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Find the values of $m$ and $n$ such that the quadratic equation $\frac{x^2}{m} + \frac{x}{n} + 1 = 0$ has two distinct real roots. | nvidia |
Determine the number of complex numbers of the form $z = \cos\theta + i\sin\theta$, where $0 \le \theta < 2\pi$, that are roots of the equation $z^3 + az^2 + bz + c = 0$, where $a, b, c$ are integers and $a$ is prime. | nvidia |
The sum of the first $n$ positive integers is equal to $k^2 + k$, where $k$ is a positive integer. Find the smallest possible value of $n$ such that $k$ is a prime number. | nvidia |
Let $z$ be a complex number such that the sum of its real and imaginary parts is equal to $1$, while the sum of the absolute values of $z$ and its conjugate $\overline{z}$ is equal to $3$. What is the value of $|z|$? | nvidia |
Count the number of positive divisors of 360 that are in the form $p^k$, where $p$ is a prime number and $k$ is a positive integer. | nvidia |
Find the maximum value of $a^3b^3$, given that $a+b=6$ and $a$ and $b$ are positive real numbers. | nvidia |
Let c be a real number randomly selected from the interval $[-5,5]$. Then, p and q are two relatively prime positive integers such that p/q is the probability that the equation $x^4+9c^2=(2c^2-6c)x^2$ has $\textit{at least}$ two distinct real solutions. Find the value of $p+q$. | nvidia |
Find the smallest possible value of the sum of two positive 3-digit integers whose greatest common divisor is 5. | 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 |
Solve for the sum of all possible values of $x$ when $3^{x^2+4x+4}=9^{x+2}$. | nvidia |
Find the value of $x$ such that the terms $\frac{1}{x^2}$, $\frac{1}{x}$, and $x$ form a geometric sequence. | nvidia |
Let $x$ and $y$ be real numbers such that
\[
\begin{vmatrix} 1 & x & y \\ 1 & 2x & 3y \\ 1 & 3x & 4y \end{vmatrix}
= 2016.\]Find $xy$. | nvidia |
How many positive integer divisors of $3003^{3003}$ are divisible by exactly 3003 positive integers? | nvidia |
Find the positive integer $m$ such that
$$\arctan\frac{1}{2} + \arctan\frac{1}{7} + \arctan\frac{1}{8} + \arctan\frac{1}{m} = \frac{\pi}{3}.$$ | nvidia |
Let $f$ be a function defined on the positive integers, such that
\[f(xy) = f(x) + f(y)\]for all positive integers $x$ and $y.$ Given $f(10) = 14$ and $f(40) = 20,$ find $f(500).$ | nvidia |
Find the largest positive integer $b$ such that $b$ is a multiple of $7$ and $b$ is a multiple of $15.$ | nvidia |
Given that $O$ and $H$ are the circumcenter and orthocenter of triangle $ABC$ respectively, and that $AB=AC$, calculate the length of side $BC$ if $\angle BAC=120^\circ$, $OH=6$ units, and the length of the altitude from $A$ to side $BC$ is $\sqrt{3}$ units. | nvidia |
Find the largest positive integer such that the sum of its digits is less than or equal to 10, and its decimal representation forms a geometric progression (i.e., each digit after the first is obtained by multiplying the previous digit by a fixed constant). | nvidia |
The numbers $a$, $b$, and $c$ are pairwise relatively prime positive integers such that $a^2+b^2=c^2$. Find the least possible value of $a+b+c$. | nvidia |
In a 3x3 grid, place the numbers 1 through 9 such that the sum of the numbers in each row, column, and diagonal is maximized. What is the maximum sum that can be achieved? | nvidia |
In a regular hexagon $ABCDEF$, the point $G$ is the intersection of the diagonals $\overline{AC}$ and $\overline{DF}$. The coordinates of the vertices $A$ and $D$ are $A=(7,13)$ and $D=(10,17)$. Find the sum of the absolute values of all possible slopes for the diagonal $\overline{CE}$. | nvidia |
Let $r$ be a real number, $|r| < 2,$ and let $z$ be a complex number such that
\[z^2 + \frac{1}{z^2} = r.\]Find $|z^2|.$ | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g((x + y)^2) = g(x)^2 + 2xg(y) - y^2\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(2),$ and let $t$ be the sum of all possible values of $g(2).$ Find $m \times t.$ | nvidia |
Let's suppose there are two positive real numbers, a and b. Find the minimum value of the following expression:
(ab(a+b)/a-b)^2
for real numbers a > 1 and b > 1. | nvidia |
Find all values of $k$ for which the equation $kx^2-4x+2=0$ has two distinct real solutions. | nvidia |
Determine the smallest possible degree of $p(x)$ in the rational function $\frac{p(x)}{x^4+2x^2+1}$ such that the graph of the function has a slant asymptote. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 1.$ Find the minimum value of
\[\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}} + \frac{1}{\sqrt{z}}.\] | nvidia |
In a triangle, the length of the hypotenuse is $3x+2$, and the lengths of the other two sides are $x+5$ and $2x-3$. The area of the triangle is 24 square units. What is the length of the shortest side of the triangle? | nvidia |
Determine the number of distinct prime factors of the sum of the positive divisors of $400^2$. | nvidia |
Given a positive integer $n$, let $x$ and $y$ be real numbers such that $x^2 + y^2 = n$ and $xy$ is a positive integer. Find the maximum possible value of $xy$ in terms of $n$. | nvidia |
Given the polynomial $f(x) = (x-2)(x-3)(x-5)(x-7)$, find the sum of the coefficients of all the terms in the expansion of $f(x)$ that have an even exponent. | nvidia |
Let $g$ be a function defined on the set of positive integers such that
$$\log_2x \cdot g(y) + \log_2y \cdot g(x) = g(xy)$$
for all positive integers $x$ and $y.$
Find the least positive integer $n$ such that $g(n) = 6g(1).$ | nvidia |
The sum of the interior angles of a polygon with $n$ sides is less than $2700^\circ$. What is the largest whole number value of $n$ that satisfies this inequality? | nvidia |
Find the sum of all integers $m$ such that $\binom{12}{3} + \binom{12}{5} + \binom{12}{7} = \binom{14}{m}$. | nvidia |
In the diagram, $\triangle PQR$, $\triangle QRS$ and $\triangle RST$ are isosceles triangles with $PR = 12$. All the angles of each triangle are integers in degrees, and the sum of the angles in each triangle is 180 degrees. The sum of the interior angles of the quadrilateral $PQST$ is 370 degrees. Find the perimeter o... | nvidia |
Find the smallest positive integer $a$ such that there exists an integer $b$ satisfying the equation $12a + 7b = 151$. | nvidia |
Determine $x$ such that $\sqrt{5x-2}+2=x$. | nvidia |
Given that $d = c-5$, $c = b+3$, $b = a-1$, and $a = 2$, evaluate the expression \[ \frac{a+3}{a+2} \cdot \frac{b+2}{b+1} \cdot \frac{c-2}{c-3} \cdot \frac{d+1}{d-1}, \] provided that none of the denominators are zero. | nvidia |
Find the product of the squares of the roots of $2x^4+5x^3-6x^2-8x+3=0$. | nvidia |
A five-digit number has the form $abcba$. If the sum of its digits is 15, find the value of $b$ such that the number is divisible by 11. | nvidia |
The graph of $y=f(x)$ is shown below, with $1$ unit between grid lines. Assume $f(x)$ is defined only on the domain shown.
What is the sum of all integers $c$ for which the equation $f(x)=c$ has exactly $6$ solutions?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm... | nvidia |
Find the value(s) of $x$ such that $8xy-12y+2x-3=0$ is true for all values of $y$. | nvidia |
Let $\otimes$ be a binary operation defined on positive real numbers such that $a\otimes b=ab^2$ for all $a$ and $b$. Determine $x$ if $2\otimes x=(2\otimes3)\otimes(3\otimes x)$. | nvidia |
Find the number of functions $f(n),$ taking the positive integers to the real numbers, such that
\[f(a + b) + f\left(\frac{a}{b}\right) = f(a) f(b) + 1\]for all positive integers $a$ and $b$, where $a > b$. | nvidia |
A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that for each complex number $z$, $f(z)$ is equidistant from both $z$ and the origin. Given that $|a+bi|=8$, find $b^2.$ | nvidia |
In a certain library, there are books with three-digit catalog numbers. How many of these catalog numbers contain at least one digit that is a multiple of 3? (For example, the numbers 153 and 540 are two catalog numbers that contain at least one digit that is a multiple of 3, but 226 is not.) | nvidia |
A sequence of five positive integers has the property that when the five numbers are added to their pairwise products taken two at a time, the resulting set contains exactly nine distinct elements. Find the smallest possible sum of these nine elements. | nvidia |
Consider the set of all possible sums of the digits of the natural numbers in base $b$. Determine the smallest base $b$ such that there exists at least one number in the set which is not representable as a product of a single-digit number and another number. | nvidia |
Find the sum of the eight smallest positive integer multiples of 7 that are also multiples of 3. | nvidia |
The product of two positive consecutive integers is 506. What is their sum?
Problem: In a triangle, the length of the hypotenuse is 10 inches, and the lengths of the other two sides are consecutive integers. Find the sum of the lengths of these two sides. | nvidia |
Suppose that $a$ and $b$ are positive integers such that $0.\overline{ab} + 0.\overline{ba} = \frac{35}{99}$. Find the largest possible value of $a + b$. | nvidia |
Find the sum of the first seven prime numbers that have a units digit of 7.
Another problem inspired by this one:
Find the sum of the first seven prime numbers that have a units digit of 7 and are also quadratic residues modulo 13.
Note: A quadratic residue modulo 13 is a number that is congruent to a perfect square... | nvidia |
Find the positive real number $y$ such that $\lceil y \rceil \cdot y = 135$. Express $y$ as a decimal. | nvidia |
Find all positive integers $n$ for which the equation $\cos^2 x + n\cos x + 1 = 0$ has solutions for $x$ in the interval $[0, 2\pi)$. | nvidia |
Consider the equation \[\lceil x^3 \rceil - \lceil x \rceil^3 = 53.\] Find the largest integer $n$ such that $n \leq x < n+1$. | nvidia |
Consider a sequence of numbers defined by $b_1 = 1$ and $b_n = b_{n-1} + n^2$ for $n > 1$. Find the smallest $n$ such that the sum of the digits of $b_n$ is divisible by 7. | nvidia |
Find the sum of all x that satisfy the equation
$$2^{\log_2(x)}+3x=2^{\log_2(x)+2\log_2(x-1)}+x+2$$ | nvidia |
If $a$,$b$, and $c$ are positive real numbers such that $a(b+c) = 152$, $b(c+a) = 162$, and $c(a+b) = 170$, then find $abc.$ | nvidia |
Let $g$ be a function such that $g(x^2+x+1)=3x-1$ for all real numbers $x$. Find $g(5)$. | nvidia |
Determine the number of positive divisors of the number $32$. | nvidia |
Let $x,y,z$ be non-negative real numbers such that $x+y+z=3$ and $xy+yz+zx=2$. Find the maximum value of $\max\{x,y,z\}$. | nvidia |
Find the value of $x_2(x_1+x_3)$ for the cubic equation $(x+1)(x^2-x-6)=0$, where $x_1<x_2<x_3$ are the real roots of the equation. | nvidia |
How many positive four-digit integers have the property that the sum of their digits is the same when read forwards and backwards? | nvidia |
Determine the number of positive integers less than 100 that are not divisible by any of the prime numbers between 30 and 40. | nvidia |
A real number $b$ is chosen randomly and uniformly from the interval $[-15, 20]$. Find the probability that the roots of the polynomial
\[x^3 - 3bx^2 + (3b + 1)x - 1\]are all real. | nvidia |
How many values of the digit $B$ are there such that the number $34B56$ is divisible by $3$ and the number $145B$ is divisible by $5$? | nvidia |
What is the smallest positive integer that can be written in the form $2002m + 44444n$, where $m$ and $n$ are integers? | nvidia |
Find the sum of all values of $x$ such that the expression $\frac{x^3-3x^2-4x+12}{x^2-3x-4}$ is equal to zero. | nvidia |
Let $p(x)$ be a polynomial of degree 2007 such that $p(x) = p(2x)$ for all $x$ in the interval $[0, 1]$. Find the number of distinct real roots of the equation $p(x) = 0$ in the interval $[0, 2]$. | nvidia |
Determine the radius of a circle inscribed in a right triangle with legs of length 7 and 10, and with the hypotenuse of length 11. The circle is tangent to all three sides of the triangle, and the center of the circle lies on the line connecting the incenter and the circumcenter of the triangle. | nvidia |
The sum of two consecutive even integers is 48. Find the smaller integer. | nvidia |
A sequence is defined recursively by the following formula:
$$a_1=\frac{1}{2}, \quad a_n=1-\frac{1}{a_{n-1}+1}.$$
Prove that this sequence converges and find its limit. | nvidia |
What is the sum of the five smallest distinct positive integer multiples of 7 that are also multiples of 3? | nvidia |
Find the smallest positive integer $n$ such that the product of its distinct prime divisors is equal to 30. | nvidia |
Find the smallest positive four-digit number that is divisible by each of the four smallest prime numbers. | nvidia |
A list of five positive integers has all of the following properties:
$\bullet$ The only integer in the list that occurs more than once is $8,$
$\bullet$ its median is $9,$ and
$\bullet$ its average (mean) is $10.$
What is the largest possible integer that could appear in the list? | nvidia |
Find the smallest positive integer that can be expressed as the sum of a perfect square, a perfect cube, and a perfect fifth power, such that the sum of its digits is equal to the sum of the digits of the perfect square, the perfect cube, and the perfect fifth power. | nvidia |
Determine the smallest positive integer $n$ such that $x^3 + x + n$ has only one rational root. | nvidia |
Let $g(x)$ be a function defined on the set of integers, such that
$$g(g(x))=x.$$Given that $g(99)=12$, find $g(12)$. | nvidia |
Find the smallest positive integer $x$ such that the product of $3^{2017}$, $5^{2019}$, and $x$ is a perfect square. | nvidia |
Find the sum of the squares of the odd integers from 1 to 99, inclusive. | nvidia |
Find all values of $p$ for which the product of the solutions of
\[7x^2 - x + p = 0\]equals the positive difference between the reciprocals of these solutions. Enter all possible values of $p,$ separated by commas. | nvidia |
Find all real values of $b$ for which the equation $$x^3 + bx^2 + x + 1 = 0$$ has at least two distinct complex roots that are conjugates of each other. | nvidia |
Find the area of the triangle with vertices $(2,3),$ $(-3,0),$ and $(5,8).$ | nvidia |
The quadratic function $f(x) = x^2 - 4x - 5$ has two real roots. Find the number of integers $n$ such that $f(n) > 0$ and $f(n+1) < 0$. | nvidia |
In three-dimensional space, find the number of lattice points that have a distance of 3 from the origin.
Note: A point is a lattice point if all its coordinates are integers. | nvidia |
A museum charges $ \$10$ for adult admission and $ \$7$ for children's admission. On a particular Saturday, the museum collected a total of $ \$1420$ in admission fees from at least one adult and at least one child. Among all the possible ratios of adults to children at the museum that day, find the one that is close... | nvidia |
Let $s(n)$ denote the sum of the digits of the integer $n$. Find the smallest positive integer $n$ such that $s(n) = s(n^2) = s(n^3) = \cdots = s(n^{100})$. | nvidia |
Determine the number of three-digit positive integers that have a units digit equal to the sum of the other two digits. | nvidia |
A function $g(x)$ satisfies the condition $g(x+y) = g(x) + g(y)$ for all real numbers $x$ and $y$, and $g(x) \neq 0$ for all real numbers $x$. Find $g(2)$ if $g(1) = 3$. | nvidia |
Consider the quadratic equation $x^2 + (2k-1)x + k^2 = 0$, where $k$ is a real number. Determine all possible values of $k$ such that the equation has two distinct real roots and the product of the roots is less than the sum of the roots. Express your answer in interval notation. | nvidia |
In a regular octahedron with edge length 1, there are 8 equilateral triangles each with side length $\sqrt{2}$ whose vertices are also vertices of the octahedron. The sum of the areas of these triangles can be expressed in the form $a\sqrt{b}$, where $a$ and $b$ are positive integers. Find $a + b$. | nvidia |
Let $P$ be a point inside a tetrahedron $ABCD$ such that the volume of $PBCD$ is $\frac{1}{6}$ of the volume of $ABCD$. Find the ratio of the area of the tetrahedron's cross-section at $P$ to the area of the base $BCD$. | nvidia |
In the year 2025, the International Mathematical Olympiad will take place in France. Let $F$, $R$, and $A$ be distinct positive integers such that the product $F\cdot R\cdot A=2025$. What is the largest possible value of the sum $F+R+A$? | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $a^2b^2c^2 = 1.$ Compute the maximum value of
\[\frac{a^2b}{c} + \frac{b^2c}{a} + \frac{c^2a}{b}.\] | nvidia |
Consider the graph of the rational function $\frac{x^2-4x+3}{x^3-2x^2-x+2}$. Let $p$ be the number of holes in the graph, $q$ be the number of vertical asymptotes, $r$ be the number of horizontal asymptotes, and $s$ be the number of oblique asymptotes. Find $2p+3q+4r+5s$. | nvidia |
Let $S$ be the sum of all positive integers $n$ such that $n^2+12n-2007$ is a perfect square. Find the remainder when $S$ is divided by $1000.$
| nvidia |
Find a real number $x$ such that $2^x\cdot 3^x\cdot 5^x=216$. | nvidia |
What is the positive integer $N$ for which $$ 22^2 \times 55^2 = 10^2 \times N^2 \ ? $$ | nvidia |
Circle $C$ has its center at $(3, -8)$ and a radius of $12$ units. What is the sum of the $x$-coordinates of the two points on circle $C$ that are also on the $x$-axis? | nvidia |
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