 and all the integers that are 1 more than a multiple of 5, how many integers remain in the set? | nvidia |
Let $f(x)$ be a quadratic function such that $f(x+1)-f(x-1)=8x$ for all real numbers $x$. Find $f(5)-f(3)$. | nvidia |
Determine the number of positive integers less than 1000 that have digits which increase from left to right. | nvidia |
The sum of two non-zero numbers $a$ and $b$ is equal to the product of their reciprocals. If the sum of $a$ and $b$ is equal to $6$, then find the product of $a$ and $b$ when $\frac{1}{a}-\frac{1}{b}=2$. | nvidia |
In the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,$ where $a$ and $b$ are positive constants, consider any chord $\overline{AB}$ that passes through the focus $F = (c,0),$ where $c > 0.$ Find the equation of the locus of all points $P$ such that angles $\angle APF$ and $\angle BPF$ are equal. | nvidia |
Find the smallest positive integer that has exactly 14 distinct prime factors. | nvidia |
Let $f(x)$ be a function that satisfies the following equation for all real numbers $x$:
$$f(ax+b) = f(29x),$$
where $a$ and $b$ are constants. Suppose that $f(x)$ is not a constant function and that $f(0) \neq f(1)$. Find the smallest possible value of $a$ such that $f(x)$ is a periodic function with period 1. | nvidia |
Let $f$ be a function that takes real numbers as input and returns their inverses, and let $g$ be a function that takes real numbers as input and returns their squares. Find $f(g(5))$. | nvidia |
A set of six positive integers has all of the following properties:
• The sum of the two smallest integers is 15,
• its median is 12,
• its average (mean) is 16.
What is the largest possible integer that could appear in the set? | nvidia |
Find the number of distinct positive integers that are divisors of at least one of the numbers $40$ and $72$, but not both. | nvidia |
What is the smallest positive integer with exactly 15 positive integer divisors? | nvidia |
Determine the real number $\alpha$ such that the function $\sin^2 x + \sin^2 (x + \alpha)$ has the maximum value of $\frac{3}{2}$. | nvidia |
Find the sum of the reciprocals of the positive divisors of 24, except 1 and 24 themselves, if you are given that the sum of the reciprocals of the positive divisors of 6 (again excluding 1 and 6) is $\frac{1}{2}$. | nvidia |
Find all real numbers $a$ such that the roots of the polynomial
$$x^3 - 6x^2 + 21x + a$$form an arithmetic progression and are not all real. | nvidia |
Find the sum of the squares of all positive integers that are divisible by 4 and less than 20. | nvidia |
Suppose $x$ and $y$ are integers such that $xy+5x+4y=-5$. Find the greatest possible value of $y$. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(2) = 2$ and
\[g(xy + g(x)) = xg(y) + g(x)\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g \left( \frac{1}{3} \right),$ and let $t$ be the sum of all possible values of $g \left( \frac{1}{3} \right).$ Find $m \times t.$ | nvidia |
Find the largest prime number that can be expressed as the sum of three distinct perfect squares. | nvidia |
A hexagon is inscribed in a rectangle so that the vertices of the hexagon bisect the sides of the rectangle. The perimeter of the rectangle is 120 centimeters. What is the number of square centimeters in the area of the hexagon? | nvidia |
A quadrilateral $ABCD$ is inscribed in a circle of radius $r$ so that its diagonals $AC$ and $BD$ intersect at right angles. If $AC=s$ and $BD=t$, then, for all permissible values of $s$ and $t$:
$\textbf{(A)}\ s^2+t^2\le4r^2 \qquad \textbf{(B)}\ s^2+t^2=4r^2 \qquad \textbf{(C)}\ s^2+t^2 \ge 4r^2 \qquad\\ \textbf{(D)}... | nvidia |
In an isosceles triangle, the angle bisector of the vertex angle intersects the opposite side at a point that divides it into segments of lengths 15 and 20. If the perimeter of the triangle is 160, find the area of the triangle. | nvidia |
The value of the repeating decimal $0.\overline{11}$ can be expressed as a fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Determine the sum of the numerator and denominator of this fraction. | nvidia |
What is the least positive integer that is divisible by three distinct primes? | nvidia |
Determine all possible values of $n$ such that the least common multiple of $n$, $n+1$, and $n+2$ is equal to their product. | nvidia |
How many positive three-digit integers with each digit less than 7 are divisible by 5 and have exactly two digits that are prime numbers? | nvidia |
Find the largest area of a rectangle inscribed in a semicircle with radius 100, given that the sides of the rectangle are parallel to the diameter of the semicircle. | nvidia |
Find the point on the graph of the equation $y=4x$ that is equidistant from the points $C(-4,0)$ and $D(0,6)$. | nvidia |
The graph of \[\frac{(x-2)^2}{3^2} - \frac{(y+1)^2}{2^2} = 1\] represents a hyperbola. Find the equation of the circle that passes through the vertices of the hyperbola and has its center at the center of the hyperbola. | nvidia |
Find the smallest positive four-digit number that is divisible by each of the four smallest prime numbers. | nvidia |
Find the smallest value of $b$ such that the equation $\frac{2\sqrt{b^2+x^2}-b^2-x^2}{\sqrt{b^2+x^2}+1}=1$ has exactly one solution for $x$. | nvidia |
Find the greatest $a$ such that $\frac{7\sqrt{(2a)^2+(1)^2}-4a^2-1}{\sqrt{1+4a^2}+3}=2$. | nvidia |
In the equation $y=x^3+ax^2+bx+c$, determine $a$ such that the sum of the squares of the roots of the equation is equal to 21, given that the sum of the roots is 6 and the sum of the products of the roots taken two at a time is 11. | nvidia |
Find the smallest integer $n$ such that the product of all prime factors of $n$ is equal to $105$. | nvidia |
Find the number of positive integers less than or equal to 1000 that are divisible by 3 and have exactly two digits that are the same. | nvidia |
Let $a$ and $b$ be two positive integers such that $a\geq 2$ and the base-$a$ representation of $b$ consists only of 1's. What is the smallest possible value of $a$ for which $b$ is divisible by the number formed by the base-$10$ representation of $b$? | nvidia |
Let $f(x)$ be the function defined as the sum of the digits of a positive integer $x$. For example, $f(1729)=1+7+2+9=19$. The $\textit{order}$ of an input $x$ with respect to $f$ is the smallest positive integer $m$ such that $f^{(m)}(x)=f^{(m+1)}(x)$. Find the order of $496$ with respect to this function $f$. | nvidia |
Suppose $x$ and $y$ are integers such that $xy+5x+4y=-5$. Find the greatest possible value of $y$. | nvidia |
Find the number of positive integers whose cube roots are within the range of 10.2 and 10.3. | nvidia |
Let $a, b, c, d$ be complex numbers such that $|a|=|b|=|c|=|d|=5$. Find the possible values of $|a+b+c+d|$. | nvidia |
In the triangle $ABC$ the ratio of the length of the side $AB$ to the length of the side $BC$ is equal to the ratio of the length of the side $AC$ to the length of the side $AB$ and is equal to the ratio of the length of the side $BC$ to the length of the side $AC$. Find the cosine of the angle $B$. | nvidia |
Determine the largest power of 10 that divides the product of all integers from 1 to 1000. | 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 |
Compute the number of ways to arrange the letters in the word "committee" such that the vowels are together. | nvidia |
Find the number of positive integers less than 10^6 that have exactly 3 digits 1 and exactly 2 digits 0, and whose digits add up to 15. | nvidia |
Determine the maximum possible value of the product
\[\frac{a}{2b} \cdot \frac{b}{4c} \cdot \frac{c}{8a},\]where $a,$ $b,$ and $c$ are positive real numbers. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $x^2+y^2+z^2=1$. Find the minimum value of
\[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{xyz}.\] | nvidia |
In the equation $x^2 + y^2 = 1$, where $x$ and $y$ are real numbers, determine the value of $x$ when the point $(x, y)$ on the unit circle is $120^\circ$ counterclockwise from the point $(1, 0)$. | nvidia |
How many three-digit positive integers have at least two digits the same? | nvidia |
Let A be a 2 x 2 matrix whose elements are all positive integers and whose determinant is 3. What is the smallest possible value of the sum of all elements of A? | nvidia |
Let $g(x)$ be a function defined for all positive real numbers $x$ such that
\[g(xy) = g(x) + g(y)\]for all positive real numbers $x$ and $y,$ and $g(x) \neq 0$ for all positive real numbers $x.$ Find $g(2).$ | nvidia |
Given that $2 + 3i$ is a root of the polynomial $x^2 + ax + b$, find the value of $b$ if the sum of the roots of the polynomial is equal to the product of the roots. | nvidia |
Find all real numbers $a$ such that the area of the parallelogram formed by the vectors $a \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ -4 \end{pmatrix}$ is equal to 15 square units. Enter all the solutions, separated by commas. | nvidia |
A rectangle is inscribed in a right triangle with legs of lengths 6 and 8, such that one vertex of the rectangle is at the right angle of the triangle, and the opposite vertex lies on the hypotenuse. Determine the maximum possible area of the rectangle. | nvidia |
Find the number of ordered triples $(x,y,z)$ of real numbers such that $x + y = 2$ and $xy - z^2 = 1.$ | nvidia |
Determine the values of the positive integer $m$ for which the quadratic equation $mx^2+12x+m=0$ has rational solutions. | nvidia |
Determine the sum of all single-digit replacements for $z$ such that the number ${24{,}z38}$ is divisible by 6. | nvidia |
In the equation $x^2+y^2=36$, determine the value of $k$ such that the line $y=kx$ is tangent to the circle. | nvidia |
Find the smallest positive integer $m$ such that $m^2 \equiv 34399 \pmod{15}$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x + y = 10.$ Find the minimum value of $\frac{1}{x} + \frac{1}{y}.$ | nvidia |
Find the largest three-digit integer whose digits are distinct and form a geometric sequence, but with the additional constraint that the middle digit is equal to the average of the other two digits. | nvidia |
The numbers $2468$, $1817$ and $2134$ have something in common: each is a $4$-digit number beginning with $2$ that has exactly two digits with a difference of $1$. How many such numbers are there? | nvidia |
Let $\overline{CD}$ be a diameter of a circle with diameter 2. Let $E$ and $F$ be points on one of the semicircular arcs determined by $\overline{CD}$ such that $E$ is the midpoint of the semicircle and $CF=\frac{2}{3}$. Point $G$ lies on the other semicircular arc. Let $h$ be the length of the line segment whose endpo... | nvidia |
A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is t... | nvidia |
Find the number of integers that are divisors of at least one of the two numbers, $24$ and $30$. (A divisor of a number is an integer that can divide the number without a remainder. For example, the divisors of $6$ are $1, 2, 3,$ and $6$.) | nvidia |
Find the value of $x$ such that $\frac{x}{3} + \frac{2}{5} = \frac{3}{2}$. | nvidia |
The function $g(x)$ is defined on the real numbers such that $g(g(x)) = 2x^2 + 1$ and $g(x^2 + 1) = x^2 - 1$ for all real numbers $x.$ Find $g(5).$ | nvidia |
Find the only positive real number $y$ for which $\displaystyle \frac{y-3}{7} = \frac{3}{y-7}$. | nvidia |
Given that $a$, $b$, and $c$ are positive integers such that $a < b < c$, find the largest possible value of $c$ such that $c \leq 1000$ and the product $$(a^3 + b^3 + c^3) \cdot [(a+1)^3 + (b+1)^3 + (c+1)^3]$$ is a perfect cube. | nvidia |
Find the least positive integer $n$ such that $5616n$ is a multiple of $11609$. | nvidia |
In the figure below, $ABCD$ is a rectangle with $AD=4$ and $CD=5$. The points $E$, $F$, and $G$ are on the sides of the rectangle such that $\triangle EFG$ is equilateral. What is the area of $\triangle EFG$?
[asy]
fill((0,1)--(4,0)--(2,5)--cycle,lightgray);
for(int i=1; i < 5; ++i){
for(int k=1; k < 4; ++k){
draw((... | nvidia |
The geometric mean of an even number of consecutive positive integers is $y$. Find the product of the smallest and largest of the integers in terms of $y$. | nvidia |
The expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $1:2:3$ that can be written in the form\[{n\choose k} : {n\choose k+1} : {n \choose k+2}\]Find the sum of all possible values of $n+k$.
| nvidia |
We define a function $f(x)$ such that $f(11)=34$, and if there exists an integer $a$ such that $f(a)=b$, then $f(b)$ is defined and
$f(b)=3b+1$ if $b$ is odd
$f(b)=\frac{b}{2}$ if $b$ is even.
What is the smallest possible number of integers in the domain of $f$? | nvidia |
Find the sum of all $y$ such that the expression $\frac{y^2-3y-2}{y^2-3y+2}$ is undefined. | nvidia |
Find the largest base $b$ such that the three-digit number $123_b$ can be expressed as the sum of two two-digit numbers in base $b$. | nvidia |
Find the sum of the positive factors of 30 that are multiples of 3. | nvidia |
Determine the value of $x$ that satisfies $\sqrt[5]{x\sqrt{x^3}}=3$. | nvidia |
Find the smallest positive integer such that when its digits are read in reverse order, the resulting number is a multiple of the original number. | nvidia |
A circle of radius $r$ centered at $(0,b)$ intersects the line $x=4$ at $P$ and $Q$ with $P$ lying on the $y$-axis. If the area of the region bounded by the line $x=4$, the $x$-axis, and the circular arc $PQ$ is twice the area of the circular sector $PQO$, find $b$ in terms of $r$. | nvidia |
Find the maximum value of $g(x) = \cos^2(2x) + \sin(4x)$ over all real numbers $x$. | nvidia |
Given that the quadratic equation $ax^2+bx+c=0$ has roots $2$ and $-3$, find the value of $\frac{b^2+c^2}{a^2}$. | nvidia |
A scalene triangle is projected onto the three faces of a unit cube such that the area of its projections on each face is equal to the area of the corresponding face of the cube. Find the perimeter of the triangle. | nvidia |
In a 4x4 grid, each cell is filled with one of the numbers 1, 2, 3, or 4. For each of the 16 cells, a sum of four numbers is computed, where the four numbers are the numbers in the cell and its three adjacent cells (up, right, and down). What is the minimum possible value of the sum of these 16 sums? | nvidia |
What is the largest whole number that must divide the sum of the squares of any three consecutive positive integers? | nvidia |
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