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The sum of two positive integers $a$ and $b$ is 1001. What is the largest possible value of $\gcd(a,b)$? | nvidia |
Let $a$ and $b$ be positive integers such that $a+b=24$.
Find the maximum value of $a \cdot b$. | nvidia |
Find the maximum value of
$$\frac{\sqrt{ax^2+by^2}+cz}{\sqrt{x^2+y^2+z^2}}$$
over all positive real numbers $a, b, c, x, y, z$ subject to the constraint $ax^2+by^2+cz^2=1.$ | nvidia |
Let $S$ be the set of positive integers $n$ such that when $\frac{1}{n}$ is expressed as a decimal, the sum of the first $n$ digits after the decimal point is divisible by $n$. What is the largest element of $S$? | nvidia |
What is the greatest number $n$ such that $n!$ can be expressed as the product of $n$ consecutive positive integers? | nvidia |
The number of sides of a polygon is equal to the number of diagonals of the polygon. Find the number of sides of the polygon. | nvidia |
Find the value of $a$ such that $\log_{a^2}2x=3$ and $x=9$. | nvidia |
Find the sum of the $2007$ roots of $(x-1)^{2007}+2(x-2)^{2006}+3(x-3)^{2005}+\cdots+2006(x-2006)^2+2007(x-2007)$.
| nvidia |
Determine all integers $n$ such that the number $5^n+7^n$ is divisible by $12$. | nvidia |
What is the sum of all integer values of $N$ such that $N^2$ is strictly between 120 and 160? | nvidia |
Find all positive integers $n$ such that $\phi(n)$ is a prime number, where $\phi(n)$ denotes Euler's totient function. | 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 |
Find the sum of the largest and smallest divisors of the number 24 that are relatively prime to each other. | nvidia |
Let $r_1$ and $r_2$ be the roots of the equation $x^2+ax+b=0$. Determine the value of $b$ such that $r_1^3+r_2^3=8$. | nvidia |
Given that the point $(9,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=\frac{f(2x)}2+2$. What is the sum of coordinates of that point? | nvidia |
A rectangle has integer dimensions and a diagonal length that is also an integer. Find the smallest possible perimeter of such a rectangle where the length of the diagonal is a prime number. | nvidia |
Find all real numbers $a$ such that the equation
\[ \sqrt[3]{x} = \frac{a}{1 + \sqrt[3]{x}}\]
has exactly two distinct solutions. | nvidia |
Let $g(x)$ be a function defined on the real numbers such that
$$g(x)=\begin{cases} x^2+d &\text{ if } x>2\\
x+e &\text{ if }x\leq 2.
\end{cases}$$
If $g(1)=-1$, $g(2)=0$, and $g(3)=12$, find $d+e$. | nvidia |
The polynomial $x^3+ax^2+bx+c$ has three positive integer roots that form an arithmetic progression. If the product of the roots is 1350 and the sum of the roots is 30, find the middle root. | nvidia |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}^n = \mathbf{I}.\] | nvidia |
Given a positive real number $x$, find the smallest value of $x$ that satisfies the equation:
$$\frac{x^2}{2} + \frac{1}{x} = \frac{x^2 + 1}{\sqrt{x^2 + 1}}$$ | nvidia |
Find the product of all positive integers less than or equal to 12 that have an odd number of positive divisors. | nvidia |
Four fair six-sided dice are stacked up. What is the smallest number of sides that must be visible such that the sum of all the numbers on the visible sides is at least 20? | nvidia |
Let $a$ and $b$ be positive integers such that
$$a^b-b^a=2a+b,$$
$$a^b+b^a=a+2b.$$
Determine the ordered pair $(a,b)$. | nvidia |
Find the number of divisors of 216 that are not divisible by 3. | nvidia |
A box contains a certain number of light bulbs, some of which are defective. When three bulbs are selected randomly without replacement, there is a probability of exactly $\frac{1}{3}$ that all three are defective. Determine the smallest possible number of light bulbs in the box that is consistent with this data. | nvidia |
A circle is divided into n equal arcs. If the measure of each arc is 45 degrees, find the measure of the central angle subtended by a chord that connects two points on the circle that are 3 arcs apart. | nvidia |
Determine the value of $c$ such that the inequality $x^2+cx+5<0$ has exactly one integer solution. | nvidia |
Find the length of the diagonal of a rectangle whose sides are two consecutive even integers, given that its area is equal to the perimeter of the rectangle increased by 32. | nvidia |
What is the sum of all the positive integers less than or equal to $20$ that are relatively prime to $20$? | nvidia |
A parabola has its vertex at $(2,3)$ and an axis of symmetry of $x=2$. What is the sum of the $x$-coordinates of the two points on the parabola that are also on the line $y=5$? | nvidia |
Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positi... | nvidia |
Find the number of positive integers $n,$ $1 \le n \le 1000,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
Given the equation
$$5x^2-kx+1=0$$
determine $k$ such that the difference of the roots be equal to unity.
Write another problem inspired by this one:
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
Let $f(x) = x^3 + ax^2 + bx + c$ be a cubic polynomial. Suppose the sum of the roots of $f(x)$ is $-2$, the sum of the squares of the roots is $12$, and the sum of the cubes of the roots is $-20$. Find the value of $a$. | nvidia |
The sum of the squares of three consecutive positive integers is equal to 50. Determine the smallest of these integers. | nvidia |
Find the number of 5-digit numbers that start with 3 and have exactly three identical digits. | nvidia |
If $a$, $b$, and $c$ are positive integers such that $a^2 \cdot b^2 \cdot c^2 = 2^4 \cdot 3^2 \cdot 5^2$, how many different values of $a \cdot b \cdot c$ are possible? | nvidia |
What is the sum of all the four-digit positive integers that end in 5? | nvidia |
Given that $x$ is a positive multiple of $3$, $x^3 > 30$, but $3x < 21$, find the value of $x$. | nvidia |
Let n be a positive integer. The number of positive divisors of $n^3$ that are less than $n^2$, divided by the number of positive divisors of $n^2$ that are less than $n$, equals 5. Determine n. | nvidia |
A circle with center $O$ and radius 2 is given. Find the smallest possible distance between the center $O$ and a point $A$ that lies outside the circle, given that the distance from $A$ to any point on the circle is greater than or equal to the distance from $A$ to a fixed point $P$, which lies inside the circle and ha... | nvidia |
How many of the integers between 1 and 2000, inclusive, can be expressed as the sum of the cubes of two nonnegative integers? | nvidia |
Find the value of $x$ such that $0.\overline{x234} = \frac{9}{11}$. | nvidia |
Find all values of $x$ such that $\lceil x \rceil - x = 0.7$ and $x$ is a non-negative real number. | nvidia |
If $p$, $q$, and $r$ are digits and $0.pqr$ can be expressed as $\frac{1}{n}$ where $n$ is an integer such that $0<n\le8$, then what is the smallest possible value of $p+q+r$? | nvidia |
Consider the set of consecutive positive integers {1, 2, 3,..., 8}. The elements of this set are to be divided into two subsets, each containing four elements. The sum of the elements of one subset is 20, and the sum of the elements of the other subset is 20 as well. What is the maximum possible value of the product of... | nvidia |
The product of the areas of two similar triangles is equal to 1125. If one triangle has sides that are 3/5 times the length of the corresponding sides of the other triangle, what is the area of the larger triangle? | nvidia |
If $x$, $y$, and $z$ are three positive integers such that $\sqrt{x} + \sqrt{y} = \sqrt{z}$, then prove that $x$, $y$, and $z$ form a Pythagorean triplet. Use this result to conclude that there are infinitely many triplets of positive integers that satisfy this property. | nvidia |
Find the coordinates of the point $P$ that maximizes the sum of the squares of the distances from $P$ to the vertices of the triangle with vertices $(0,0)$, $(1,2)$, and $(3,1)$. | nvidia |
In triangle $ABC$, the altitude from $A$ to $BC$ divides $BC$ into segments of lengths $x$ and $y$, where $x$ and $y$ are positive integers and $x>y$. If the area of triangle $ABC$ is $\sqrt{5}$, what is the length of side $AB$? | nvidia |
Determine the number of natural numbers from 1 to 1000 that are neither perfect squares nor perfect cubes. | nvidia |
A set of eight positive integers has the following properties:
$\bullet$ The only two integers that occur more than once in the list are $3$ and $5,$ each occurring twice,
$\bullet$ the list contains at least two even integers,
$\bullet$ its median is $6,$ and
$\bullet$ its average (mean) is $7.$
What is the l... | nvidia |
There are four points that are $5$ units from the line $y=13$ and $13$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? | nvidia |
In a circle with center O, points A and B are diametrically opposite. Point C is on the circle so that angle AOC measures 60 degrees. The point P is on the line AB such that CP is perpendicular to AB. Find the ratio of the length of CP to the radius of the circle. | nvidia |
Let $a$ and $b$ be two positive real numbers such that $a + b = 1$. Determine the smallest possible value of $\frac{a}{b^2} + \frac{b}{a^2}$. | nvidia |
Suppose that $a$ and $b$ are positive integers such that $(a+bi)^3 = 10+15i$. Find $a+bi$. | nvidia |
Determine the number of distinct circles that can be drawn on a coordinate plane such that each circle passes through the points (1, 1) and (1, -1) and its center lies on the line y = x + 1. | nvidia |
Find the largest integer $m$ such that $m\leq 2023$ and the sum of the areas of the squares with side lengths equal to the terms of the sequence $a_n = \sqrt{n}$, $n = 1, 2, 3, \ldots, m$, is an integer. | nvidia |
Given the equation $x^2+2mx+n=0$, where the roots are $\alpha$ and $\beta$, determine $m$ such that $\alpha+\beta=4$. | nvidia |
Someone observed that $8! = 10 \cdot 11 \cdot 12 \cdot 13$. Find the largest positive integer $m$ for which $m!$ can be expressed as the product of $m - 4$ consecutive positive integers. | nvidia |
Find the number of solutions to the equation
$$x^2+y^2+z^2-w^2=81,$$
where x, y, z and w are positive integers and $(x,y,z,w)$ is identified with $(u,v,t,s)$ whenever $\{x,y,z\}=\{u,v,t\}$ and $w=s$. | nvidia |
Determine all positive integers $a$ and $b$ such that $(a+b)^3-a^3-b^3$ is divisible by $7$. | nvidia |
The units digit of the product of two numbers is equal to the units digit of their sum. The product of the two numbers is 386. What is the sum of the two numbers? | nvidia |
Let $m$ be the product of all positive integers less than $4!$ which are invertible modulo $4!$. Find the remainder when $m$ is divided by $4!$.
(Here $n!$ denotes $1\times\cdots\times n$ for each positive integer $n$.) | nvidia |
Find the minimum value of
\[x^2 + xy + y^2\]over all real numbers $x$ and $y$ satisfying the condition $x + y = 1$. | nvidia |
A deck of $n$ cards, numbered from $1$ to $n$, is shuffled. What is the probability that the number on the card at the top of the deck is greater than the number on the card at the bottom of the deck? | nvidia |
Find the sum of all positive integers $n$ less than or equal to 100 such that $n$ can be expressed as the product of exactly two distinct prime numbers. | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $a$ is a divisor of $b$ and $b$ is a divisor of $c$. If the sum of the divisors of $a$ is $15$, the sum of the divisors of $b$ is $32$, and the sum of the divisors of $c$ is $45$, what is the value of $\dfrac{c}{ab}$? | nvidia |
How many integers $N$ less than $1000$ can be written as the sum of $j$ consecutive positive odd integers from exactly 5 values of $j\ge 1$?
| nvidia |
When two standard dice are rolled, the numbers $x$ and $y$ are obtained. Find the probability that $xy = 30$. | nvidia |
Find the value of m/n such that the square of the length of each side of a square inscribed in the circle whose equation is x^2+y^2=1 is m/n, where m and n are relatively prime positive integers. The length of the diagonal of the square that passes through the center of the circle is 1 unit. Determine m+n. | nvidia |
A circle is inscribed in a triangle with side lengths $a$, $b$, and $c$. Two of the sides, $a$ and $b$, are each divided into $n$ equal segments. Let $p(a, b)$ be the probability that a randomly chosen point on one of the segments of $a$ and a randomly chosen point on one of the segments of $b$ are on the same side of ... | nvidia |
There are 2023 positive integers with both their sum and the sum of their reciprocals equal to 2024. Let y be one of these integers. Find the minimum value of y + 1/y. | nvidia |
What is the sum of all of the solutions of the equation $\frac{4x}{20}=\frac{5}{x}$? | nvidia |
Determine the highest power of 3 that divides the sum of all numbers from 1 to 216. | nvidia |
Let $f$ be a function from the set of positive integers to the set of positive integers such that
(i) $f(1)=1$
(ii) For every $n>1$, there exists a positive integer $k>1$ such that $f(n)=f\left(\frac{n}{k}\right)+f(k)$
Let $N$ be the number of possible values of $f(2019)$, and let $M$ be the maximum possible value o... | nvidia |
Find the sum of all values of $y$ such that $3^{y^2-2y-1} = 9^{y - 3}$. | nvidia |
Find the largest integer $n$ such that $e^n$ is less than 2023 times $\pi$. | nvidia |
Find the sum of all the positive integers less than or equal to 100 that are not divisible by 7. | nvidia |
Let $x$, $y$, and $z$ be real numbers such that
\begin{align*}
(x+y+z)^2&=9,\\
x(y+z)+y(x+z)+z(x+y)&=10.
\end{align*}
Find $xyz$. | nvidia |
The angles of elevation of the top of a tower from two points at distance 20 and 18 meters from the base of the tower and on the same straight line through the base are complementary. Prove that the height of the tower is 12 meters. | nvidia |
Compute the sum of the roots of the equation \[x\sqrt{x} - 6x + 7\sqrt{x} - 1 = 0,\]given that all of the roots are real and nonnegative. | nvidia |
Let $a$ be a three-digit number such that there is no repeating digit.
If one wants to add a two-digit number $x$ to $a$, and then a one-digit number $y$ to the sum, such that the final number is the three-digit number that is given by the digits of $a$ in reverse order, how many possible values of $a$ are there? | nvidia |
Find a six-digit integer that is divisible by 111, if the sum of its digits is 27 and the fourth digit is twice the first one. | nvidia |
Let $N$ be a positive integer such that the number of digits of $2^N$ is 1000 more than the number of digits of $5^N$. How many elements of the set $\{2^k\cdot 5^j | k,j \in \mathbb{Z}, 0 \le k,j \le N \}$ have a units digit of 0? | nvidia |
Find a positive integer that is divisible by 21 and has a cube root between 30 and 30.5. | nvidia |
Determine all real $t$ such that for any non-negative real numbers $a,$ $b,$ $c,$ and $d$ satisfying $a^2+b^2+c^2+d^2=1$, the inequality
\[(a^2+b^2)\sin t+(c^2+d^2)\cos t>\frac{1}{\sqrt{2}}\]
holds. | nvidia |
In triangle $ABC$, let $I$ be the incenter and $G$ be the centroid. Given that $IG = 8$ and the area of triangle $ABC$ is 32, find the value of $a^2 + b^2 + c^2$. | nvidia |
In a regular octagon inscribed in a circle, the length of one side is equal to the radius of the circle. Find the perimeter of the octagon if the area of the circle is $256\pi$ square units. | nvidia |
There are 2011 positive numbers with both their sum and the sum of their reciprocals equal to 2012. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}.$
Here's another problem:
A set of $n$ non-zero real numbers is such that the sum of the squares of the numbers is equal to the sum of the r... | nvidia |
Problem:
Three congruent circles are drawn inside an equilateral triangle such that they are tangent to each other and to the sides of the triangle.
If the side length of the triangle is 18, determine the radius of one of the circles. | 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 |
What is the smallest positive integer $n$ such that $\log_{5}{n}$ is an integer? | nvidia |
[asy] draw((0,0)--(2,2)--(5/2,1/2)--(2,0)--cycle,dot); MP("A",(0,0),W);MP("B",(2,2),N);MP("C",(5/2,1/2),SE);MP("D",(2,0),S); MP("a",(1,0),N);MP("b",(17/8,1/8),N); [/asy]
In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice that of angle $B$, and the measures of segments $AD... | nvidia |
How many positive integers will have a remainder of 3 when divided into 97, if it is known that these integers are all prime numbers? | nvidia |
Find the number of positive divisors of $M$ if $M = 2^5 \cdot 3^2 \cdot 11^3 \cdot 13$. | nvidia |
The sides of the acute triangle $ABC$ are known to have lengths $AB=7$, $BC=8$ and $CA=9$.
If point $P$ inside the triangle $ABC$ is such that
the distances $AP$, $BP$ and $CP$ from the vertices to $P$ are equal, determine $\sin\angle BAC$. | nvidia |
The equation $x^2 + kx + m = 0$ has two distinct solutions $x_1$ and $x_2$ such that the sum of their squares is equal to twice the product of $x_1$ and $x_2$. Find the value of $k$ in terms of $m$. | nvidia |
Find the value of $x$ such that the four-digit number $\underline{x24x}$ is divisible by 12. | nvidia |
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