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What is the sum of all positive integer values of $n$ such that $\frac{n+18}{n}$ is an integer? | nvidia |
Given that $y = \frac{x-3}{x+2}$ and $z = \frac{x+5}{x-7}$, evaluate the expression $$
\frac{1+z}{1-y} \cdot \frac{2-y}{2+z} \cdot \frac{4+z}{4-y},
$$
assuming that $x = -3$ and none of the denominators are zero. | nvidia |
The lengths of the medians of a triangle with side lengths $a$, $b$, and $c$ satisfy the equation $\ds \frac{m_1}{m_2} + \frac{m_2}{m_3} + \frac{m_3}{m_1} = 3.$
Prove that the given triangle is equilateral. | nvidia |
Let the difference between the sum of the positive divisors of two different positive integers of the form $3^i$ and $7^j$ be equal to $600$. What is $\left| i-j \right|$? | nvidia |
Find all integer values of $a$ so that the polynomial
\[x^3 + 3x^2 + ax + 7 = 0\]has at least one integer root. Enter all possible values of $a,$ separated by commas. | nvidia |
Find the sum of all numbers $x$ satisfying $x+25/x = 10.$ | nvidia |
What is the smallest positive integer $n$ such that $3^n \equiv 1 \pmod{19}$? | nvidia |
A number $m$ is called a "digital root" if it is equal to the sum of its digits. Determine the smallest positive digital root that is divisible by 9. | nvidia |
What is the smallest positive integer n such that there exist n distinct strings of length 4, each consisting of exactly two 0's and two 1's, such that no two strings are identical and no string is a rotation of another? | nvidia |
Determine the least positive integer $n$ such that $12n + 15n$ and $20n + 25n$ have a common divisor greater than 1. | nvidia |
Consider the set of all integers between 1000 and 10000 (inclusive) that have at least two digits of 7. Find the remainder when the sum of all the numbers in this set is divided by 4. | nvidia |
Let $f(x)$ be a function such that
$f(x+y)+f(x-y)=2f(x)f(y)$ for all real numbers $x$ and $y.$
If $f(1)=2$ and $f(2)=5$, find $f(3)$. | nvidia |
If $n$ is a positive integer, then let $f(n)$ be the sum of the digits of $\frac{1}{5^{{}^n}}$ that are to the right of the decimal point. What is the smallest positive integer $n$ such that $f(n) > 10$? | nvidia |
$n$ coins are simultaneously flipped. The probability that at most one of them shows tails is $\frac{3}{16}$. Find $n$. | nvidia |
Find a real number $a$ such that the matrix
$$\left[\begin{array}{ccc}a&1&1\\1&a&1\\1&1&a\end{array}\right]$$is the matrix of a magic square. | nvidia |
Consider two circles with radii $r_1$ and $r_2$ centered at $(0, 1)$ and $(0, -1)$ respectively.
Find the value of $k$ such that the graphs of these circles intersect the graph of $y = kx^2$ at exactly one point each. | nvidia |
How many 10-digit positive integers have all digits either 1 or 2, and have two consecutive 1's that are not at the beginning or end of the integer? | nvidia |
Let $P(x)=1-x+x^2-x^3+\cdots+x^{2008}$, where $x$ is a complex number. Find the sum of all values of $x$ such that $P(x)P(-x)=3$. | nvidia |
If $x$ and $y$ are positive real numbers such that $\log_x y = 2$ and $\log_y x = 3$, find $\log_{10} (xy)$. | nvidia |
A certain integer $n$ can be represented as a three-digit number in base $b$ for some positive integer $b$. If $n$ is divisible by 7 in base 10, find the largest possible value of $n$ that is also divisible by 7 when read as a number in base $b$. | nvidia |
The product of two positive integers is 24. The positive difference of these two integers is 4. What is the sum of the two integers? | nvidia |
Find the smallest positive integer $m$ such that $3m \equiv 7 \pmod{11}$. | 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 |
Count the number of 6-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6 exactly once, such that the sum of the digits in the even positions is greater than the sum of the digits in the odd positions. | nvidia |
Let $a$, $b$, $c$ be real numbers such that
$a^2-2b+3c=50$
$b^2-2c+3a=37$
$c^2-2a+3b=23$.
Compute $a+b+c$. | nvidia |
$E$, $F$, $G$, and $H$ are points on a circle, and segments $\overline{EG}$ and $\overline{FH}$ intersect at $Q$, such that $EQ=10$, $QG=3$, and $FH=8$. Find $FQ$, given that $FQ < HQ.$ | nvidia |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | nvidia |
Find the minimum possible value of the expression
\[\frac{x^2}{y+z} + \frac{y^2}{z+x} + \frac{z^2}{x+y},\]
where $x,$ $y,$ and $z$ are positive real numbers such that $x+y+z=1$. | nvidia |
Let $P(x)$ be a polynomial such that
\[P(a + b) P(a - b) = P(a)^2 - P(b)^2\]for all real numbers $a$ and $b$. Find the number of distinct real roots of $P(x)$. | nvidia |
Find the sum of all integers between 10 and 50 that are not divisible by 3. | nvidia |
Emily has two eight-sided dice. One of the dice is fair, and the other die is biased so that it comes up seven with probability $\frac{3}{4}$ and each of the other seven sides has probability $\frac{1}{28}$. Emily chooses one of the two dice at random and rolls it four times. Given that the first three rolls are all se... | nvidia |
The set $T = \{1, 2, 3, \ldots, 99, 100\}$ contains the first $100$ positive integers. After the perfect squares and the numbers that are one more than a perfect square are removed, how many integers remain in the set $T$? | nvidia |
Let $b_1 = b_2 = 1.$ For $n > 2,$ let $b_n$ be the number of positive integers $k$ such that
\[b_{n-1}b_{n-2} = \sum_{i=1}^{k} i^2.\]
Compute the product $b_1 \cdot b_2 \cdot b_3 \cdot \dots \cdot b_{50}.$ | nvidia |
Point $Q$ lies on the line $y = 2$ and is 15 units from the point $(-2, 4)$. Find the sum of all possible $x$-coordinates that satisfy the given conditions. | nvidia |
A circle is inscribed in a regular hexagon, and a chord of the circle is drawn such that it is tangent to two adjacent sides of the hexagon. If the length of the chord is 6 cm, find the radius of the circle. | nvidia |
What is the largest whole number $n$ that can be added to 6531 such that the resulting number is still divisible by 9? | nvidia |
Find the positive integer n for which the sum of the digits in the number 9^n is the greatest. | nvidia |
Find all positive values of $c$ so that the inequality $x^2-6x+c<0$ has real solutions for $x$. Express your answer in interval notation. | nvidia |
Compute the sum of all the positive integer divisors of 144. | nvidia |
Compute the number of ways to arrange the letters in the word "STATISTICS" such that the letters "S" and "T" are never adjacent. | nvidia |
Let $\{a_n\}$ be a geometric sequence such that
$$
\frac{a_1+a_2+\dots+a_n}{a_{n+1}+a_{n+2}+\dots+a_{2n}}=\frac{3}{7}
$$
for any positive integer $n$.
What is the value of the common ratio? | nvidia |
Given that the degree measure of the sum of the interior angles of a convex polygon with $n$ sides is 1800, what is the degree measure of the sum of the exterior angles of a concave polygon with $n+2$ sides? | nvidia |
What is the smallest positive integer that has exactly eight distinct positive factors? | nvidia |
The reality game show Survivor is played with 16 people divided into two tribes of 8. In the first episode, two people get homesick and quit. If every person has an equal chance of being one of the two quitters, and the probability that one person quits is independent of the probability that any other person quits, wha... | nvidia |
In the complex plane, the point $A$ represents the number $3-4i$, and the point $B$ represents the number $-2+7i$. Find the length of the segment $AB$. | nvidia |
Find the smallest possible value of $(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$, given that $x$, $y$, and $z$ are positive real numbers such that $x^2+y^2+z^2=1$. | nvidia |
Let $x$ and $y$ be nonzero real numbers such that
\[(4 - 5i)(x + yi)\]is a real number. Find $\frac{x}{y}.$ | nvidia |
In the expression $\left(3x^2+2x+5\right)\left(2x-1\right)$, find the coefficient of the $x^3$ term and the constant term, and then find the sum of these two coefficients. | nvidia |
Given that the sum of two fractions is $\frac{3}{4}$, and one of the fractions has a numerator that is twice as large as its denominator, find the value of the other fraction.
Express your answer in the form of $\frac{a}{b}$. | nvidia |
Find the sum of the squares of the solutions of $x^2-13x+4=0$. | nvidia |
A bookshelf contains 10 books, 5 of which are fiction and 5 of which are non-fiction. The books are arranged randomly on the shelf. What is the probability that the two books on either end of the shelf are of different genres? | nvidia |
For how many positive integers $n$ does $1+3+5+\cdots+(2n-1)$ evenly divide $15n$? | nvidia |
Find the smallest positive integer $k$ such that the function $f(x) = \sin x + \cos kx$ has a period of $2\pi$. | nvidia |
Let triangle $ABC$ be an equilateral triangle in the xy-plane. Given that the length of the median $AM$ is $12$, and that the line passing through $A$ and the center of the circumcircle of the triangle is parallel to the line $y=3x+5$, find the area of triangle $ABC$. | nvidia |
Find the value of $a$ such that the matrix
$$\begin{pmatrix} 1 & 1 \\ a & 1 \end{pmatrix}$$
has an inverse that is equal to its transpose. | nvidia |
Let $g$ be a function from the set of positive integers to the set of positive integers, such that $g(a) \times g(b) = g(a+b) \times g(a \times b)$ for all positive integers $a$ and $b$. If $g(1)=1$, find the number of distinct possible values of $g(5)$. | nvidia |
An irregular hexagon is formed by joining the vertices of six identical equilateral triangles. Find the ratio of the area of the hexagon to the area of one of the triangles. The triangles are joined so that their sides form the edges of the hexagon. | nvidia |
There exists a positive real number $x$ such that $ \tan (\arccos (x)) = x $. Find the value of $x^2$. | nvidia |
Find the greatest common divisor of the set of positive integers that are one less than a perfect square and divisible by $7$. | nvidia |
What is the least positive integer $n$ such that $2^n$ is a divisor of $\frac{(2n)!}{n!}$? | nvidia |
What is the smallest positive integer $n$ such that the sum of the digits of $n$ is equal to the number of divisors of $n$? | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(0) = 0$ and
\[g(x^2 + y^2) = (x + y) (g(x) - g(y))\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(1),$ and let $t$ be the product of all possible values of $g(1).$ Find $m \times t.$ | nvidia |
Let $A$ and $B$ be the two sets of positive divisors of the numbers $a=3^3 \cdot 5^3 \cdot 7^3$ and $b=3^2 \cdot 5^2 \cdot 11^2$.
Find the sum of all positive integers which are simultaneously elements of the sets $A$ and $B$. | nvidia |
Find the number of rational numbers of the form $p/q$, where $p$ and $q$ are positive integers with $gcd(p,q) = 1$, between $\sqrt[3]{15}$ and $\sqrt[3]{30}$. | nvidia |
Let $f(n)$ be the sum of the positive integer divisors of $n$. For how many values of $n$, where $1 \le n \le 25$, is $f(n)$ prime? | nvidia |
Let $S$ be the set of all nonnegative integers $n$ such that $n^2$ has the same remainder as $n$ when divided by 24. Find the sum of all elements in $S$ that are less than 1000. | nvidia |
Consider a cube with side length 1 and vertices at $(0,0,0)$, $(0,0,1)$, $(0,1,0)$, $(0,1,1)$, $(1,0,0)$, $(1,0,1)$, $(1,1,0)$, and $(1,1,1)$. Let $A=(a_1,a_2,a_3)$ be a point inside the cube such that the sum of the squared distances from $A$ to the vertices of the cube is minimized. Find the coordinates of $A$. | nvidia |
In a regular hexagon, the radius of the circumscribed circle is twice the radius of the inscribed circle. Find the ratio of the area of the circumscribed circle to the area of the inscribed circle. | nvidia |
Let $x$ be a real number such that $x^3+4x=8$. Determine the value of $x^7+64x^2$. | nvidia |
Problem:
Points $A$ and $B$ are symmetric with respect to the line $y = x$. The coordinates of point $A$ are $(x, y)$. Find the coordinates of point $B$ if the area of the rectangle formed by the points $(0,0)$, $(0,y)$, $(x,y)$, and $(x,0)$ is equal to the area of the rectangle formed by the points $(0,0)$, $(0,x)$, $... | nvidia |
If $x$, $y$, and $z$ are non-zero complex numbers such that $|x+y+z|=|x|+|y|+|z|$ and $|x|=|y|=|z|$, find $\left|\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\right|.$ | nvidia |
Let $m \ge 3$ be an integer and let $S = \{3,4,5,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.
| nvidia |
Determine the value of $x$ that satisfies $\sqrt[5]{x\sqrt{x^3}}=3$. | nvidia |
The numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there?
The numbers $2341$, $2005$ and $2468$ have something in common: each is a $4$-digit number beginning with $2$ that has exactly two identi... | nvidia |
Find the minimum value of
\[x^2 + xy + y^2\]over all real numbers $x$ and $y.$ | nvidia |
Given that $z$ is a complex number such that $|z| = 1$, find the value of $z$ that satisfies the equation $(z^2)^{-1/2} = \overline{z}$.
(Here, $\overline{z}$ denotes the complex conjugate of $z$.) | nvidia |
Nine tiles are numbered $1, 2, 3, \cdots, 9,$ respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
| nvidia |
The number $2.\overline{517}$ can be expressed as a fraction $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Determine the sum of the digits of $a$. | nvidia |
Define a positive integer n to be a factorial tail if there is some positive integer m such that the decimal representation of m! ends with exactly n zeroes. How many positive integers less than 1992 are not factorial tails?
New Problem:
A sequence of positive integers is called a "bubble" if each term is the sum of t... | nvidia |
A segment with a central angle of $\frac{\pi}{3}$ is cut from a circle with a radius of 9. The radius of the circle inscribed within the segment is:
$$\textbf{(A)}\ 3 \tan \frac{\pi}{6} \qquad \textbf{(B)}\ 3 \sin \frac{\pi}{6} \qquad \textbf{(C)}\ 3 \cos \frac{\pi}{6} \qquad \textbf{(D)}\ 3 \cot \frac{\pi}{6} \qquad ... | nvidia |
Find two three-digit numbers whose product is 123456 and have the property that their decimal representations reversed are integers divisible by 7. Determine their sum. | nvidia |
Find the number of distinct positive integers that can be expressed as the sum of two distinct factors of 48. | nvidia |
Consider the polynomial $P(x)=ax^3+bx^2+cx+d$. If the sum of the squares of the coefficients of $P(x)$ is equal to the sum of the squares of the coefficients of $P(x+1)$, what is the value of $a+b+c+d$? | nvidia |
A combinatorial equation $f(n)=\sum_{i=0}^{k}(-1)^{i}\binom{n}{i}$ is given, where $n$ and $k$ are positive integers and $f(n)$ denotes the number of subsets of a set $\{1, 2, \dots, n\}$ that contain at most $k$ elements.
Determine the largest possible value of $n$ such that $f(n) < 10^5$. | nvidia |
Find the matrix $\mathbf{B}$ such that $\mathbf{B}^3=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}.$ | nvidia |
Find the sum of all integers between 10 and 99 whose squares remain the same when their digits are reversed. | nvidia |
In a right triangle with integer length sides, the area is equal to 120 square units. Find the maximum length of the hypotenuse. | nvidia |
A five-digit positive integer starts with 1 and ends with 5. If the number formed by the three middle digits is divisible by 6, how many such five-digit integers are possible? | nvidia |
Find the sum of all values of x for which the expression $\frac{x-3}{x^2+2x+1}$ is equal to zero. | nvidia |
There exists a positive real number $x$ such that $\sin(\arccos(x)) = \frac{x}{\sqrt{2}}$. Find the value of $x^2$. | nvidia |
Find the sum of cubes of all roots of the equation
$$x^3-7x^2+3x-1=0$$
if it is known that the sum of the roots is equal to 7. | nvidia |
Let $f(n)$ return the number of distinct ordered pairs of positive integers $(a, b)$ such that for each ordered pair, $a^2 + b^2 = n$. What is the smallest positive integer $n$ for which $f(n) = 5$? | nvidia |
Find the sum of the reciprocals of the divisors of $24$. | nvidia |
Compute the number of ways to arrange the letters of the word "COMPUTER" such that the vowels appear in alphabetical order. | nvidia |
Let $a,$ $b,$ $c$ be real numbers such that $a + b + c = 0.$ Find the set of all possible values of $ab + ac + bc.$ | nvidia |
Find the sum of the reciprocals of the solutions of $x^2-13x+4=0$. | nvidia |
The diagonals of a regular hexagon meet at a point $O$, and the circumcircle of the hexagon has a radius of 6. If the area of triangle $AOB$ is 12, where $A$ and $B$ are adjacent vertices of the hexagon, find the area of the hexagon. | nvidia |
Find the largest possible value of $z$ for which the equation $x^2 + y^2 = z^2$ has exactly four ordered pairs of integer solutions $(x,y)$ with $x \geq 0$ and $y \geq 0$. | 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 |
Determine the number of integer solutions for the equation $2x+3y=7$ such that $0\le x \le 20$ and $0\le y \le 20$. | nvidia |
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