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Let $a$ be a real number such that $a > 0$ and
\[\prod_{n=1}^{\infty} \left(1 + \frac{2n+1}{a^n}\right) = \frac{27}{16}.\]Find $a$. | nvidia |
Determine the range of values of $a$ such that the inequality $ax^2+bx+c>0$ is satisfied for all real values of $x$. | nvidia |
Find the value of $x$ such that the sum of the squares of the numerator and denominator of the fraction $\frac{x^2+x}{x+1}$ is equal to the square of the whole fraction. | nvidia |
The equations
\begin{align*}
y&=3x\\
x&=my^2-1
\end{align*}define two curves on the plane. Determine the value of $m$ such that these curves intersect at exactly one point. | nvidia |
Find all pairs of positive integers (m, n) such that the quadratic equation $x^2 - mx + n = 0$ has two roots, one of which is a perfect square and the other is a perfect cube. | nvidia |
In a rectangular prism with integral dimensions, the sum of the lengths of the edges is $m$, and the sum of the areas of all six faces is $n$. Determine $n$ in terms of $m$. | nvidia |
Find the smallest positive integer $n$ such that the sum of the squares of the first $n$ positive integers is greater than 1000. | nvidia |
Two of the roots of the equation \[px^3+qx^2+rx+s=0\]are $2$ and $-1.$ Given that $p \neq 0,$ compute $\frac{q+r}{p}.$ | nvidia |
If $y$ is an even number, find the largest integer that always divides the expression
\[(6y-3)(4y+2)(2y-4)\] | nvidia |
Determine the number of whole numbers $x$ such that the inequality $2\sqrt{x}+3 \leq 7$ holds. | nvidia |
The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $10$. Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle. | nvidia |
How many 4-digit numbers can be formed using the digits 1, 2, 3, and 4, with the condition that the first digit cannot be 1, and the last digit must be even? Consider that some of the digits can be repeated. | nvidia |
Find the largest value of $x$ such that the expression $\frac{x^3-6x^2+11x-6}{x^2-4x+4}$ is negative. | nvidia |
Let $a$ and $b$ be two positive integers, and define $S(a,b)$ to be the set of positive integers that are not expressible as the sum of a multiple of $a$ and a multiple of $b$. Find the smallest positive integer $k$ such that $|S(3,k)| < 1000$. | nvidia |
Let $a$, $b$, and $c$ be three distinct positive integers such that $ab+bc+ca=38$, $a^2+b^2+c^2=93$, and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{8}$. Find $abc$. | nvidia |
In the equation $x^2+bx+c=0$, the sum of the squares of the roots is equal to the product of the roots. If the sum of the roots is -4, determine the value of $b+c$. | nvidia |
Compute the number of ways to choose a 5-letter word from the letters in the word "listen" such that the word contains exactly one repeated letter. | nvidia |
How many different positive integers can be represented as the sum of two perfect squares? | nvidia |
Let $S$ be the set of all positive integers $n$ such that $\sqrt{n}$ is not an integer. For each $n \in S$, let $f(n)$ be the smallest positive integer $k$ such that $n+k$ is a perfect square. Evaluate
\[\sum_{n \in S} \frac{1}{2^{f(n)}}.\] | nvidia |
Given that $\triangle ABC$ is an isosceles triangle with base $BC=50$, and point $E$ lies on line $AB$ such that $AE$ is perpendicular to $AB$. If the altitudes from $B$ and $C$ intersect at point $E$, determine the length of segment $BE$. | nvidia |
Find all real values of $x$ that satisfy $\frac{1}{x+1} + \frac{3}{x+7} \ge \frac23.$ (Give your answer in interval notation.) | nvidia |
Given that $2a+b=6$ and $a+2b=7$, find the value of $5a^2+3ab+5b^2$. | nvidia |
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 312^\circ.$ | nvidia |
Point $A$ is located inside $\triangle BCX$ such that $\overline{AX}\parallel\overline{BC}$ and $AX:BC=2:3$. Determine the ratio of the area of $\triangle AXC$ to the area of $\triangle ABC$. | nvidia |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | nvidia |
Find the minimum value of the expression $\frac{2x^2+3x+4}{x^2-2x+1}$ for real $x$. | nvidia |
In a quadrilateral, the perpendicular bisectors of the sides are tangent to a circle. Two opposite vertices of the quadrilateral are endpoints of a diameter of the circle, and the other two vertices are on the circle. Find the area of the quadrilateral in terms of the radius of the circle. | nvidia |
In a class of 32 students, 5 students scored an A on a math test. Which of the following fractions is closest to the proportion of students who scored an A?
$$
\frac{1}{6} \qquad \frac{1}{7} \qquad \frac{1}{8} \qquad \frac{1}{9} \qquad \frac{1}{10}
$$ | nvidia |
Find the sum of the squares of the first ten positive integers that are not divisible by 3. | nvidia |
Let $f(n)$ denote the number of 3-colorings of the vertices of the cube such that no two adjacent vertices have the same color. Let $g(n)$ denote the number of 3-colorings of the vertices of the octahedron (8 triangular faces, 12 edges, 6 vertices) such that no two adjacent vertices have the same color. Find $f(n)+g(n)... | nvidia |
Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$. | nvidia |
Let $\mathbf{A}$ be a $2\times 2$ matrix such that
$\mathbf{A}=\begin{pmatrix}\cos \frac{\pi}{14}&\sin\frac{\pi}{14}\\\sin\frac{\pi}{14}&-\cos\frac{\pi}{14}\end{pmatrix}$
and let $\mathbf{v}$ be a unit vector such that
$\mathbf{v}=\begin{pmatrix}\cos\frac{\pi}{10}\\\sin\frac{\pi}{10}\end{pmatrix}.$
Find the smallest po... | nvidia |
For a positive integer $m$ and nonzero digits $d$, $e$, and $f$, let $D_m$ be the $m$-digit integer each of whose digits is equal to $d$; let $E_m$ be the $m$-digit integer each of whose digits is equal to $e$, and let $F_m$ be the $3m$-digit (not $m$-digit) integer each of whose digits is equal to $f$. What is the gre... | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=18$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=6$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | nvidia |
The function $g(x)$ takes positive real numbers to real numbers, such that
\[xg(y) + yg(x) = g \left( \frac{x}{y} \right)\]for all positive real numbers $x$ and $y.$ Find all possible values of $g(50).$ Enter all possible values, separated by commas. | nvidia |
Let $x$ and $y$ be distinct real numbers such that
\[
\begin{vmatrix} 1 & 16 & 25 \\ 5 & x & y \\ 5 & y & x \end{vmatrix}
= 0.\]Find $x + y$. | nvidia |
Find the greatest integer n such that the sum of the least powers of 3 that divide the integers between 1 and 3^n (inclusive) is a perfect cube. | nvidia |
Find the smallest positive integer that can be expressed as the sum of two perfect squares in three different ways. | nvidia |
Problem:
Simplify the expression $\ds\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+...}}}$
and express it in the form $a+b\sqrt[3]{c}$ where $a,b,c$ are integers and $c$ has no factors that are perfect cubes of any positive integer other than 1. Find $a+b+c$. | nvidia |
Solve the equation $\sqrt{2x+1}=\sqrt{x+2}$, where $x$ is a real number. | nvidia |
Find the number of positive three-digit integers whose digits are consecutive terms of an arithmetic sequence. | nvidia |
In a parallelogram $ABCD$, the coordinates of the vertices are $A(3,0)$, $B(3,8)$, $C(x,8)$, and $D(x,0)$. A diagonal of the parallelogram intersects the line $y=2x-6$ at point $E$. If the area of $\triangle BDE$ is 15, determine the positive difference of the $x$ and $y$ coordinates of point $E$. | nvidia |
Find the number of positive integers not exceeding 100 that are either perfect squares or perfect cubes, but not both. | nvidia |
In a triangle, the sum of the lengths of the three sides is $x+y+z$. The area of the triangle can be expressed as $xy=36$, $xz=48$, and $yz=64$. If the perimeter of the triangle is minimized, what is the value of $x+y+z$? | nvidia |
In a regular hexagon, a circle is inscribed so that it touches all six sides of the hexagon. If the radius of the circle is 6 cm, find the area of the region outside the circle but inside the hexagon. | nvidia |
Find the quotient of two negative integers such that their product is 1008 and their difference is 48. | nvidia |
Find the smallest positive integer $k$ such that $
z^{10} + z^9 + z^6+z^5+z^4+z+1
$ divides $z^k-1$. | nvidia |
Let $f(x)$ be a monic polynomial with rational coefficients such that the roots of $f(x^2)$ are all integers and $-3, \pm5, \pm11, \pm12$ are roots of $f(x)$. What is the smallest possible degree of $f(x)$? | nvidia |
What is the largest integer that can be written in two different ways as the sum of three distinct integers? | nvidia |
A triangle is formed with one vertex at the vertex of the parabola $y=2x^2-3$ and the other two vertices at the intersections of the line $y=s$ and the parabola. If the area of the triangle is between $16$ and $128$ inclusive, find all possible values of $s$. Express your answer in interval notation. | 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 |
Find the number of divisors of the number 3^5 * 2^2 * 11 that are not multiples of 11. | nvidia |
Find the minimum value of
\[a^2 + b^2 + c^2 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c},\]
where $a$, $b$, and $c$ are positive real numbers such that $abc=1$. | nvidia |
In $\triangle RED$, $\measuredangle DRE=75^{\circ}$ and $\measuredangle RED=45^{\circ}$. $RD=1$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC}\perp\overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA=AR$. Then $AE=\fr... | nvidia |
A box contains $100$ colored balls, where each ball is colored with one of five distinct colors. Every two balls with the same color have the same weight, but the weights of the balls with different colors are all distinct. How many balls should be taken out of the box at random in order to guarantee that at least two ... | nvidia |
In a triangle, the lengths of the sides are given by $a$, $b$, and $c$. Prove that the expression $(a+b+c)^2 - (a^2+b^2+c^2)$ is always equal to $2(ab+bc+ca)$. | nvidia |
Let $\omega$ be a complex number with $|\omega|=2$. There is a positive real number $\lambda>1$ such that the points $\omega$, $\omega^2$ and $\lambda\overline{\omega}$ are vertices of an equilateral triangle. Find $\lambda$. | nvidia |
A base-5 integer is divisible by the base-10 number 7 if and only if the alternating sum of its digits is divisible by 7. Determine which of the base-5 integers 1103, 2123, and 3454 is divisible by 7. | nvidia |
Given that the point $(2,3)$ is on the graph of $y=g(x)$, there is one point that must be on the graph of $3y=2g(2x)-4$. What is the difference of the coordinates of that point? | nvidia |
What is the positive integer $N$ for which $$ 22^2 \times 55^2 = 10^2 \times N^2 \ ? $$ | nvidia |
In the expansion of $(x+1)^n$, it is known that the coefficients of the $x^k$ and $x^{k+1}$ terms are in a 2:3 ratio, and that the coefficient of the $x^{k+2}$ term is 165. Determine the value of n. | nvidia |
Find the area of the triangle bounded by the lines $y=2x,$ $y=-3x,$ and $y=4$. | nvidia |
How many positive integers $n$ have the property that $\frac{1}{n}$ can be expressed as a repeating decimal with a period of exactly 4 digits and a non-zero ten-thousands digit? | nvidia |
Find a three-digit positive integer that is divisible by 2, 3, 4, and 5, and when its digits are reversed, the resulting number is divisible by 7. | nvidia |
Find the sum of all integer values of $m$ such that $\frac{15}{3m + 2}$ is an integer. | nvidia |
Prove that for any positive integer $n$,
$\left(\dfrac{a_1^{b_1}}{a_2^{b_2}}\right)^0+\left(\dfrac{a_3^{b_3}}{a_4^{b_4}}\right)^0+\dots+\left(\dfrac{a_{2n-1}^{b_{2n-1}}}{a_{2n}^{b_{2n}}}\right)^0=n$,
where $a_i$ and $b_i$ are any non-zero integers for $i=1,2,\dots,2n$. | nvidia |
In the figure below, the graph of $y=x^2-4x+6$ is drawn. The graph of $y=k(x-2)^2+2$ is also shown, where $k$ is a constant. Find the value of $k$ such that the graphs of these two functions intersect at exactly one point. [asy]
import cse5; import olympiad;
size(200);
add(grid(10,10));
draw((0,0)--(10,0),linewidth(1.2... | nvidia |
Find the smallest positive integer $n$ such that $6n$ is a palindrome number, where $n$ is a two-digit number. | nvidia |
A lucky integer is a positive integer which is divisible by the sum of its digits. What is the least positive multiple of 9 that is not a lucky integer? | nvidia |
Find the smallest possible value of the radius of a circle that circumscribes a triangle whose sides are 3, 4, and 5. | nvidia |
Let $x$ and $y$ be real numbers such that $x^6-y^4=y^2-x$, $x^3+y^3 = 20$, and $x > y$. Determine the value of $x^3-y^2$. | nvidia |
A convex polygon with 7 sides is inscribed in a circle of radius 1. The lengths of the consecutive sides are such that each is twice the length of the previous one, starting with the shortest side with length 1. Find the length of the diagonal that connects the vertices of the shortest and the longest sides. | nvidia |
Find the number of terms in the sequence $1, 2, 4, 8, \ldots, 2^n$, where $n$ is a positive integer and the sequence is formed by multiplying the previous term by 2. | nvidia |
Determine $a$ such that the expression $\dfrac{\sqrt[5]{3^a}}{\sqrt[7]{3^{a+1}}}$ is equal to $\dfrac{1}{3}$. | nvidia |
A circle with center O and radius 5 cm is inscribed in a square. A line segment PQ, with P on the circle and Q on the square, is 10 cm long. Find the sum of all possible whole number values of the length of the line segment OQ. | nvidia |
Find all real numbers $x$ such that \[3 \le \frac{x}{2x-5} < 8.\](Give your answer in interval notation.) | nvidia |
Find the number of positive integers $n \le 500$ such that $n$ can be expressed as the difference of two perfect squares. | nvidia |
Given a positive integer $m$, the number $162m^5$ has exactly $162$ positive integer divisors, including $1$ and $162m^5$. What is the smallest possible value of $m$? | nvidia |
Determine the number of squares with positive area whose vertices are at points $(x,y)$ in the coordinate plane, where $x$ and $y$ are integers between $1$ and $7$, inclusive. | nvidia |
Find the smallest positive integer $n$ such that the polynomial $x^n + x^{n-1} + x^{n-2} +... + x + 1$ has a root that is also a root of the polynomial $x^4 + x^3 + x^2 + x + 1$. | nvidia |
Find the largest value of $d$ such that $-2$ is in the range of $f(x)=2x^2+4x+d$. | nvidia |
Find the smallest prime number that divides the sum of the squares of the first 10 positive integers. | nvidia |
Find the smallest value of $n$ such that $2x^2 + nx - 225$ can be expressed as the sum of two perfect squares with integer coefficients. | nvidia |
Find the largest possible value of x such that the sum of the squares of three consecutive integers, x, x+1, and x+2, is less than or equal to 2010. | nvidia |
Let $P(z)$ be a polynomial with real coefficients such that no zeroes of $P(z)$ lie on the unit circle. Prove that for every positive integer $n$, there exists a complex number $w$ on the unit circle such that $|P(w^n)| < 1/n$. | nvidia |
What is the smallest positive integer $n$ such that $2^n$ is a factor of $(2n)!$? | nvidia |
Given that the center of a circle is at (7,1) and it passes through the point (2,1), find the length of its tangent drawn from the origin. | nvidia |
Let a sequence be defined such that for n >= 3 the nth term is the sum of all the digits of the (n-1)st term, and the first two terms are 1 and 2. Determine the smallest positive integer k such that the kth term is 9. | nvidia |
A square is inscribed in a circle such that the diagonals of the square intersect at the center of the circle. If the coordinates of two adjacent vertices of the square are $(5, 2)$ and $(8, -1)$, what is the equation of the circle? | nvidia |
Find the smallest positive integer $k$ such that $\frac{k}{k+37}$ is equal to a terminating decimal. | nvidia |
Consider the graph of the function $\frac{x^2-4x+4}{x^3-x^2-2x}$. Let $p$ be the number of x-intercepts, $q$ be the number of y-intercepts, $r$ be the number of vertical asymptotes that are also roots of the numerator, and $s$ be the number of horizontal asymptotes that are also lines of symmetry for the graph. Determi... | nvidia |
Given that $a$, $b$, and $c$ are positive integers such that $a$ is the sum of two different primes, $b$ is the sum of two different perfect squares, and $c$ is the sum of a prime and a perfect square, what is the smallest possible value of $a+b+c$? | nvidia |
What is the sum of all positive integers less than 100 that are squares of perfect squares? | nvidia |
Let p be a prime number. Find the number of positive integer divisors of (p^3) * (2p^2) that are not divisible by p. | nvidia |
Find the sum of the digits of the number obtained by concatenating the palindromic numbers from 1 to 100, inclusive. | nvidia |
Consider the sequence $a_n$ where $a_1=1$ and $a_{n+1}=\frac{a_n+1}{2}$ for $n\geq 1$. Prove that the sequence converges and find its limit. | nvidia |
Increasing the radius of a cylinder by $6$ units increased the volume by $y$ cubic units. Increasing the height of the cylinder by $6$ units also increases the volume by $y$ cubic units. If the original height is $2$, then the original radius is:
$\text{(A) } 2 \qquad \text{(B) } 4 \qquad \text{(C) } 6 \qquad \text{(D)... | nvidia |
The repeating decimal $0.\overline{135}$ can be represented as the sum of two fractions $\frac{a}{b}$ and $\frac{c}{d}$, where $a, b, c,$ and $d$ are positive integers with no common factors and $b \neq d$. Determine the value of $a + b + c + d$. | nvidia |
Determine all possible values of $t$ such that the following system of equations has a solution:
$$\sqrt{t}+\frac{1}{z}=8$$
$$z-\frac{1}{\sqrt{t}}=11$$
(Enter your answer as a comma-separated list.) | nvidia |
Find the number of positive integers $n$ such that the greatest common divisor of $n^2+1$ and $n^2+2$ is a prime number. | nvidia |
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