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Given the expression $\log_{y^3}{x}\cdot\log_{x^4}{y^2}\cdot\log_{y^5}{x^6}\cdot\log_{x^7}{y^8}$, simplify it in terms of $\log_y{x}$ and find the constant $a$ such that the expression is equal to $a(\log_y{x})^2$. | nvidia |
Let $P(x)$ be a polynomial with integer coefficients such that $P(1) = 60$ and $P(x)$ is divisible by $(x-1)^2.$ Find the number of divisors of $P(1)$ that are not divisible by 3. | nvidia |
Find all distinct pairs of positive integers $(m, n)$ such that $m + n = 6$ and the sum of the digits of $mn$ is 9. | nvidia |
Find the integer $n$, $0 \le n \le 11$, such that \[n \equiv 10389 \pmod{12}.\] | nvidia |
Let $x$ be the number of positive integers less than $50$ that are divisible by both $3$ and $5$. Let $y$ be the number of positive integers less than $50$ that are divisible by $3$ but not by $5$. Compute $x \cdot y$. | nvidia |
Let $A$ be the sum of the first 2006 terms of the series $1-1+1-1+1-1+...$ and let $B$ be the sum of the squares of the first 2006 terms of the same series. What is $A+B$? | nvidia |
Find the area of the region
$$\left\{\left(\sqrt{x}+1\right)^2+i\left(\sqrt{x}-1\right)^2: x \in [0, \infty)\right\}$$
that lies inside the square with vertices at $0$, $1$, $1+i$, and $i$. | 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 |
Let $x$ and $y$ be two positive numbers such that their sum is 100 and the difference of their squares is 3600. Find the product $xy$. | nvidia |
Consider a polynomial of the form $P(x) = (x^2 + ax + b)^m$, where $a$ and $b$ are constants and $m$ is a positive integer. It's given that the coefficients of the polynomial $P(x)$ form a geometric sequence with a common ratio of $2$. Find the sum of all possible values of $m$ and the degree of the polynomial $P(x)$. | nvidia |
Let $Q$ be the product of the first $200$ positive even integers. Find the largest integer $m$ such that $Q$ is divisible by $2^m$. | nvidia |
Determine all real values $y$ such that the equation $$\lfloor y \rfloor^2 + y^2 = 3y\lfloor y \rfloor$$ holds true. | nvidia |
Consider a sequence of positive integers that are multiples of both 4 and 6. Let $c$ be the number of terms in this sequence that are less than 60. Let $d$ be the number of positive integers that are less than 60 and a multiple of 8 and a multiple of 3. Compute the difference between the sum of the first $c$ terms of t... | nvidia |
What is the smallest integer value of $c$ such that the graph of $y=x^3+cx^2+3x-1$ has exactly two $x$-intercepts? | nvidia |
How many positive three-digit integers have an even number of positive factors? | nvidia |
Let $p(x)=x^2+bx+c$ be a quadratic polynomial such that $p(1)=-2$, $p(-1)=8$. Determine the value of $p(3)$. | nvidia |
In a set of seven consecutive integers, the smallest integer is greater than one-third of the sum of the seven integers. What is the largest integer that could be in the set? | nvidia |
The terms of the sequence $(b_i)$ defined by $b_{n + 2} = \frac {b_n \cdot b_{n+1} + 2020} {b_{n} + b_{n+1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $b_1 \cdot b_2$. | nvidia |
A square with side length $x$ is inscribed in a circle. Two smaller squares are then inscribed in the two regions inside the circle but outside the large square, such that each smaller square shares a side with the large square. If the sum of the areas of the two smaller squares is equal to one-third the area of the la... | nvidia |
Determine the smallest integer $k$ such that
$k\cdot \left\lceil 7.3 \right\rceil > 56$. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a^2 + b^2 = 25.$ Find the maximum value of $a^2b^2.$ | nvidia |
Let $a_n$ be the number of ways to arrange $n$ distinct objects in a circle such that no two adjacent objects are consecutive in the original ordering. Find the value of $a_{10}$ modulo $100$. | nvidia |
The sum of two positive integers is 50 and their difference is 12. What is the value of the positive difference of the squares of the integers? | nvidia |
The quotient of two positive integers is $\frac{3}{4}$ and their sum is 42. What is the value of the smaller of the two integers? | nvidia |
The sum of the y-coordinates of the vertices of a triangle in the Cartesian plane equals 15. Find the sum of the y-coordinates of the centroids of the triangle. | nvidia |
Find the smallest positive integer $n$ such that $2^n$ has at least 100 positive divisors that are perfect squares. | nvidia |
Find the smallest positive integer $n$ such that $n!$ is divisible by the fourth power of every prime less than or equal to $n$. | nvidia |
Find the smallest number $n$ such that any $n$ people sitting at a round table will necessarily contain at least one group of three people who all know each other or all do not know each other. | nvidia |
Determine $m$ such that the product of the digits of the number $456m$ is divisible by 45. | nvidia |
What is the smallest three-digit integer whose digits are distinct and form an arithmetic sequence, with the additional condition that the sum of the digits is a perfect square? | nvidia |
A square has side length equal to $2^4 \cdot 3^2 \cdot 5^3$ meters. How many square tiles with sides equal to $2^a \cdot 3^b \cdot 5^c$ meters can be used to cover the square, where $a, b, c$ are positive integers? | nvidia |
In an isosceles triangle $ABC$ with $AC=BC$, the vertices $A$ and $B$ are inscribed in a circle $O$, and the vertex $C$ lies outside the circle. The circle intersects side $AB$ at point $M$, and the tangents to the circle at points $A$ and $B$ intersect at point $N$. Find the ratio $\frac{AC}{MN}$. | nvidia |
A right-angled triangle has two sides of lengths $a$ and $b$ with $a > b$ and hypotenuse of length $c$ that satisfy the equation $a^2 - b^2 = 4ab$. Given that the solutions $a$ are in the form $a = x + y\sqrt{z}$, where $x$, $y$, and $z$ are positive integers and the expression is completely simplified, what is the val... | nvidia |
Let $a$, $b$ and $c$ be positive integers such that $\gcd(a,b) = \gcd(a,c) = \gcd(b,c) = 1$.
Determine the largest possible value of $\gcd(a^3,b^2c)$. | nvidia |
Find the smallest positive integer $b$ such that $x^6 + bx^3 + 1$ is always composite for all integers $x$. | nvidia |
Let $a,$ $b,$ $c,$ $d$ be positive integers such that
$$2^a+3^b+7^c=11^d+13$$
How many possible values of $d$ are there? | nvidia |
Find the largest value of $m$ such that $2x^2 + mx + 50$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
The difference between two numbers is 9, and the sum of the squares of each number is 153. What is the value of the product of the two numbers? | nvidia |
In the equation
$$4x^2-kx+3=0$$
determine $k$ such that the sum of the squares of the roots be equal to 3.5. | nvidia |
Let $a,b,c,d$ be positive real numbers such that
$\begin{vmatrix} a & b \\ c & d \end{vmatrix}=-1$.
Find
$\begin{vmatrix} \sqrt{a} & \sqrt{b} \\ \sqrt{c} & \sqrt{d} \end{vmatrix}$. | nvidia |
The number -15/2 is the sum of the solutions of the equation $(3x+5)(2x-9) = 0$. Find the sum of the solutions of the inequality $(3x+5)(2x-9) \geq 0$. | nvidia |
Given that $a+b+c=10$ and $a^2+b^2+c^2=40$, find the value of $(a+b+c)^2-3(a^2+b^2+c^2)$. | nvidia |
At the 2007 Math Olympics, Team Canada won $17$ out of a possible $100$ medals. Which one of the following is closest to the fraction of medals that they won? $$
\frac{1}{4} \qquad \frac{1}{5} \qquad \frac{1}{6} \qquad \frac{1}{7} \qquad \frac{1}{8}
$$ | nvidia |
There are three consecutive numbers $n$, $n+1$, and $n+2$. The square of the largest number is equal to the sum of the squares of the other two numbers. Determine the middle number. | nvidia |
If $m$ and $n$ are positive integers such that the polynomial $x^2-mx+n$ has only positive integer roots, determine the least possible value of $n$ such that $m-n$ is equal to 25. | nvidia |
Let $a$, $b$ be positive integers, and let $m$ be a prime number greater than $5$ such that $\gcd(a,b)=1$ and $\gcd(a,m)=1$. Determine the largest possible value of $\gcd(a+b,m)$ in terms of $m$. | nvidia |
Compute the number of ordered triples of integers $(x,y,z)$ with $1\le x<y<z\le 100$ such that $x^2+y^2+z^2$ is a perfect square. | nvidia |
How many distinct ways can the seven letters of the word "Laughed" be arranged so that no two vowels are adjacent? | nvidia |
Find the smallest positive integer n such that the sum of the entries in the nth row of Pascal's triangle is greater than 1000. | nvidia |
Find a positive integer that is divisible by 18 and has a square root between 32 and 32.5. | nvidia |
Find the value of $b$ so that the lines with the given equations are parallel. \begin{align*}
y &= -3x+5 \\
4y+bx &= 11.
\end{align*} | nvidia |
Let $C$ equal the number of three-digit integers that are multiples of 3. Let $D$ equal the number of three-digit integers that are multiples of 2. Find $C+D$. | nvidia |
Determine $x$ such that $\log_{1/9}x=\frac{1}{2}$ and find the value of $x$ that satisfies this equation. | nvidia |
In the five-digit integer $12,\!A4B$, what is the largest possible sum of the digits $A$ and $B$ so that the five-digit integer will be divisible by 9? | nvidia |
A pyramid with square base $ABCD$ is cut by a plane that intersects the edges $AB$, $BC$, $CD$, and $DA$ at points $K$, $L$, $M$, and $N$, respectively, and divides the pyramid into two smaller pyramids. If the volume of the smaller pyramid containing vertex $A$ is one-seventh that of the larger pyramid, find the ratio... | nvidia |
Determine the value of $x$ that satisfies $\sqrt[5]{x\sqrt{x^3}}=3$. | nvidia |
Find the sum of all integers between 1 and 100 that are divisible by 5 or 7. | nvidia |
A sequence $a_1, a_2, a_3, \ldots$ is called recursively centered if $a_{n+1} = a_n^2 + 1$ for all positive integers $n$, and $a_1$ is a positive integer. If $a_n$ is a recursively centered sequence, what is the maximum number of terms $a_n$ that can be equal to a perfect square? | nvidia |
Let $a$ and $b$ be positive integers such that the number
$$N=\sqrt{a+\sqrt{b}}-\sqrt{a-\sqrt{b}}$$
is an integer. Find the minimum possible value of $N$. | nvidia |
A positive integer is called "remarkable" if the sum of its digits is equal to the product of its distinct prime divisors. Find the smallest remarkable number that is divisible by 7. | nvidia |
A right circular cone is inscribed in a right circular cylinder such that the radius of the base of the cone is one-third of the radius of the base of the cylinder, and the height of the cone is equal to the height of the cylinder. What is the ratio of the volume of the cone to the volume of the cylinder? | nvidia |
In a random tournament of 16 teams, each team plays exactly one game against each of the other teams. The probability that for every positive integer k<5, no collection of k teams contains exactly one team that wins all its games is m/n, where m and n are relatively prime positive integers. Find m+n. | nvidia |
Find the number of different possible values of the perimeter $p<3000$ for a quadrilateral $ABCD$ with positive integer side lengths, right angles at $B$ and $C$, $AB=3$, and $CD=AD$. | nvidia |
A portion of the graph of a quadratic function $f(x)$ is shown below.
Let $g(x)=-f(x)$ and $h(x)=f(-x)$. If $a$ is the number of points where the graphs of $y=f(x)$ and $y=g(x)$ intersect, and $b$ is the number of points where the graphs of $y=f(x)$ and $y=h(x)$ intersect, then what is $10a+b$?
[asy]
size(150);
real ... | nvidia |
Let $z$ be a complex number with the property that the sum of the squares of the distances between $z$ and the vertices of the triangle with vertices $1+i, 1-i,$ and $-1$ is equal to $7.$ What kind of curve is the locus of all such complex numbers $z$? | nvidia |
Find the greatest integer that is less than $\frac{1}{100} \cdot \frac{\sum\limits_{k=1}^{18} \frac{1}{k!(19-k)!}}{\frac{1}{19!}}$. | nvidia |
Let $R$ be a rectangle with a length of $3$ and a width of $2$. Three points are chosen independently at random inside $R$. The probability that the area of the triangle formed by the three points is at most $1$ is $\dfrac{d-e\pi}{f}$, where $d$, $e$, and $f$ are positive integers with $\gcd(d,e,f)=1$. What is $d+e+f$? | nvidia |
A planar graph $G$ has $18$ vertices, $40$ edges, and $22$ faces, where each face is a triangle or a pentagon. A chord of $G$ is a line segment connecting two non-adjacent vertices that lie on the same face. Determine the maximum number of chords that $G$ can have. | nvidia |
Find the sum of the integers from -25 to 75, inclusive. | nvidia |
The midpoints of the sides of a triangle with vertices $(2,5)$, $(10,1)$, and $(6,9)$ form a new triangle. Find the sum of the squares of the coordinates of the vertex of the new triangle opposite the side that corresponds to the original side with endpoints $(2,5)$ and $(10,1)$. | nvidia |
Let $n \ge 2$ be an integer and let $T = \{1, 2, 3, \ldots, n\}$. Find the smallest value of $n$ such that for every partition of $T$ into three subsets, at least one of the subsets contains integers $x$, $y$, and $z$ (not necessarily distinct) such that $x + y = z$. | nvidia |
Let $g$ be a function taking the positive integers to the positive integers, such that
(i) $g$ is decreasing (i.e. $g(n + 1) < g(n)$ for all positive integers $n$)
(ii) $g(m + n) = g(m) + g(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m$ and $n$ are consecutive integers, then $g(m) = 2g(n)$ or $g(n) = 2g(m... | nvidia |
A circle is inscribed in a triangle with a given perimeter. The radius of the circle is 3 units and the area of the triangle is 30 units squared. Find the length of the longest side of the triangle. | nvidia |
Find the smallest positive integer $n$ such that there exists a convex polygon with $n$ sides having exactly $3$ right interior angles. | nvidia |
What is the sum of all integer values of $n$ such that $\frac{20}{2n - 1}$ is an integer? | nvidia |
Define a regular $n$-sided spiral to be the union of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that
the points $P_1, P_2,\ldots, P_n$ are coplanar and no three of them are collinear,
each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint,
all of t... | nvidia |
Let \(a,b,c\) be real numbers such that \(abc=1\). Prove that
\[\frac{a-1}{a+1}+\frac{b-1}{b+1}+\frac{c-1}{c+1}=3\left(\frac{a-1}{a+1}\cdot \frac{b-1}{b+1}\cdot\frac{c-1}{c+1}\right).\] | nvidia |
Find the sum of all the odd integers between 1 and 100 that are not divisible by 3. | nvidia |
Determine the number of integers n for which the cube of the sum of the digits of n is equal to the sum of the cubes of the digits of n^2. | nvidia |
Find a cubic polynomial, in $x,$ with rational coefficients such that $1+i\sqrt{2}$ is a root of the polynomial and the sum of the coefficients is equal to 5. | nvidia |
What is the largest five-digit number that can be formed using each of the digits 1, 2, 3, 4, and 5 exactly once, such that the sum of the first two digits is greater than the sum of the last two digits? | nvidia |
Find the value of x for which the sum of the squares of the real and imaginary parts of $3-x + (4+xi)$ is equal to 59. | nvidia |
In a sequence of points $(x_n, y_n)$, where $x_n$ are consecutive positive integers starting from 1 and $y_n$ are positive integers, it is known that the ratio of the slope between consecutive points is constant. Given that the points $(1, 6)$, $(2, 10)$, and $(3, 15)$ are in the sequence, determine the value of $y_6$ ... | nvidia |
Let $f(x)=2^x$ and $g(x)=3x-4$. The composition of the two functions can be written as $f(g(x))=2^{3x-4}$. Find the inverse of the composition of the two functions and determine the value of the inverse at $x=1$. | nvidia |
Determine $a$, such that the repeating part of the decimal representation of the fraction
$\ds\frac{a}{99}$
be equal to the non-repeating part of the decimal representation of the fraction
$\ds\frac{a}{9999}$. | nvidia |
What is the product of the positive integer divisors of a number that is equal to the sum of its own digits? | nvidia |
What are the last two digits in the sum of factorials of the first 100 positive integers? | nvidia |
Let $a$ and $b$ be two positive integers such that $ab$ is a perfect cube and $a+b$ is a perfect square. Determine the smallest possible value of $a+b$. | nvidia |
A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| nvidia |
Find the least positive integer that is divisible by 4, 5, and 7. | nvidia |
A quadratic function $f(x)$ has the property that $f(f(x)) = f(x)$ for all values of $x$. Determine the sum of all values of $x$ such that $f(x) = x$. | nvidia |
Consider the set of all positive integers whose base-10 representation consists only of 1's and 2's. What is the smallest positive integer that is not in this set? | nvidia |
Determine the value of $b$ for which the number $56b8$ is divisible by $45$. | nvidia |
Determine all real numbers $a$ such that for every non-zero real number $b$, we have
$$\frac{a^2+b^2}{a+b} \geq \frac{ab}{2}.$$
Express your answer in interval notation. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that their arithmetic mean, geometric mean, and harmonic mean form a geometric progression, that is,
$\frac{a+b+c}{3}, \sqrt[3]{abc}, \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$
form a geometric progression. Find $a+b+c$. | nvidia |
Find the largest value of $m$ such that $3x^2+mx+70$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
The diagonals of a regular hexagon are the graphs of two functions that intersect at six points. If the area of the hexagon is 48 square units, what is the sum of the $x$-coordinates and $y$-coordinates of the intersection points? | nvidia |
The sum of the positive integers $a$, $b$, and $c$ is 24. What is the maximum possible value of the product $abc$? | nvidia |
Find the equation of the plane that passes through the point $(2,3,1)$ and is perpendicular to the line defined by the equation $x=t, y=2t-1, z=3t+2$. Enter your answer in the form $Ax+By+Cz+D=0$, where $A, B, C, D$ are integers such that $A>0$ and $\gcd(|A|,|B|,|C|,|D|)=1$. | nvidia |
Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and
\[f(x - y) = \sqrt{f(xy) + 2}\]for all $x > y > 0.$ Determine $f(2009).$ | nvidia |
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