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Problem:
Consider a function $g(x)$ defined on the domain $\{x_1,x_2,x_3,x_4\}$, such that the graph of $y=g(x)$ consists of four points that form a convex quadrilateral of area $48$.
Find the area of the convex quadrilateral formed by the graph of $y = 3g\left(\frac{x}{2}\right)$. | nvidia |
Compute the value of $x$ such that $\sqrt[4]{x} + \sqrt[4]{16x} = 10$. | nvidia |
In the equilateral triangle $ABC$ with side length $24$, points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the circumscribed circle of triangle $ABC$. Find the maximum possible length of $BD+EC$. | nvidia |
For all positive integers $n$, what is the largest integer that always divides into the sum of $n$, $n^2$, and $n^3$? | nvidia |
A number has the property that the sum of its digits is equal to the number of its divisors. What is the smallest such number that is greater than 20? | nvidia |
Find the number of positive four-digit integers of the form $\_\_75$ that are divisible by 9 and have exactly two digits in common with the number $2021$. | nvidia |
What is the smallest positive integer that has exactly six positive integer divisors and is divisible by 7? | nvidia |
Find the sum of the values of $x$ that satisfy the equation $|2x|+5=21$. | nvidia |
Given that the image of the point with coordinates $(2,3)$ under the reflection across the circle with equation $(x-2)^2 + (y-3)^2 = 4$ is the point with coordinates $(6,5)$. Find the equation of the line that passes through the center of the circle and the midpoint of the segment connecting the point with its image. | 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 smallest positive integer $n$ such that the product of its digits is divisible by the number $n$ itself. | nvidia |
Let $p(x)$ be a polynomial with real coefficients such that
\[p(x) = p(\cos \theta) + p(\sin \theta)\]
for all real $\theta$. If $p(x)$ is a cubic polynomial and $p(0) = 1$, find $p(x)$. | nvidia |
Let $$x$$ be a real number such that $$\arctan x = 3\arctan \frac{x}{3},$$
then compute $$\cos(\arctan x).$$ | nvidia |
Let $C$ be a circle with a center at the origin and radius $5$. Find the value of $p$ such that the two circles
$$x^2 + y^2 = 5^2$$
and
$$(x-3)^2 + (y-p)^2 = 3^2$$
are tangent externally. | nvidia |
Consider a function $g(x)$ that maps the set of positive integers to the set of positive integers, satisfying the condition:
\[g(\gcd(m,n)) + g(\text{lcm}(m,n)) = g(m) g(n) + 2\]
for all positive integers $m$ and $n$. Determine the number of such functions. | nvidia |
The roots $x_1$ and $x_2$ of the equation $x^2-3ax+a^2=0$ are such that $x_1^2+x_2^2=1.75$. Determine $a$.
Let $P(x)$ be a polynomial with integer coefficients. Given that there are distinct integers $a, b, c, d$ such that $P(a)=P(b)=P(c)=P(d)=3$, find the minimum possible value of $P(0)$.
Do you want me to generate ... | nvidia |
Given that the foci of the ellipse $2x^2+5y^2+12x-20y+42=0$ are on the major axis, find the distance from the center of the ellipse to either of the foci. | nvidia |
A box contains three different colored balls: red, blue, and green. The balls are drawn one by one, and after each draw, the ball is returned to the box. The probability that the red ball is drawn is $\frac{1}{7}$, the probability that the blue ball is drawn is $\frac{2}{7}$, and the probability that the green ball is ... | nvidia |
The blue hyperbola shown is the graph of the equation $xy = 2x + 3y - 9$. Find the area of the rectangle with opposite vertices at the center of the hyperbola and on the asymptotes of the hyperbola.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+... | nvidia |
Consider the functions $f(x)$ and $g(x)$ with graphs as shown below.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=ti... | nvidia |
Find all possible values of the expression
\[\frac{a^2+b^2+c^2}{|a^2+b^2+c^2|} + \frac{ab+bc+ca}{|ab+bc+ca|}.\]Enter all possible values, separated by commas. | nvidia |
Let $P(x) = (1+x)^{2010}$. Find the sum of all the real coefficients of the expansion of $P(x)$ such that when you divide it by $x^2+1$, the remainder is equal to $0$. | nvidia |
Let $a$ and $b$ be real numbers. If the polynomial $x^2 + ax + b$ has exactly one real root and $a = 2b - 1$, find the value of the sum of all possible values of $b$. | nvidia |
In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is 56. Find the area of the polygon.
[asy]
unitsize(0.5 cm);
draw((3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(6,2)--(6,3)--(7,3)--(7,4)--(6,4)--(6,5)--(5,5)--(5,6)--(4,6)--(4,7)--(3,7)... | nvidia |
Compute the ordered pair of positive integers $(a,b)$ such that
\begin{align*}
2a^b&=b^a+3,\\
a^b+1&=b^a-2.
\end{align*} | nvidia |
Find the sum of the reciprocals of the roots of $x^2-13x+4=0$. | nvidia |
Determine the largest positive integer $n$ for which $n^2+13$ is divisible by $n+2$. | nvidia |
The sum of the squares of the divisors of the number 20! is equal to 950. How many positive integers less than 20 are there whose squares divide 20!? | nvidia |
How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999?$ For example, both $121$ and $211$ have this property.
$\mathrm{\textbf{(A)} \ }226\qquad \mathrm{\textbf{(B)} \ } 243 \qquad \mathrm{\textbf{(C)} \ } 270 \qquad \mat... | nvidia |
Let $f(x)=2x+1$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f(x^{-1})$. | nvidia |
Let $z$ be a complex number such that $|z|=1$ and $|z^4+z^2+1|=0$. Find $\left|\left(z^2+2\right)^3\right|$. | nvidia |
The graphs of three functions, $r(x)$, $s(x)$, and $t(x),$ are shown here on one set of axes: [asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.... | nvidia |
Alicia, Brian, Carlos, and David each randomly select a real number between 0 and 1, inclusive, and write it on a slip of paper. The numbers are then sorted in ascending order to form a list $L$. If $x$ is the number written by Alicia, let $P(x)$ be the probability that $x$ is either the second smallest or second large... | nvidia |
The solutions $x_1$ and $x_2$ of the equation
$$x^2-2x\cos\alpha+1=0$$
are such that
$|x_1|-|x_2|=\tan\frac{\alpha}{2}$.
Determine the value of $\alpha$. | nvidia |
If the point $(3,6)$ is on the graph of $y=g(x)$, and $h(x)=(g(x))^2$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? | nvidia |
Let $p(x)=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5$ be a polynomial with nonnegative real coefficients such that $p(1)=4$ and $p(0)=0$. Find the minimum value of $a_0+a_2+a_4$ as the coefficients $a_0,a_1,a_2,a_3,a_4,a_5$ vary over all nonnegative real numbers. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 2.$ Find the minimum value of
\[\frac{1}{a} + \frac{1}{b} + \frac{1}{c}.\] | nvidia |
Problem:
The sum of the squares of the solutions of the equation $x^2 - 12x + 36 = 0$ is $m$ times the sum of the solutions. Find $m$. | nvidia |
Find a prime number $p$ such that the digits of the number $p^2$ form an arithmetic progression. What is the smallest possible number of digits in $p^2$? | nvidia |
The domain of the function $f(x) = \arccos(\log_{k}(px))$ is a closed interval of length $\frac{1}{2024}$, where $k$ and $p$ are positive integers and $k>1$. Find the smallest possible value of $k+p$. | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2+kx+p=0$$
are such that
$x_1^2+x_2^2=0$.
Determine $k$. | nvidia |
Find the maximum value of $a^b$, where $a$ and $b$ are positive integers and $a + b < 10$. | nvidia |
Find all values of $b$ for which the points $(0,0,0),$ $(2,b,0),$ $(0,2,b),$ and $(b,0,2)$ are coplanar. | nvidia |
Let $p$ be a prime number greater than 3. Prove that there are at most two possible values of $a$ in the range $1<a<p-1$ such that $a^3 \equiv 1 \mod p$. | nvidia |
What is the largest positive integer $n$ such that $\sqrt[n]{2021}$ is an integer? | nvidia |
The function $f(x)$ is a periodic function of period $2\pi$ with domain $0\leq x\leq 2\pi$ and range $-1\leq f(x)\leq 1$. The graph of the function is shown below, where one unit on the graph represents $\pi/4$ units of $x$. For how many values of $k$, with $-1\leq k\leq 1$, does the equation $f(x)=k$ have exactly 4 so... | nvidia |
Let $a$ be a positive integer. Consider the sequence of real numbers $x_n$ such that $x_1=a$ and $x_{n+1}=x_n-\lfloor x_n\rfloor$ for all $n\geq 1$.
Find all possible values of $a$ such that $x_n=0$ for some $n\geq 1$. | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
Find the smallest three-digit prime number such that reversing the digits of the number produces a composite number. | nvidia |
Find the value of $k$ such that the equation $\sec (x^2+k) = x^4 + 2x^2 + 1$ has exactly one solution. | nvidia |
Determine the value of $a$ such that the points $(2, 3)$, $(-2a+4, a)$, and $(a-3, -1)$ are vertices of an equilateral triangle. | nvidia |
Determine the value of a such that $x^3+ax^2+7x-9$ is exactly divisible by $x-1$. | nvidia |
Given the equation
\[\frac{(x^2-2x+1)(2x^2+bx+3)}{x^2+2x-3} = x^2-4x+7,\]
find the constant $b$ such that the equation holds for all $x$ except $x = -3$ and $x = 1$. | nvidia |
Let $\sigma(n)$ denote the sum of the positive divisors of $n$, including 1 and $n$. Define $T(n)$ by $T(n) = \sigma(1) + \sigma(2) + \cdots + \sigma(n)$. Find the smallest positive integer $n$ such that $T(n)$ is divisible by $n$. | nvidia |
Let $x$, $y$, and $z$ be distinct real numbers that sum to $0$. Compute \[ \dfrac {xy+yz+zx}{x^2+y^2+z^2}. \] | nvidia |
There are four points that are $5$ units from the line $y=13$ and $13$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? | nvidia |
A circle is centered at the point $(a,b)$, and the tangent line to the circle at the point $(-4,3)$ is given by the equation $y=-x/3+5/3$. Given that the circle intersects the line $y=x$ at two points, and that the distance between these two points is $\sqrt{10}$, find the value of $a+b$. | nvidia |
A circle of radius $r$ is inscribed in a square with side length $s$. A hexagon with side length $x$ is inscribed in the circle, such that the center of the circle is also the center of the hexagon. If the ratio of the area of the hexagon to the area of the square is $\frac{3}{4}$, find the ratio of the length of the d... | nvidia |
Find the number of terms in the expansion of \((2x+y)^{12}\) that contain \(x^{10}\). | nvidia |
Compute the value of $x$ such that $\sin(\arctan x) = \frac{2}{3}$. | nvidia |
A certain regular tetrahedron has three of its vertices at the points $(0,1,2),$ $(4,2,1),$ and $(3,1,5).$ Find the coordinates of the fourth vertex, given that they are also all integers. | nvidia |
What is the smallest positive integer $n$ such that there exist at least 10 different $n$-digit positive integers that are divisible by 7 and have exactly three distinct digits? | nvidia |
Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $x+y+z=10$, $xy+yz+xz=25$, and $xyz=8$. Find the value of $x^2+y^2+z^2$. | nvidia |
Problem:
Find the smallest positive integer $k$ such that for all nonnegative integers $n$, $r_k(3n) = r_k(n)$.
(Note: $r_k(n)$ denotes the remainder when $n$ is divided by $k$) | nvidia |
Find the sum of the digits of the product of all the prime numbers that are factors of the numbers in the sequence $1!$, $2!$, $3!$, $4!$, $5!$, $6!$, $7!$, $8!$, $9!$. | nvidia |
Let $x_1$, $x_2$, $\ldots$, $x_n$ be a sequence of positive integers such that $\sum\limits_{i=1}^{n} ix_i = n!$ for some positive integer $n$. Find the smallest value of $n$ such that there exists a sequence of $x_i$ that satisfies the given condition and the following inequality:
\[x_1^2 + x_2^2 + \dots + x_{n}^2 \ge... | nvidia |
Find all real numbers $a$ for which there exists a nonzero, 2-dimensional vector $\mathbf{w}$ such that
\[\begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix} \mathbf{w} = a \mathbf{w}^T.\]Enter all the solutions, separated by commas. | nvidia |
Let $f(x)$ be a function that takes a positive integer as input and returns a positive integer as output. The function satisfies the following properties:
$f(1) = 1$
$f(ab) = f(a) + f(b)$ for all positive integers $a$ and $b$
Determine the value of $f(1728)$. | nvidia |
A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are... | nvidia |
A monic quartic polynomial has a root $1-2i$. If the product of its real roots is $25$ and the sum of its real roots is $-10$, find the quartic polynomial. | nvidia |
What is the product of the digits of the largest three-digit number whose digits are all distinct prime numbers? | nvidia |
The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real who... | nvidia |
A magic square is an array of integers such that the sum of the numbers in each row, column, or diagonal is the same. The numbers from 1 to 15 are arranged in a magic square, such that the sum of the numbers in each row, column, or diagonal is 15. How many different magic squares can be formed using the numbers from 1 ... | nvidia |
Find the number of values of $k$ for which the circle $x^2+y^2=k^2$ intersects the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ at exactly one point. | nvidia |
Let $x$, $y$, $z$, and $w$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}x^2+y^2&=&z^2+w^2&=&2023,\\ x^2y^2&=&z^2w^2&=&250000.\end{array}\]
If $T=xy+zw$, compute the value of $\lceil T\rceil$. | nvidia |
How many positive integers less than 100 have an odd number of positive divisors? | nvidia |
If the point $(3,6)$ is on the graph of $y=g(x)$, and $h(x)=(g(x))^2$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? | nvidia |
Two standard six-sided dice are rolled and the numbers on the upper faces are added. What is the probability that the sum is a prime number? | nvidia |
Find the smallest positive integer $n$ such that the sum of the proper positive integer divisors of $n$ is equal to the product of the proper positive integer divisors of $n$. | nvidia |
The sum of the digits of the number $2^{2005}$ is divided by $9$. Determine the remainder. | nvidia |
A bag contains ten balls, some of which are red and the rest of which are yellow. When two balls are drawn at random at the same time, the probability that both balls are red is $\frac{1}{15}$. How many balls in the bag are red? | nvidia |
In a right triangle, the length of the hypotenuse is 6 units and the sum of the squares of the lengths of the other two sides is 12 square units more than the area of the triangle. Find the length of the shorter leg. | nvidia |
Determine the number of subsets of the set $\{1, 2, 3, \dots, n\}$ such that the sum of the elements in each subset is less than or equal to $n^2$. | nvidia |
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 2 by 3 by 4 units. Given that the volume of this set is $\displaystyle {{m+n\pi}\over p}$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m+n+p$. | nvidia |
Find the value of $x$ such that $\frac{x^3-2x^2-5x+1}{x^2+1}=-\frac{13}{8}$. | nvidia |
Let $a$ and $b$ be positive real numbers such that
\[\frac{1}{a^2 + 1} + \frac{1}{b^2 + 1} = \frac{1}{2}.\]Find the minimum value of $\sqrt{a^2 + b^2}.$ | nvidia |
Compute $\sum\limits_{n=1}^{\infty} \dfrac{3n-1}{2^n}$, where the sum is taken over all positive integers $n$. | nvidia |
Given an obtuse triangle ABC with ∠A>90°, what is the minimum value of the sum of the squares of the three side lengths of triangle ABC if the three angle bisectors intersect at the circumcenter of triangle ABC? | nvidia |
Find the number of positive integers that are divisible by exactly three distinct prime numbers, given that the prime numbers are less than or equal to 10. | nvidia |
If the sum of the squares of two positive integers has 8 digits, then their product has at most how many digits? | nvidia |
Three positive integers have a sum that is $10$ times the sum of their reciprocals, and the sum of the squares of the smallest two integers is equal to the largest integer. Find the largest possible value of the product of the three integers. | nvidia |
Find the sum of all positive integers less than 1000 that are cubes of perfect cubes minus the sum of all positive integers less than 1000 that are squares of perfect squares. | nvidia |
In the triangle with side lengths $\sqrt{3x-1}$, $\sqrt{3x+1}$, and $\sqrt{2x+3}$, the angle opposite to the side of length $\sqrt{3x+1}$ is $60^\circ$. Find the value of $x$ such that the triangle is a right triangle. | nvidia |
Let $a_n$ be a sequence defined as the product of all positive integers from $1$ to $n$, inclusive, plus the sum of all positive integers from $1$ to $n$, inclusive. Find the smallest positive integer $n$ such that the greatest common divisor of $a_n$ and $a_{n+1}$ is greater than $1$. | nvidia |
Find the perimeter of the triangle with vertices $(2,3),$ $(4,6),$ and $(6,4),$ given that the line passing through the first two vertices has a slope of $3/2.$ | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=18$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=6$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 1$ and
\[f(xy + f(x)) = xf(y) + f(x)\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f \left( \frac{1}{2} \right),$ and let $s$ be the sum of all possible values of $f \left( \frac{1}{2} \right).$ Find $n \times s.$
A... | nvidia |
Determine the smallest positive integer $x$ such that the product of the squares of its digits is divisible by $120$. | nvidia |
Find the smallest perfect number greater than 10.
A perfect number is a positive integer that is equal to the sum of its proper divisors, excluding itself. | nvidia |
The function $f$ takes nonnegative integers to real numbers, such that $f(1) = 1,$ and
\[f(m + n) + f(m - n) = \frac{f(2m) + f(2n)}{2}\]for all nonnnegative integers $m \ge n.$ Find the sum of all possible values of $f(10).$ | nvidia |
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