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Let f(x) be a real-valued function defined on the set of positive real numbers such that
f(x)+ f(y)= f(\frac{x+y}{xy})
for all positive real numbers x and y. Find all possible values of f(1/2). | nvidia |
Find the sum of all numbers $y$ such that $y + \frac{y^2}{y-1} = 3$. | nvidia |
When the digits in the number $2005$ are reversed we obtain the number $5002,$ and $5002 = a \cdot b \cdot c$, such that $a$, $b$ and $c$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$ and $p_3$ such that $p_1 + p_2 + p_3 = a+b+c$? | nvidia |
Let $p$ be the number formed by writing all positive integers from 1 to 100 in a row, and $q$ be the number formed by writing all positive integers from 1 to 50 in a row.
What is $\gcd(p,q)$? | nvidia |
Determine the value of $a$ such that the product $3^a \cdot 5^a$ can be expressed as some integer to the power of $2a-3$. | nvidia |
Consider the operation $\nabla$ defined by $a\nabla b=ab^2-b+a$. Determine the value of $x$ such that $x\nabla x=x\nabla (2x)$. | nvidia |
Consider a set of four spheres in 3D space, each with a different radius. The spheres are positioned such that each sphere is tangent to exactly three other spheres. Determine the maximum number of points where at least three spheres intersect. | nvidia |
Let $a_n = \frac{1}{n^2}$ for all positive integers $n$. Let $X$ be the sum of all terms of the form $a_n$ where $n$ is not an even multiple of 5, and let $Y$ be the sum of all terms of the form $a_n$ where $n$ is an even multiple of 5.
Determine $\frac{X}{Y}$. | nvidia |
Let $x,y,z$ be positive integers such that $xyz=1000$. Let $\sigma$ denote the sum of the squares of the reciprocals of $x,y,z$, that is $\sigma=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$. Determine the minimum possible value of $\sigma$. | nvidia |
Find the largest possible value of $\frac{x}{y}+\frac{y}{x}$ if there are 25 positive numbers whose product is 1, and for any two adjacent numbers $x$ and $y$, their sum is greater than 1. | nvidia |
Given a semicircle with diameter $\overline{AB}$, let $\overline{MN}$ be a chord that intersects the diameter at point $O$, where $AO:OB = 2:3$. Let $Q$ be the midpoint of $\overline{AB}$, and let $\overline{PQ}$ be perpendicular to $\overline{MN}$, intersecting it at point $P$. The ratio of the length of $\overline{PQ... | nvidia |
The sum of the squares of two consecutive positive integers is 365. Determine the smaller integer. | nvidia |
Let $x$ be a real number such that $x^2 - x - \lceil{x}\rceil = 0$. Find the value of $\lceil{x}\rceil$. | nvidia |
Solve the equation $1-x^2=0.\overline{x}$ where $x$ is a positive number between 0 and 1. | nvidia |
Find the sum of all integer values of $x$ such that $x^2$ is strictly between $100$ and $196$. | nvidia |
Prove that $\sum_{k=0}^{n}\binom{n}{k}x^ky^{n-k}=(x+y)^n$, and then use this formula to find the value of $\sum_{k=0}^{49}\binom{99}{2k+1}$. | nvidia |
Compute the product of the roots of the equation $x\sqrt{x} - 2x + 5\sqrt{x} - 3 = 0,$ given that all of the roots are real and nonnegative. | nvidia |
Let $p$ be a prime number and $a,b,c,d,e$ be five nonnegative integers, and consider a congruence in the form $x^2+y^2+ax+by+c\equiv 0\pmod{p}$. If there exists a pair $(x_0,y_0)$ with $0\le x_0,y_0<p$ such that $x_0^2+y_0^2+ax_0+by_0+c\equiv 0\pmod{p}$, then there exist other $d-1$ such pairs with both $x$ and $y$ in ... | nvidia |
Find the point in the $xy$-plane that is equidistant from the points $(0,3,0),$ $(3,2,1),$ and $(2,-1,0).$ | nvidia |
A square sheet of paper has a side length of $x$ cm, where $x$ is a positive integer. If two opposite corners of the sheet are folded so that they meet at the midpoint of the hypotenuse of a right triangle formed by the sheet, and the length of the hypotenuse is $y$ cm, where $y$ is also a positive integer, how many po... | nvidia |
Find the intersection points of the graphs of the equations $y=|x^2-4x+3|$ and $y=x^2-6x+9$, and determine which of these points lie on the graph of the equation $y=\sqrt{x^2-6x+9}$. | nvidia |
A bookshelf has 5 shelves, and the owner wants to place a certain number of books on each shelf such that the total number of books is 21. The number of books on each shelf must be a positive integer, and the owner wants to maximize the number of shelves that have a different number of books. What is the maximum number... | nvidia |
Let $S$ be the set of all functions $g : \mathbb{Z} \to \mathbb{Z}$ such that
$$g(m+n)g(m-n) = (g(m)+g(n))^2-4mn\cdot g(n)$$for all integers $m$ and $n$. Find the largest subset of $S$ such that for any two functions $g_1$ and $g_2$ in this subset, there exists an integer $k$ such that $g_1(k) \neq g_2(k)$. | nvidia |
Find the largest integer k such that k^3 is less than 127. | nvidia |
The average of a set of 5 numbers is 12. The numbers in the set are 6, 12, x, x+2, and 15. What is the largest possible value of x if all numbers in the set are positive integers? | nvidia |
Find the smallest positive integer $n$ such that $\lfloor \log_{10} (n) \rfloor + \lfloor \log_{10} (n^2) \rfloor = 12$. | nvidia |
Determine the value of $x$ such that the equation $\dfrac{2^{x+1}+2^{x-1}}{2^x+2^{-x}}=3$ holds true. | nvidia |
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Find the maximum value of the expression
\[\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{abc}.\] | nvidia |
The first three terms of an arithmetic sequence are $a$, $\sqrt[3]{b}$, and $c$, where $a$, $b$, and $c$ are positive integers. Find the largest possible value of $b$ such that the sequence has a common difference of $\sqrt[3]{7}$. | nvidia |
Determine all continuous functions $f(x)$ such that
$$f(x+1)+f(x^2)=f(x)+f(1)$$
holds for all real numbers $x$. | nvidia |
Let $a$ and $b$ be positive integers such that $a^2 + b^2 = 125.$ Find the maximum value of $a + b.$ | nvidia |
Find the sum of all real solutions to the inequality $|x^2-4| < |x^2-7| < 2$. | nvidia |
Find the equation of the line that passes through the point $(1,1)$ and is perpendicular to the line that is tangent to the curve given by the polar equation $r=2\sin(2\theta)$ at the point $(2, \frac{\pi}{4})$. | nvidia |
Find the equation of the line that passes through the points $(2,7)$ and $(5,10)$, and determine the point on this line that is closest to the origin. | nvidia |
A complex number $a+bi$ is such that $a^2+b^2=25$ and $a+b=6$. Determine $a-b$. | nvidia |
Let \(f(x)\) be a polynomial function such that \(f(0)=3\) and \(f(x^{2})=(x-1)^{2}f(x)\) for all real \(x\). Determine \(f(1)\). | nvidia |
Let $a, b, c$ be positive real numbers such that $a^2 + b^2 + c^2 = 18.$ Find the maximum value of $\sqrt[3]{abc}.$ | nvidia |
Find the sum of all real values of $m$ for which $\frac{3}{5x} = \frac{m - 2x}{9}$ has exactly one real solution. | nvidia |
Find the value of the expression
$$\frac{3^{10} \cdot 9^5}{27^3} \cdot \left(\frac{3^2}{9}\right)^{-2}$$ | nvidia |
Determine $k$ such that the graph of the equation
$$x^2+y^2+2\sqrt 3 x + 2y -3xy +k=0$$
represents two intersecting lines. | nvidia |
Find all values of $b$ such that $b+\lceil b\rceil=17.8$. | nvidia |
Let $b$ and $c$ be real numbers. If the polynomial $x^2+bx+c$ has exactly one real root and $b=c+1$, find the value of the product of all possible values of $c$. | nvidia |
Find all real numbers $x$ that satisfy the inequality $\ds\frac{x^2-4x+3}{x^2+2x-15} \ge 0$. | nvidia |
$n$ fair dice are simultaneously rolled. The probability that at least $n-1$ of them show 6 is $\frac{11}{36}$. Find $n$. | nvidia |
Determine the value of $a$ such that the quadratic equation $x^2 + 2ax + 2a = 0$ has two solutions, one of which is thrice the other. | nvidia |
Find the least positive integer such that when multiplied by 17, the result is divisible by 7 and 11. | nvidia |
Find the number of positive integers $n$ less than or equal to 100 such that the decimal representation of $\frac{1}{n}$ has a non-repeating part of length 1. | nvidia |
Compute the sum of the series $$\sum_{n=1}^{\infty} \frac{\sin(n\pi/2)}{n^2}.$$ | nvidia |
Find the largest product obtainable from two integers whose sum is 344. | nvidia |
Let $H$ be the set of polynomials of the form $$ Q(z)=z^m+d_{m-1}z^{m-1}+\cdots+d_2z^2+d_1z+25, $$where $ d_1,d_2,\dots, d_{m-1} $ are integers and $Q(z)$ has distinct roots of the form $c+id$ with $c$ and $d$ integers. How many polynomials are in $H$? | nvidia |
Find the least common multiple of 12, 15, and 18.
This problem requires a different approach to solve, as it involves finding the smallest number that all three numbers can divide into evenly, rather than finding the largest number that divides all three numbers evenly. | nvidia |
Find the number of positive divisors of $n^2$ that are not divisors of $n$, where $n$ is a positive integer with exactly 5 positive divisors. | nvidia |
Find the equation of the locus of points equidistant from the $x$-axis and the circle $(x-1)^2+y^2=4$. | nvidia |
Let $a$ and $b$ be real numbers such that $\lfloor x \rfloor = ax + b$ for all real numbers $x$ in the interval $2 \leq x < 3$. Determine $a + 2b$. | nvidia |
Find the smallest positive integer $n$ such that the polynomial $x^n - 1$ has a root in common with the polynomial $(x^2 - 2x - 1)^n + 1$. | nvidia |
Let $f(x)$ be a function that maps real numbers to real numbers. Define $g(x)$ as the maximum value of $|f(x)+2x|$ over all real numbers $y$ such that $|y-x|=1$. For what value of $x$ is $g(x)$ minimized, if it is known that $f(x)$ is an odd function and $f(0)=0$? | nvidia |
Find the largest positive integer n such that the sum of the smallest n prime numbers is a multiple of n itself. | nvidia |
Find the smallest integer $n$ such that the product of its positive divisors is equal to $2^{10}$. | nvidia |
Find the ratio of the sum of the squares of the first 20 positive integers to the sum of the cubes of the first 5 positive integers. | nvidia |
A beekeeper has $9$ beehives, each containing a certain number of bees. The number of bees in each hive is a positive integer, and the total number of bees in all the hives is $100$. The beekeeper wants to distribute the bees among the hives so that the difference in the number of bees between any two hives is minimize... | nvidia |
Let $m$ be a positive integer, and consider the set of integers $\left\{1,2,\dots,3m\right\}$. For how many values of $m$ is it possible to partition this set into three disjoint subsets, such that the sum of the elements in each subset is equal? | nvidia |
A deck of forty cards consists of four $1$'s, four $2$'s,..., and four $10$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relatively ... | nvidia |
Find the smallest positive integer $n$ such that the sum of $n^3$ and $n^2$ is a perfect square. | nvidia |
Let $f(x)$ and $g(x)$ be two functions such that
\[f(x+a) = g(x)\]for all integers $x$, where $a$ is a positive real number. Suppose that $f(x)$ is a quadratic function and $g(x)$ is a linear function, and that $f(0) = g(0) = 0$. Find the smallest possible value of $a$ such that $f(x)$ and $g(x)$ have the same leading ... | nvidia |
Let $a$ and $b$ be positive integers. Consider the equation $ax+by=100$, where $x$ and $y$ are positive integers. How many different ordered pairs $(a,b)$ will make this statement true: there are exactly $3$ positive integer solutions to the equation? | nvidia |
Two of the roots of the equation \[ax^3+bx^2+cx+d=0\]are $3$ and $-2.$ Given that $a \neq 0,$ compute $\frac{b+c}{a}.$
A new problem:
The polynomial $p(x) = ax^3 + bx^2 + cx + d$ has three roots, one of which is $x = 2$. If the sum of the other two roots is $-3$, and the product of the other two roots is $-4$, find $... | nvidia |
What is the sum of the integers in the bottom row of a triangular array of numbers where each number is the sum of the two numbers directly above it, and the top row consists of the prime numbers less than 20? | nvidia |
There are four points that are $3$ units from the line $y=7$ and $7$ units from the point $(5,7)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? | nvidia |
Given that
$$\frac{1}{(x^2+1)(x^2+4)(x^2+9)}=\frac{Ax+B}{x^2+1}+\frac{Cx+D}{x^2+4}+\frac{Ex+F}{x^2+9}$$
determine the value of $A+C+E$. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R} $ be defined by $ g(x) = x^2 + \sin x + 1 $. How many real numbers $x $ are there such that $ x \in (0, 2\pi) $ and $g(x)$ is a rational number? | nvidia |
Let $n$ be a positive integer such that $n^3 + 1$ is divisible by 11. What is the smallest positive integer $m$ such that $n^2 - m$ is divisible by 7? | nvidia |
Let $m$ and $n$ be positive integers such that
\[\frac{3}{5} < \frac{m}{n} < \frac{2}{3}\]
and $n$ is as small as possible. What is $n-m$? | nvidia |
A portion of the graph of $f(x)=ax^2+bx+c$ is shown below. The distance between grid lines on the graph is $1$ unit.
What is the value of $a+b+2c$?
[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=-... | nvidia |
Let $F_1$ and $F_2$ be the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$, where $F_1$ is on the negative $x$-axis and $F_2$ is on the positive $x$-axis. Let $P$ be the point on the ellipse such that the distance $F_2P-F_1P$ is minimal. Find the coordinates of $P$. | nvidia |
Determine the smallest number such that the sum of its digits is equal to the sum of the digits of its square. | nvidia |
The points $(0,0)$, $(a,b)$, and $(c,d)$ are the vertices of a triangle with an obtuse angle at $(0,0)$. Given that the slopes of the lines containing the sides opposite the acute angles are equal, find the value of $ac+bd$. | nvidia |
Find the sum of the squares of all integers that are both multiples of 3 and less than 20. | nvidia |
Determine all positive integers $n$ such that $2^{2^{2^{2^n}}}$ is divisible by $1000$. | nvidia |
Find the product of all constants $t$ such that the quadratic $x^2 + tx - 10$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers. | nvidia |
The hexagon with vertices $(0,0)$, $(4,1)$, $(8,6)$, $(7,10)$, $(2,9)$, and $(-1,4)$ is cut into two congruent polygons by a line through the point $(3,5)$. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime integers. Find $m+n$. | nvidia |
Find the sum of all integers $k$ such that $\binom{23}{4} + \binom{23}{5} = \binom{24}{k}$. | nvidia |
Find the smallest positive integer $m$ such that all the roots of the equation $z^6 + z^3 + 1 = 0$ are $m^{\text{th}}$ roots of unity. | nvidia |
The area of the largest possible inscribed triangle having one side as a diameter of circle $C$ is $54$ square centimeters. The two vertices of the triangle on the diameter are connected to a point $P$ on the circle such that $\angle P=90^\circ$. If point $P$ is connected to the center of the circle instead, determine ... | nvidia |
The sum of the squares of two positive integers is 90. The product of the two integers is 27. What is the sum of the two integers? | nvidia |
Let p>q>0 be positive integers such that
$\binom{p}{q}+\binom{p}{p-q}=\binom{p}{q-1}+\binom{p}{p-q+1}=\binom{p}{q+1}+\binom{p}{p-q-1}=m$
for some positive integer m. Find p+q+m. | nvidia |
The integers from 1 to 100 are written on cards. What is the minimum number of cards that need to be removed so that any two remaining cards have numbers that are relatively prime? | nvidia |
Find the number of positive integers less than 1000 that have a remainder of 5 when divided by 11. | nvidia |
Find the smallest positive integer $m$ such that $5m \equiv 924 \pmod{21}$. | nvidia |
Given a point $P$ on a square grid, how many points $Q$ on the same grid can be chosen such that the triangle formed by $P$, $Q$, and the point symmetric to $P$ about the center of the grid is a right triangle? | nvidia |
Let $g(n)$ return the number of distinct ordered pairs of positive integers $(a, b)$ such that for each ordered pair, $a^3 + b^3 = n$. Find the smallest positive integer $n$ for which $g(n) = 2$. | nvidia |
What is the smallest possible number of digits in the sum of two 5-digit whole numbers such that the sum contains at least one digit that is not present in either of the original numbers? | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $xyz=1$. Find the minimum value of the expression
$$x^4+y^4+z^4+2x^2y^2+2y^2z^2+2z^2x^2.$$ | nvidia |
Find all continuous functions $f(x)$ that satisfy the equation:
\[f(x+y) - f(x) = f(x) f(y)\]
for all real numbers $x$ and $y$. | nvidia |
Let $\alpha$ be a real number randomly selected from the interval $[0, \pi]$. Then, $m$ and $n$ are two relatively prime positive integers such that $m/n$ is the probability that the equation $\sin(x) = \sin(\alpha) + \sin(2x)$ has at least one solution in the interval $[0, 2\pi]$. Find the value of $m+n$. | nvidia |
What is the smallest value of $n$ such that the $n$th row of Pascal's Triangle contains at least three odd integers that are not 1? | nvidia |
Find the number of integers between 100 and 1000 (inclusive) that have exactly three distinct digits and are divisible by 5. | nvidia |
Given that $a$, $b$, and $c$ are positive real numbers such that $a(b+c) = 152$, $b(c+a) = 162$, and $c(a+b) = 170$, then find $abc$.
A similar problem would be:
Given that $x$, $y$, and $z$ are positive real numbers such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{12}$, $\frac{1}{y} + \frac{1}{z} = \frac{1}{15}$, and... | nvidia |
Consider the equation $x^2 + bx + c = 0$, where $b$ and $c$ are integers. The roots of the equation can be expressed as $\frac{-b \pm \sqrt{b^2 - 4c}}{2}$. Suppose the difference between the squares of the roots is equal to $4k$, where $k$ is a positive integer. Find the sum of all possible values of $b$ that satisfy t... | nvidia |
Find the product of all composite numbers between 1 and 15 that are not divisible by 3. | nvidia |
Find the positive integer $m$ for which the expression
\[\frac{m}{3} + \frac{12}{m}\] attains its minimum value. | nvidia |
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