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Find the positive difference between the two values of $x$ that satisfy the equation $\displaystyle\sin{\left(2x+\frac{\pi}{4}\right)} = \frac{\sqrt{2}}{2}$. | nvidia |
In the equation
$$ax^2-5x+c=0$$
determine $c$ such that the sum of the roots be equal to $2a$. | nvidia |
The integer $n$ is the largest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\frac{n}{15}$.
| nvidia |
Determine the sum of all positive integers with three digits that are divisible by 2. | nvidia |
Find the largest three-digit integer that is equal to the sum of its own digits multiplied by a prime number. | nvidia |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | nvidia |
Find the product of all integers between $-10$ and $10$ that are $2$ more or $2$ less than a multiple of $5$. | nvidia |
Given a cube with an edge length of $5$ units, determine the area of the largest square that can be inscribed within the faces of the cube, taking into account the positions of the square's vertices on the edges of the cube. | nvidia |
In a graph, the vertices are labeled with vectors $\mathbf{v}_1, \mathbf{v}_2,..., \mathbf{v}_n$, where each vector $\mathbf{v}_i$ has a magnitude of 1. The edges of the graph are weighted with the squared Euclidean distance between the vectors corresponding to the endpoints of the edge. The graph is fully connected, m... | nvidia |
The value of
\[\frac{n}{2} + \frac{18}{n}\]is smallest for which positive integer $n$? | nvidia |
If $m$ is a positive integer such that $6m^2-m=393120$, find $m$. | nvidia |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}^n = \mathbf{I}.\] | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x + f(y)) = f(x) + f(y)\]for all real numbers $x$ and $y.$ | nvidia |
Find the value of $x$ such that $\arctan x = \frac{\pi}{4}$. | nvidia |
Find the smallest positive integer congruent to $25 \pmod{29}$ that can be expressed as the sum of a perfect square and a prime number. | nvidia |
Find the maximum value of
\[\frac{x + 2y + 3}{\sqrt{x^2 + y^2 + 1}}\]over all real numbers $x$ and $y.$ | nvidia |
The function $g(x)$ is defined on the real numbers such that $g(g(x)) = x^2$ and $g(x+1) = x^2 + 2x + 2$ for all real numbers $x.$ Find $g(3).$ | nvidia |
Find the smallest positive integer $n$ such that the product of $n$ and 678 is a palindrome when expressed in base 8. | nvidia |
Let $S$ be the sum of all positive integers $n$ that satisfy the equation $n\cdot n! = 4320m$, where $m$ is a positive integer. Find the value of $S$. | nvidia |
The product of two numbers is 240, and the sum of their reciprocals is 1/15. What is the value of the sum of the two numbers? | nvidia |
Let $a$ be a positive real number. Consider the set of all numbers of the form $a^m\cos m\theta$, where $m$ is a non-negative integer and $\theta$ is a rational multiple of $\pi$. Let $S$ be the set of all finite sums of such numbers. Determine the values of $a$ such that every real number can be represented in a uniqu... | nvidia |
The positive integers $a$ and $b$ are such that $4ab$ and $9a+4b$ have the same prime factors.
What is the least possible value of $a+b$? | nvidia |
Determine the value of $a$ such that the function $f(x) = a^x$ has a minimum value of $\frac{1}{3}$ for $x > 0$. | nvidia |
Compute the sum of all two-digit positive integers. | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=10$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | nvidia |
Let $x$ be a positive real number. Find the minimum value of $4x^5 + 5x^{-4}.$ | 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 |
Determine $k$ such that $\log_{3}7+\log_{3}k=1$. | nvidia |
Any six points are taken inside or on a cube with side length $2$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Then $b$ is what value? | nvidia |
Let $f(x)$ be a monic polynomial of degree 6, such that $$\lim_{x\to \infty}\frac{f(x)-x^6}{x^4}=0,$$and let there exist nonzero constants $a$ and $b$ such that
$f(a)=b$,
$f(2a)=-b$,
$f(3a)=b$, and
$f(4a)=-b$.
Find the limit $\ds\lim_{x\to -\infty} f(x)$. | nvidia |
What is the smallest value of $k$ such that the product of the first $k$ prime numbers exceeds the product of the first $10$ positive integers? | nvidia |
Let $g$ be a function such that $g(2)=1$ and $g(x^2)=2g(x)$ for all $x>0$. Find $g^{-1}(16)$. | nvidia |
Let $A$ be a set of positive integers such that the sum of all the elements in $A$ is 100, and the sum of the reciprocals of all the elements in $A$ is $\frac{1}{3}$.
Find the smallest possible number of elements in $A$. | nvidia |
Let set $\mathcal{B}$ be a 60-element subset of $\{1,2,3,\ldots,80\},$ and let $T$ be the sum of the elements of $\mathcal{B}.$ Find the number of possible values of $T$. | nvidia |
How many positive three-digit integers with each digit greater than 4 are divisible by 6?
A new problem inspired by this one:
Find the number of ways to arrange the digits 1, 2, 3, 4, 5, and 6 to form a six-digit number such that the sum of the digits in the odd places is equal to the sum of the digits in the even pl... | nvidia |
A hyperbola and a circle share a focus, and the center of the circle lies on the transverse axis of the hyperbola. The hyperbola and circle intersect at two points. Given that the equation of the circle is $(x-3)^2 + y^2 = 9$, find the product of the slopes of the tangents to the hyperbola at these two points. | nvidia |
Find the number of positive integers less than 5000 that have a remainder of 3 when divided by 12 and are also multiples of 17. | nvidia |
The two sides of a square are extended by the same distance on each side, and the area enclosed between the two extensions is 24 square units. Find the length of the side of the square. | nvidia |
Determine the sum of the squares of the roots of the polynomial $x^2 + 6x + 13$, given that one of its roots is $-3 + 2i$. | nvidia |
Find all possible functions $f:\mathbb{R} \to \mathbb{R}$ such that
\begin{align*}
f(x+2) &= f(x)+f(2), \\
f(x^3) &= (f(x))^3 + 2f(x^2)
\end{align*}for all real numbers $x.$ Enter the function(s) $f(x).$ | nvidia |
Find the largest possible value of $k$ such that the equation $2x^3 - 3x^2 + kx - 1 = 0$ has exactly one real root. | nvidia |
Let $q$ be the largest prime with 3005 digits. What is the smallest positive integer $n$ such that $q^3 + n$ is divisible by 20? | nvidia |
Let $a$, $b$, $c$, $d$, and $e$ be positive integers such that $$a^4+b^4+c^4+d^4+e^4=2010.$$ Determine the possible number of different values of $a$. | nvidia |
Find the equation of the line passing through the point $(0, -1)$ that is parallel to the line containing the vertices of the triangle with vertices $(2,3),$ $(4,-1),$ and $(6,5).$ | nvidia |
In $\triangle XYZ$ the ratio $XZ:ZY$ is $5:2$. The bisector of the exterior angle at $Z$ intersects $XY$ extended at $Q$ ($X$ is between $Q$ and $Y$). The ratio $QX:XY$ is:
$\textbf{(A)}\ 1:2 \qquad \textbf{(B)}\ 2:5 \qquad \textbf{(C)}\ 5:2 \qquad \textbf{(D)}\ 2:1 \qquad \textbf{(E)}\ 9:1$ | nvidia |
The function $f(x)$ is defined for all real numbers $x$ and satisfies
$$f(x)+2f(1/x) = x^2-3x+4.$$ Let $S$ denote the sum of all real values of $x$ such that $f(x)=f(1/x).$ Compute the integer nearest to $S.$ | nvidia |
For each integer $n\geq 4$, let $a_n$ denote the base-$n$ number $0.\overline{133}_n$. The product $a_4a_5 \dotsm a_{99}$ can be expressed as $\frac{m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$? | nvidia |
Given a set of $n$ non-negative real numbers $a_1,a_2,\dots,a_n$ such that $\sum\limits_{i=1}^{n} a_i = 1$, find the maximum value of $\prod\limits_{i=1}^{n} (1+a_i)$. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}$. Determine the largest positive integer $n$ for which there exists a unique integer $m$ such that $n=a+b+c+d$, $m=a+c$, and the given inequality holds. | nvidia |
How many positive three-digit integers have an even number of positive factors? | nvidia |
The equation $\sin^2 x + \cos^2 y = 1$ holds for every real $x$. Determine all real numbers $a$ such that the equality
$\sin^2 ax + \cos^2 x = 1$
holds for all $x$. | nvidia |
The sum of the squares of two positive integers $m$ and $n$ is equal to their product: $m^2+n^2=mn$. The difference of these numbers is 18. Find $m$. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $x+y+z=1$. Find the maximum value of $xyz\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$. | nvidia |
Let $f(x)$ and $g(x)$ be two functions that satisfy the conditions below.
Assume
$a,b,c,d$ be real constants such that $\frac{a}{c}$ and $\frac{b}{d}$ are relatively prime positive integers, with $\frac{a}{c}>\frac{b}{d}>1.$ There exist $u,v>1$ such that
$$f(x)=a\cdot\left(x-\frac{b}{d}\right)^2+c\cdot\left(x-\frac{a}{... | nvidia |
What is the smallest positive integer $n$ such that there exist two positive integers $x$ and $y$ whose sum is $n$ and whose product is divisible by $n$? | nvidia |
Let $a$ and $b$ be two positive real numbers such that $a^2+b^2=3ab$. Find the value of $a+b$ when $a$ and $b$ satisfy the condition $\frac{1}{a} + \frac{1}{b} = 3$. | nvidia |
Let $f(n)$ be the number of positive integer divisors of $n$. For example, $f(30)=8$ since $1,2,3,5,6,10,15,$ and $30$ are all divisors of $30$.
What is the smallest positive integer $k$ for which $f(k)=9$? | nvidia |
Let $a$ and $b$ be positive integers such that $a^2+b^2$ is a prime number. Determine the largest possible value of $a+b$. | nvidia |
Let $S$ be a set of real numbers such that for all $x$ and $y$ in $S$, the number $x^2y$ is also in $S$. Suppose that $f(x)$ is a function such that for all real numbers $x$ in $S$,
\[f(x^2y)=f(x)(1+xyf(y)).\]
Find a real number $y$ in $S$ such that for any value of $f(1)$ and any real number $x$ in $S$,
\[f(x^2y)=f(x)... | nvidia |
Consider the function $f(x)$ defined on the set of positive integers, where $$f(x) = \begin{cases} x^2 &\quad \text{if } x \text{ is a perfect square} \\ f(x-1) + f(x+1) &\quad \text{otherwise.} \end{cases}$$ Let $S$ be the set of all values $f(x)$ can take. Find the sum of all elements in $S$ that are less than or equ... | nvidia |
A function $f$ has the property that $f(x^2+2x-1)=\sqrt{x^3+5x^2-6x+1}$ for all real numbers $x$. What is $f(10)$? | nvidia |
Given that $-4\leq x\leq-2$ and $2\leq y\leq4$, what is the largest possible value of $\frac{x+y}{x}$?
Let's try to create a new problem that requires a different approach.
Find the maximum value of $\frac{x+y}{xy}$, given that $x$ and $y$ are positive integers satisfying $x+y=10$. | nvidia |
Find the number of ways to form a 5-digit number using the digits 1, 2, 3, 4, and 5, such that the sum of the digits in the odd places (thousands and hundreds places) is equal to the sum of the digits in the even places (tens and units places). | nvidia |
Find the number of ordered 6-tuples $(b_1, b_2, b_3, \dots, b_6)$ of positive integers, such that the cube of any number in the 6-tuple is equal to the product of the other 5 numbers. | nvidia |
Find a positive integer that is divisible by 12 and whose square root is a number between 15 and 15.5. | nvidia |
Let $P(x)$ be a polynomial with rational coefficients such that for any integer $n$ the value of $P(n)$ is the product of the digits of $n$. Let $Q(x)$ be the polynomial with the leading coefficient 1 and the constant term $-2010$ such that the value of $Q(n)$ is the number with the same digits as $n$ but in reverse or... | nvidia |
Given a sequence of regular polygons, where each polygon has one more side than the previous one, find the smallest number of polygons in the sequence such that the sum of their interior angles is a multiple of 360 degrees. | nvidia |
Point $Q$ lies on the circle $x^2+y^2=25$ and is 5 units from the point $(0,0)$. Find the sum of all possible slopes of the lines passing through $Q$ and the origin. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x+y=2$, $x^2 + y^2=5$. Determine $x^3+y^3$. | nvidia |
Find the sum of the values of x that are excluded from the domain of the equation:
$$y = \frac{2x}{3x^2 + 5x - 2}$$ | 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 |
Let $$ g : \mathbb{R} \rightarrow \mathbb{R} $$ be a differentiable function defined for all real numbers. Given that for all real numbers $ x$ we have $$ g(x) + g((x - 1) ^ 2) = x ^ 3. $$ Find $ g(10) $. | nvidia |
Two positive integers $m$ and $n$ are such that $mn+2m+n=33$. Determine $m$ and $n$.
Do you want me to generate another problem? | nvidia |
The area of a rectangle is $\frac{3}{4}$ of the area of a square with the same side length as the rectangle's width. If the area of the square is 12 square units, find the length of the rectangle. | nvidia |
Find the maximum value of
\[x^3 - y^3 - 3x^2y + 3xy^2\]
over all real numbers $x$ and $y$ such that $x^2 + y^2 = 1$. | nvidia |
Let $S$ denote the set of all positive integers $n$ such that $\frac{1}{n}$ is a terminating decimal. Find the smallest positive integer $k$ such that there exists a finite subset $T$ of $S$ with the property that every positive integer less than or equal to $k$ can be expressed as a sum of elements in $T$. | nvidia |
Find the sum of the largest one-digit prime, the largest two-digit prime, and the smallest three-digit prime that is a palindrome.
Note: A palindrome is a number that reads the same forwards and backwards. | nvidia |
Find the smallest number that can be expressed as the sum of four different cubes of prime numbers. | nvidia |
What is the sum of all positive integers less than 100 that are squares of perfect squares? | nvidia |
Compute the product of the two numbers whose sum is 100 and whose difference is 8. | nvidia |
Compute the sum of all numbers in the form $\frac{a}{b}$, where $a$ and $b$ are positive integers with no common factors and $1 \leq a < b \leq 10$. | nvidia |
Let $P$ be a point inside triangle $ABC$ such that triangle $APB$, triangle $BPC$, and triangle $CPA$ all have the same perimeter. Find the ratio of the area of triangle $ABC$ to the area of triangle $APB$. | nvidia |
Find the sum of all possible positive integer values of $c$ such that the quadratic equation $x^2 + 7x + c = 0$ has rational roots. | nvidia |
Find the sum of all positive integers $n$ such that $1.2n-4.4<5.2$. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a+b+c=3$. Determine the maximum value of $\dfrac{a}{1+ab}+\dfrac{b}{1+ab}+\dfrac{c}{1+ab}$. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 1.$ Find the maximum value of
\[\frac{x^2 + y^2 + z^2}{x + y + z}.\] | nvidia |
Find all the values of $x$ that satisfy the equation
\[\sin \left(\frac{x}{2}\right) + \cos \left(\frac{x}{2}\right) = \sqrt{3}.\] | nvidia |
The positive integers from 1 to 10 are written on 10 balls. Determine the number of ways to distribute the balls into two different boxes so that each box contains exactly 5 balls and no box contains two consecutive integers. | nvidia |
Point $P$ lies on the line $x= -3$ and is 10 units from the point $(5,2)$. Find the product of all possible $y$-coordinates that satisfy the given conditions. | nvidia |
How many three-digit integers with leading digit 1 have exactly two digits that are the same? | nvidia |
Find the sum of the distinct positive divisors of 66 that are not multiples of 3. | nvidia |
Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $abc = 64.$ Find the largest possible value of $b.$ | nvidia |
What is the smallest positive integer $n$ for which there exists a partition of the set of numbers $\{1,2,\ldots,2n\}$ into $n$ non-overlapping pairs, such that the sum of the numbers in each pair is a perfect square? | nvidia |
There are two pairs $(x,y)$ of real numbers that satisfy the equation $x+y = 3xy = 4$. Given that the solutions $x$ are in the form $x = \frac{a \pm b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are positive integers and the expression is completely simplified, what is the value of $a + b + c + d$? | nvidia |
Given that the vertices of a hyperbola are at (2, 1) and (6, 5), find the equation of its transverse axis. | nvidia |
In a given triangle ABC, the points D, E, and F are chosen on the sides AB, BC, and AC respectively, such that the lines AD, BE, and CF are concurrent. The areas of the triangles AFD, BDE, and CEF are equal. If the length of AD is 5 cm, find the area of the triangle ABC. | nvidia |
In a certain geometric sequence, the sum of the first $n$ terms is 5 times the sum of the first $\frac{n}{2}$ terms, where $n$ is a positive even integer. Find the common ratio of the sequence. | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $a>b>c$. The system of equations
$$ab+c \equiv b(a+c) \pmod{c(a+b)}$$
and
$$c(a+b)+1 \equiv ac+1 \pmod{ab}$$
has a unique solution $(a,b,c)$. Find $c^{-1} \pmod{a}$. | nvidia |
Find the number of even four-digit integers that are divisible by 20 and have a digit sum equal to 18. | nvidia |
Find the sum of the values of $y$ that satisfy the equation $|2y| - 2 = 10$. | nvidia |
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