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The function $f(x)$ satisfies
\[f(x^2 - y^2) = f(x) f(y)^2\]for all real numbers $x$ and $y,$ and $f(x) \neq 0$ for all real numbers $x.$ Find $f(4).$ | nvidia |
Consider the sequence of remainders when the powers of 3 are divided by 11. Prove that this sequence is periodic and find the length of the period. | nvidia |
Let $f(x)=\sin^2(x)+\cos^2(x)+\tan^2(x)$. Prove that for any real number $x$ we have $f(x)\geq 1$, and determine for which values of $x$ we have $f(x)=1$. | nvidia |
Given that the sum of the radius and the diameter of a circle is equal to the length of its diagonal, find the radius of the circle. | nvidia |
Two circles, one with radius $4$ cm and one with radius $6$ cm, share a common chord $AB$. The circles are positioned such that their centers and the points $A$ and $B$ are all collinear. What is the least possible integral number of centimeters in the distance between the points $A$ and $B$? | nvidia |
Find the sum of all positive integer divisors of the number that is the product of the smallest three different prime numbers. | nvidia |
A library has a collection of books, and each book is classified as either fiction or non-fiction. The library also keeps track of the number of books that are checked out and the number of books that are available on the shelves. If the number of fiction books that are checked out is between 20% and 30% of the total n... | nvidia |
Find all possible values for $y$ such that
\[\sqrt[3]{y+1}\cdot\sqrt[3]{y+2}\cdot\sqrt[3]{y+3}\cdot\sqrt[3]{y+4}=2.\] | nvidia |
Find the number of nonzero integers $m$ such that $\frac{1}{m^2} \geq \frac{1}{64}$ and $\frac{1}{m} + \frac{1}{2m} \leq \frac{1}{4}$. | nvidia |
Let $n$ be a positive integer such that $n^2-2n+1$ is divisible by a prime number $p$. Compute the sum of all possible values of $p$ less than $n$. | nvidia |
Consider a unit circle centred at the origin of a coordinate system.
A point on the circle is represented by the polar coordinates $(1,\theta)$.
There is a line passing through the point that forms an angle of $\pi/4$ with the positive direction of the x-axis.
Find the area of the triangle bounded by the radius to the... | nvidia |
Find the sum of all positive integer values of $b$ such that the quadratic equation $3x^2 + 5x + b = 0$ has two distinct real roots. | nvidia |
Find the number of distinct ways to arrange the letters of the word "STATION" such that the letters "T" and "I" are always together. | nvidia |
Find the value of the series $1+2x+3x^2+\cdots+nx^{n-1}+\cdots$ where $|x|<1$ and $x$ is a complex number. | nvidia |
The sum of the reciprocals of two consecutive positive integers is 13/84. What is their product? | nvidia |
Find the largest positive integer less than 300 that is both a multiple of 5 and has exactly three factors. | nvidia |
Let $p(x)$ be a polynomial with integer coefficients, and let $n$ be a positive integer. If the sum of the squares of the coefficients of $p(x)$ is equal to the sum of the squares of the coefficients of $p(x+n)$, and $p(0)=0$, find the largest possible value of the sum of the coefficients of $p(x)$. | nvidia |
Two positive integers $m$ and $n$ are chosen such that $m$ is the smallest positive integer with only two positive divisors and $n$ is the largest integer less than $100$ with exactly three positive divisors. What is $m+n$? | nvidia |
Let $f(n)$ be the sum of the proper positive integer divisors of $n$. For how many values of $n$ does $f(n)$ equal $n$, given that $2 \le n \le 50$? | nvidia |
Determine the number of ways to color the squares of a 3x3 chessboard with black and white colors such that no two adjacent squares have the same color. | nvidia |
In the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{2}$, both $x$ and $y$ are positive integers and $x < y$. Find the sum of all possible values for $y - x$. | nvidia |
Find the number of complex numbers $z$ that satisfy the inequality $|z^2 + 3z + 2| \leq 1$. | nvidia |
Find the smallest positive integer $n$ such that the sum of the digits of $n$ is congruent to $3 \pmod{5}$ and $n$ is divisible by 7. | nvidia |
Find the smallest positive integer $m$ such that
\[\sum_{k=1}^{m} \sin(k^3 \theta) \cos(k^2 \theta)\]
is an integer, where $\theta = \frac{\sqrt{2}}{2013}$. | nvidia |
A polynomial $P(x)$ of degree $5$ has the property that the roots of the polynomial $Q(x) = P(x) + (x-1)^5$ are all real and distinct. Find the probability that the roots of $P(x)$ are also all real. | nvidia |
Find the number of integers between 1 and 120 (inclusive) that are not coprime to 120. | nvidia |
In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of... | nvidia |
Find the largest real number $k$ such that
$$x^2 + y^2 + z^2 + 4 \ge kx + y + 2z$$ for all real numbers $x, y, z.$ | nvidia |
A regular polygon has a certain number of diagonals that is equal to the number of its sides. If a certain number of vertices are removed from the polygon, the number of diagonals of the new polygon is 3 times the number of its sides. What is the number of vertices that were removed from the original polygon? | nvidia |
The sum of the squares of the interior angles of a triangle is 540 degrees. What is the sum of the squares of the exterior angles of the same triangle? | nvidia |
Let $T(n)$ be the sum of the digits of a positive integer $n$, excluding any digits that are divisible by 3. For example, $T(1927) = 10$. For a particular positive integer $n$, $T(n) = 542$. Which of the following could be the value of $T(n+3)$?
$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 18\qquad\textbf{... | nvidia |
The sum of the squares of two consecutive positive even integers is 488. What is their product? | nvidia |
An equilateral triangle is inscribed in a circle with a side length of 6. Using the center of the circle as a vertex, a pyramid is constructed with the vertices of the triangle as its base. Compute the maximum height of the pyramid such that it is tangent to the circle, with no part of the pyramid extending outside the... | nvidia |
Find the smallest positive integer whose base-6 representation has exactly 5 digits and contains all 5 digits different. | nvidia |
Find the sum of the squares of the solutions to the equation $x^4-4x^3+5x^2-4x+1=0$, given that the sum of the reciprocals of the solutions is $\frac{3}{2}$. | nvidia |
Evaluate the product \[ (a-10) \cdot (a-9) \cdot \dotsm \cdot (a-1) \cdot a, \] where $a=2$. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a + b = 1.$ Find the set of all possible values of $\frac{1}{a^2} + \frac{1}{b^2}.$ | nvidia |
Let a parabola
$$y=(x-3)^2$$
and a line $l$ be given. The line $l$ intersects the parabola at two points $A$ and $B$ and forms two segments $AM$ and $BM$ with a point $M(0,1)$. Find the maximal length of the segment $AB$ if the line $l$ is such that the sum of the squares of the lengths of the segments $AM$ and $BM$ is... | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=7$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | nvidia |
How many positive integer divisors of $2006^{2006}$ are divisible by exactly 2006 positive integers? | nvidia |
Find the largest integer that can be expressed as a sum of two distinct prime numbers, each less than 20, and whose product is divisible by 15. | nvidia |
Let $f(x)$ be a polynomial such that
$$f(x^2)=x^2(x^2+1)(x^2+2),$$
for all real numbers $x$. Compute the value of $f(4)$. | nvidia |
Find the smallest positive integer that can be expressed as a sum of two perfect squares in three different ways, and express it in base 7. | nvidia |
Let $g : \mathbb{N} \to \mathbb{N}$ be a function such that
\[g(g(m) + n) = g(m) \cdot g(n) + g(m + n)\]for all positive integers $m$ and $n.$
Let $p$ be the product of all possible values of $g(5),$ and let $q$ be the number of possible values of $g(5).$ Find $p - q.$ | nvidia |
A sequence of positive integers has the property that the sum of the squares of any two consecutive terms is a multiple of the product of the two terms. What is the greatest possible value of the second term? | nvidia |
Let $\mathbf{A}$ be a matrix such that
$\mathbf{A}^2=\begin{pmatrix} 3 & 7\\ 2 & -2 \end{pmatrix}$
Find the matrix $\mathbf{B}$ such that
$\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}=\mathbf{A}^{-1}$ | nvidia |
Let $a$ be a real number such that $a>0$ and
\[\sum_{n=1}^{\infty} n \cdot \frac{a^n}{2^n} = 6.\]
Find $a$. | nvidia |
In a rectangular prism, the length, width, and height are integers such that the sum of the lengths of all edges is $240$ and the sum of the areas of all six faces is $1560$. Find the volume of the prism. | nvidia |
Let $x,$ $y,$ $z$ be positive real numbers such that $xyz = 8.$ Find the minimum value of $x + 2y + 4z.$ | nvidia |
Find the sum of the solutions of the equation $\frac{x}{x+1} = \frac{1}{x-1}.$ | nvidia |
Given the cubic equation $x^3+ax^2+bx+c=0$ where a, b, and c are integers and the equation has exactly two real roots, find the condition on a, b, and c under which both real roots are integers. | nvidia |
Find the smallest value of the natural number $n$ such that the sum of all the positive divisors of $n$ is greater than $2n$. | nvidia |
How many positive integer values of $x$ satisfy the inequality $x^3 \leq 2^x$? | nvidia |
Given the function $f(x)=2x^2-5$, find the value of $x$ such that $f^{-1}(x)=3$. | nvidia |
A polynomial function $f(x)$ is called "self-centered" if it satisfies the condition $f(f(x))=f(x)$ for all $x$ in its domain.
Find all quadratic functions $f(x)$ that are self-centered and have a non-zero constant term. | nvidia |
In a regular pentagon with side length 8, a circle is inscribed such that it is tangent to all five sides of the pentagon. A triangle is formed by connecting the center of the circle to two vertices of the pentagon. Find the area of this triangle, rounded to the nearest hundredth. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 5.$ Find the minimum value of
\[\frac{1}{a} + \frac{4}{b} + \frac{9}{c}.\] | nvidia |
Find the largest possible value of $n$ such that the polynomial $x^3 + 2x^2 - x + n$ is divisible by $x^2 + 3x + 2$. | nvidia |
Find the value of $x$ such that $\ds\f{(x-1)!x!}{(x+2)!}=\f{1}{20}$. | nvidia |
For how many ordered pairs of positive integers $(a,b),$ with $b<a\le 50,$ are both $\frac ab$ and $\frac{a+2}{b+2}$ integers? | nvidia |
A standard eight-sided fair die is rolled four times. The probability that the sum of the cubes of all four numbers rolled is a multiple of 8 is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | nvidia |
On the graph of the function $f(x)=\frac{1}{x}$, a triangle is drawn with its vertices at three distinct points. Two of these points are $(\frac{1}{3}, 3)$ and $(\frac{1}{2}, 2)$. Find the sum of the distances from the third vertex to the other two vertices if the slope of the line through $(\frac{1}{3}, 3)$ and the th... | nvidia |
How many three-digit numbers are there that have the property that the sum of their digits is a multiple of 5? | nvidia |
The function $g(x)$ satisfies
\[x^2g(y) = y^2g(x)\]for all positive real numbers $x$ and $y.$ If $g(8) = 12,$ find $g(2).$ | nvidia |
Square $ABCD$ has sides of length 1. Points $E$ and $F$ are on $\overline{BC}$ and $\overline{CD},$ respectively, so that $\triangle AEF$ is equilateral. A square with vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\overline{AE}.$ The length of a side of this smaller square is $\frac{a-\sqrt... | nvidia |
Given that $a$, $b$, and $c$ are positive integers, find the smallest possible value of $c$ such that there exist solutions to the system of equations
$$x \equiv y \pmod a,$$
$$2x \equiv z \pmod b,$$
$$3y \equiv c \pmod {ab},$$
subject to the constraint that $x$, $y$, and $z$ are integers. | nvidia |
Find the sum of all odd numbers in the interval [9, 55] that are divisible by 3. | nvidia |
What is the sum of all real numbers that are tripled when subtracted from their reciprocals? | nvidia |
Determine the value of $x$ such that $\cos x^\circ = \cos 236^\circ$. | nvidia |
What is the largest four-digit number that can be formed using only the digits from the first five rows of Pascal's triangle, with the restriction that no digit can be repeated more than twice? | nvidia |
The sum of the areas of two squares with sides of consecutive integers is 253. Find the side length of the larger square. | nvidia |
Given the complex numbers $z_1$, $z_2$, $z_3$, such that $|z_1|=|z_2|=|z_3|=\sqrt{2}$ and $z_1+z_2+z_3=0$, find the median of the set $\{|z_1|^2, |z_2|^2, |z_3|^2, |z_1+z_2|^2, |z_1+z_3|^2\}$. | nvidia |
Let $g(n)$ be the sum of the digits of $\frac{1}{7^n}$ that are to the right of the decimal point. What is the smallest positive integer $n$ such that $g(n) > 15$? | nvidia |
Determine the minimum value of the expression
\[2(a + x)\left(\sqrt{x^2 + b^2} - x\right),\]
in terms of positive real numbers $a$ and $b$. | nvidia |
Find the positive real value of $t$ that satisfies $|t+2i\sqrt{3}| |6-4i| = 26$. | nvidia |
A regular hexagon with side length 4 cm is inscribed in a circle. Find the radius of the largest circle that can be inscribed inside the hexagon. | nvidia |
Find the largest value of $x$ that satisfies the inequality $|x-3| + |x-5| < 4$. | nvidia |
What is the smallest positive integer that can be expressed as the sum of two different perfect squares in two different ways? | nvidia |
Find the sum of all constants $k$ such that the quadratic $x^2 + kx - 24$ can be factored in the form $(x+c)(x+d)$, where $c$ and $d$ are integers. | nvidia |
The sum of the squares of three consecutive odd positive integers is twice their product. What is the product of the three integers? | nvidia |
Find the sum of the positive whole number divisors of 243. | nvidia |
How many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\frac14$? | nvidia |
The difference between two numbers is 9, and the sum of the squares of each number is 153. What is the value of the product of the two numbers? | nvidia |
Determine the number of four-digit numbers with digits 1, 2, 3, and 4, such that no digit repeats and the number is divisible by 5. | nvidia |
A regular hexagon is inscribed in a circle. Let $V$ be the set of all vertices of the hexagon and $M$ be the set of all midpoints of the sides of the hexagon. A pair of elements, one from $V$ and one from $M$, are selected at random. What is the probability that the two chosen points are the endpoints of a radius of th... | nvidia |
A sequence of integers consists of $5$ blocks of consecutive integers, each block consisting of $7$ integers. The first block is $(a+1,a+2,...,a+7)$ and the last block is $(a+29,a+30,...,a+35)$. Determine $a$ if the sum of all integers in the sequence is equal to $1050$. | nvidia |
Compute the ordered pair of positive integers $(x,y)$ such that
\begin{align*}
x^y+1&=y^x,\\
2x^y&=y^x+7.
\end{align*} | nvidia |
Find the number of positive integers less than or equal to 200 that are multiples of 5 or 7, but not both. | nvidia |
In a triangle with vertices $(a, b)$, $(c, d)$, and $(e, f)$, the midpoints of the sides are connected to form a smaller triangle. If the coordinates of the midpoints are $(2, 3)$, $(4, 1)$, and $(6, 5)$, find the sum of the coordinates of the vertices of the original triangle. | nvidia |
A sequence is defined recursively by
$a_1=2021$,
$a_2=10000$,
$a_n=\frac{a_{n-1}+a_{n-2}}{2}$ for all positive integers $n$ with $n>2$.
Let
$a=\lim_{n\to \infty}a_n$. Find J(a, a, a). | nvidia |
Let $x$ and $y$ be real numbers such that
$2(x^2 + y^2) = x + y.$
Find the maximum value of $x - y.$ | nvidia |
The product of two consecutive two-digit positive integers has a units digit of 0. What are the possible values of the tens digits of these integers? | nvidia |
Find a positive integer $n$ such that the decimal $0.75$ can be expressed as the fraction $\frac{2n^2+1}{5n^2+2n+3}$. | nvidia |
Find the sum of the absolute values of the determinants of all $3\times3$ matrices consisting of the digits $1,2,3,4,5,6,7,8,9$, each used only once, if the digits in each row form a geometric sequence. | nvidia |
The set $A$ consists of positive integers that are multiples of $13$, while the set $B$ consists of positive integers that have $7$ as a factor. Find the least common multiple of all numbers that are elements of both sets $A$ and $B$. | nvidia |
A parabola with its vertex at the point $(0, -3)$ has a focus that lies on the line $y = 5$.
Determine the equation of the directrix of the parabola. | nvidia |
Palindromes are numbers that read the same backwards and forwards, like 5665. What is the least possible positive four-digit palindrome that is divisible by 3? | nvidia |
What is the number of different three-digit integers that can be formed using all the digits 1, 2, and 3 exactly once, such that the sum of the hundreds and units digits is a multiple of 4? | nvidia |
A polygon is inscribed in a circle such that each side is tangent to a smaller circle inside the larger circle. If the radius of the larger circle is twice the radius of the smaller circle, and the polygon has 12 sides, find the number of distinct chords of the larger circle that are tangent to the smaller circle. | nvidia |
The sum of the cubes of three consecutive integers is 216. Determine the middle integer of the three integers. | nvidia |
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