instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Find the value of $x$ that satisfies the equation $\sqrt{x^2 + \sqrt{2x+1}} = x+1$. | nvidia |
A moving robot starts at the point $(6,6)$ and moves until it hits one of the coordinate axes for the first time. When the robot is at the point $(x,y)$, it moves at random to one of the points $(x-1,y)$, $(x,y-1)$, or $(x-2,y-1)$, each with probability $\frac{1}{3}$, independently of its previous moves. The probabilit... | nvidia |
Let $x$ and $y$ be positive integers such that $x+y<10$. What is the sum of all possible integer values of $x-y$? | nvidia |
The numbers from 1 to 150, inclusive, are placed on the squares of a 15x10 grid, such that no perfect power shares a row or column with another perfect power. What is the minimum number of rows and columns that must be cleared to ensure that no perfect power remains on the grid? | nvidia |
Find the sum of all the integers from 1 to 100 that are not divisible by 2 or 3. | nvidia |
Find the smallest positive five-digit number that is divisible by each of the five smallest prime numbers. | nvidia |
For how many ordered pairs of positive integers $(x,y),$ with $x \le 50$ and $y < x,$ are both $\frac{x}{y}$ and $\frac{x+2}{y+2}$ integers? | nvidia |
Find the number of positive integers less than 1000 that have at least one digit equal to 7. | nvidia |
Let $x$ be a positive number, such that $10x^2+2x$ is a 3-digit number. How many integer values of $x$ exist, such that $10x^2+2x$ is divisible by 4? | nvidia |
What is the smallest positive integer $n$ such that $n$ and $2n+1$ are both perfect squares? | nvidia |
Count the number of three-digit palindromes that can be formed using the digits 0 through 9, with the restriction that the first digit cannot be 0. | nvidia |
Problem:
A positive integer $N$ can be represented as the sum of two distinct perfect squares in two different ways:
$$N=a^2+b^2=c^2+d^2,$$
where $a$, $b$, $c$, and $d$ are positive integers, $a>b$, and $c>d$. Furthermore, it is known that $\gcd(a,b)=\gcd(c,d)=1$. Let $x$ be the number of prime factors in the prime ... | nvidia |
Let x, y, z be three positive real numbers such that
$$(x^2+xy+y^2)(y^2+yz+z^2)(z^2+zx+x^2)=x^2y^2z^2$$
Determine the value of $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$$ | nvidia |
What is the least positive integer $n$ such that $n^2 - n$ is divisible by some but not all integer values of $k$ when $1\leq k \leq n$? | nvidia |
What is the sum of all integer values of $n$ such that $\frac{20}{2n - 1}$ is an integer? | nvidia |
Let $g$ be a function that maps positive integers to positive integers, satisfying the following conditions:
(i) $g(n + 1) > g(n)$ for all positive integers $n$,
(ii) $g(mn) = g(m) + g(n)$ for all positive integers $m$ and $n$, and
(iii) for any prime number $p$, if $g(p)$ is even, then $g(p^2) = 2g(p)$.
Find the sma... | nvidia |
Determine the smallest positive integer $n$ such that the number of distinct prime factors of $n^2+1$ is greater than the number of distinct prime factors of $n$. | nvidia |
Find the number of positive integer solutions to the equation $2^a \cdot 3^b \cdot 5^c = 3600$, where $a, b,$ and $c$ are natural numbers. | nvidia |
Consider a set of all four-digit positive integers with the following property: the sum of their digits in the tens and thousands places is equal to the sum of their digits in the ones and hundreds places. Compute the remainder when the sum of all such integers is divided by 1111. | nvidia |
Find all values of $a$ for which the points $(0,0,0),$ $(1,a,0),$ $(0,1,a),$ and $(a,0,1)$ are coplanar. | nvidia |
Let $A$ be a positive integer such that $A$ and $2A$ both end in the same sequence of three digits $XYZ$ when written in base 12, where $X$ is not equal to zero. Find the two-digit number $XY$ (in base 12). | nvidia |
Two hedgehogs found three baskets, each with 900 strawberries. The hedgehogs then each ate an equal number of strawberries. If 2/9 of the strawberries were remaining out of all the strawberries that were there, calculate the total number of strawberries that each hedgehog ate. | nvidia |
Find the smallest positive integer $n$ such that $13n \equiv 5 \pmod{16}$. | nvidia |
Given that the polynomial $x^2-kx+16$ has only positive integer roots, find the average of all distinct possibilities for $k$. | nvidia |
Find the number of positive three-digit integers that are divisible by 7 and have a sum of digits equal to 12. | nvidia |
In the plane, two circles are given, centered at $A$ and $B$ with radii $r$ and $s$, respectively. The circles intersect at two points $P$ and $Q$ such that $AP$ is perpendicular to $BQ$. Find the length of the chord $PQ$. | nvidia |
Find the product of all real numbers that are not less than 5 and not greater than 8. | nvidia |
Given that $a$ and $b$ are positive real numbers such that
\[\sin \left( \sqrt{a + b} \right) = \frac{\sin \sqrt{a} + \sin \sqrt{b}}{2},\]find the maximum possible value of $\sin \sqrt{a}$. | nvidia |
Find the product of all $y$ that satisfy the equation $\frac{4y^2}{y^2+4} = \frac{y}{y-2} + \frac{3}{y+2}.$ | nvidia |
Find the number of different triangles that can be formed with a perimeter of 10 units, where each side must have integral length. | nvidia |
The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real who... | nvidia |
Find the number of ways to arrange the digits of the number 12345 such that the sum of the digits in the even positions is equal to the sum of the digits in the odd positions. | nvidia |
Let $A$, $B$, and $C$ be points on the coordinate plane, such that $\triangle ABC$ is an equilateral triangle with side length $6$. Let $P$ be a point inside $\triangle ABC$ such that the distances $PA$, $PB$, and $PC$ are distinct integers. Find the smallest possible perimeter of $\triangle PAB$. | nvidia |
Given that $a$ and $b$ are positive integers with $\text{lcm}(a,b) = 42$, what is the smallest possible value of $\text{lcm}(a+7,b+7)$? | nvidia |
Find the greatest value of $b$ such that the inequality $b^3-7b^2+14b-8\geq0$ holds true for all real values of $b$. | nvidia |
Find the sum of the x-coordinates of the solutions to the system of equations y=|x^2+3x-7| and y=sqrt(20-x^2). | nvidia |
Determine the number of positive integers $n$ with at most four digits such that the congruence $n^2\equiv 4\pmod{41}$ holds. | nvidia |
When two fair 6-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that the two digit number $ab$ (where $a$ and $b$ are digits) and $a$ and $b$ are all divisible by 3? | nvidia |
Compute: $55\times1212-15\times1212$ . | nvidia |
Find the equation of the parabola that has its vertex at the origin, its axis of symmetry along the y-axis, and passes through the point (2, 3). | nvidia |
Find all positive integers $n$ such that the number of its divisors is equal to the product of the prime divisors of $n$ counting multiplicities. | nvidia |
The letters of the English alphabet are placed on the faces of three standard dice, with one letter on each face. The dice are tossed, and the letters $a,b,c$ are obtained. Find the probability that $a,b,c$ are the first three letters of a valid English word. | nvidia |
Find the number of positive integers $n$ such that the inequality $2n^2 + 5n - 7 \leq 0$ holds true. | nvidia |
Find the set of all negative values of $b$ so that the inequality $2x^2+bx+1\leq 0$ has exactly two integer solutions for $x$. Express your answer in interval notation. | nvidia |
A regular hexagon is cut into four congruent triangles by drawing three diagonals from the vertices of the hexagon. A smaller regular hexagon is then inscribed in each triangle. The ratio of the area of one of the small hexagons to the area of the entire regular hexagon is $\frac{m}{n}$, where $m$ and $n$ are relativel... | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 1.$ Find the minimum value of
\[\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a}.\] | nvidia |
Let $b_1=1$, and for each positive integer $n>1$ let $b_n$ be the integer nearest to $\sqrt{b_{n-1}+n}$. Find the least positive integer $n$ such that $b_n$ is a perfect square. | nvidia |
Find the smallest positive integer $n$ such that the number of prime divisors of $\frac{n!}{n^2}$ is greater than the number of digits in $n^2$. | nvidia |
Define a positive integer $n$ to be a factorial remainder if there is some positive integer $m$ such that the remainder when $m!$ is divided by $1000$ is exactly $n$. How many positive integers less than $1000$ are not factorial remainders? | nvidia |
Find the smallest positive integer $n$ such that $3n+7$ leaves a remainder of 5 when divided by 11. | nvidia |
Simplify the expression: $\sqrt{9x^2} \div 3 \cdot x$ for all positive real numbers $x$. | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=7$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | nvidia |
What is the sum of all positive integers less than 200 that are cubes of perfect cubes? | nvidia |
In the quadratic equation x^2 + bx + c = 0, where b and c are positive integers, determine the largest possible value of c such that the roots of the equation are reciprocals of each other. | nvidia |
Find the positive real number $x$ such that $\lfloor x \rfloor \cdot x = 70$. Express $x$ as a decimal. | nvidia |
Let $a, b, c$ be positive real numbers so that
\begin{align*}
a^2+b^2&=12,\\
b^2+c^2&=8,\\
c^2+a^2&=10.
\end{align*}
Find $abc$. | nvidia |
In a polyhedron, the number of vertices is half the number of faces, and the number of edges is equal to the number of faces minus two. If the sum of the number of faces, edges, and vertices of this polyhedron is 54, determine the number of faces. | nvidia |
How many distinct four-digit positive integers have digits that alternate between even and odd? | nvidia |
The hyperbola $x^2-y^2=1$ has two foci $F_1$ and $F_2$. Let $A$ and $B$ be the points on the hyperbola such that the triangles $F_1AF_2$ and $F_1BF_2$ have a right angle at $A$ and $B$, respectively. Find the distance between $A$ and $B$. | nvidia |
Determine the smallest positive integer that has exactly four different prime divisors. | nvidia |
Let $m$ be a positive integer. For how many values of $m$ does there exist an integer $n$ such that $n$ is a divisor of $m$ and $\varphi(m) + \varphi(n) = m$? | nvidia |
How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides $8,$ $15,$ and $17$? | nvidia |
Find the largest base $b$ such that the sum of the digits of $1000$ in base $b$ is not equal to $10$. | nvidia |
Let $g$ be a function mapping integers to integers such that
\[g(mn) + g(m + n - 1) = g(m) g(n) + 1\]
for all integers $m$ and $n.$
Let $k$ be the number of possible values of $g(3),$ and let $t$ be the sum of all possible values of $g(3).$ Find $k \times t.$ | nvidia |
How many 4-digit numbers can be formed using only the digits 1, 2, 3, 4, 5, 6, and 7, such that no two consecutive digits are the same and the number is divisible by 3? | nvidia |
Let $b_1,$ $b_2,$ $\dots$ be a sequence of positive integers such that
\[b_n = b_{n - 1} + \sqrt{b_{n - 1} + n}\]for all $n > 1.$ Find the smallest possible value of $b_1$ such that the sequence is strictly increasing. | nvidia |
Find the number of digits that repeat in the decimal representation of $\frac{1}{73}$. | nvidia |
Find the number of pairs of positive integers $(x, y)$ such that $x$ and $y$ are relatively prime and their product is 555. | nvidia |
Let $g$ be a function defined on the positive real numbers, such that
\[g\left(\frac{x+y}{2}\right) = \frac{g(x) + g(y)}{2}\]for all positive real numbers $x$ and $y.$ Given that $g(1) = 3,$ $g(3) = 7,$ and $g(5) = 11,$ find $g(2).$ | nvidia |
What is the largest positive integer $m$ such that $4m$ is a perfect cube and $3m+1$ is a perfect square? | nvidia |
Find the number of positive integer solutions to the inequality $|7x-4| \leq 23$ that are also solutions to the congruence $x \equiv 3 \pmod 4$. | nvidia |
Positive real numbers $a$ and $b$ satisfy $a^2+b^2=16$ and $a^2b^2=\frac{2019}{4}$. Find $ab$. | nvidia |
Compute the number of distinct subsets of size $3$ that can be formed from the elements $\left\{-\frac{1}{4}, \frac{7}{10}, \pi,\frac{5}{6},-11,49,-99,72,\sqrt{8}, 45\right\}$. | nvidia |
Compute the largest integer $k$ such that $2004^k$ divides the number of ways to arrange $2004$ objects in a circle. | nvidia |
Find all ordered pairs $(a,b)$ such that
$\begin{cases}
2a-3b=1, \\
b^2-a^2=-12, \\
\end{cases}$
and determine the value of $3(a_1+a_2+a_3+a_4)+2(b_1+b_2+b_3+b_4)$. | nvidia |
Find all possible real values of $a$ and $b$ such that the function $f(x) = ax^3 + bx^2 + 1$ satisfies the equation $f(x) + f(-x) = f(x^2)$. | nvidia |
Let $a$ and $b$ be two positive integers such that the equation $x^2+ax+by=0$ has two distinct integer roots. Find the number of ordered pairs $(a,b)$ such that the quadratic equation $x^2+bx+ay=0$ also has two distinct integer roots. | nvidia |
Let $G(x)$ be a function such that $G(2) = 5$ and
\[\frac{G(x+2)}{G(x-2)} = \frac{x^2+4x+4}{x^2-4x+4}\]for $x \in \mathbb{R}$ such that both sides are defined. Find $G(10)$. | nvidia |
Find the four-digit integer $pqrs$ such that both $pqrs$ and $pqrs^3$ end in the same sequence of four digits, where $p$ is not zero. | nvidia |
Let $T$ be the set of all positive integers. Let $g : T \to T$ be a function such that
\[g(x) \times g(y) = g(x + y) + g(xy)\]for all $x,$ $y \in T.$
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 $m + t.$ | nvidia |
What is the smallest positive integer $n$ such that $2^{2n}-1$ is divisible by 5? | nvidia |
In the equation $0.8 = \frac{y}{186+x}$, $x$ and $y$ are positive integers. Find the largest value of $x$ that satisfies the equation. | nvidia |
Let's consider the following geometric shape - an octagon, for which we know that the sum of interior angles is equal to 1080 degrees.
Determine the product of the tangents of the angles between the diagonals and the extensions of the sides of the octagon. | nvidia |
Find the number of integers $m$ such that
\[1+\left\lfloor\dfrac{123m}{124}\right\rfloor=\left\lceil\dfrac{122m}{123}\right\rceil.\] | nvidia |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \cos 170^\circ & -\sin 170^\circ \\ \sin 170^\circ & \cos 170^\circ \end{pmatrix}^n = \mathbf{I}.\] | nvidia |
Let $a$ and $b$ be positive integers such that $\frac{a^2+b^2}{2ab}=\frac{b}{a}$.
If $\gcd(a,b)=1$, find $b$. | nvidia |
In the decimal representation of positive integers, what is the smallest number of digits that can appear in a positive integer with a digit sum of 2009? | 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 |
What is the smallest positive integer that has exactly 5 positive divisors? | nvidia |
Given that the lines $y=2$, $y=5$, $x=1$, and $x=a$ make a square, find the value of $a$ that maximizes the area of the square. | nvidia |
Compute the value of $x$ such that
$\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\cdots\right)\left(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots\right)=1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\cdots$. | nvidia |
Let $S_n$ denote the set of all possible values of $\sum_{1 \le i < j \le n} a_i a_j$, where $a_1,$ $a_2,$ $\dots,$ $a_n$ are all equal to $1$ or $-1$. Determine the smallest positive integer $n$ for which $S_n$ contains at least one even number. | nvidia |
Find the sum of the positive factors of 72. | nvidia |
Among all pairs of real numbers $(a, b)$ such that $\cos \cos a = \cos \cos b$ with $-15 \pi \le a, b \le 15 \pi$, Maya randomly selected a pair $(A, B)$. Compute the probability that $A = B$. | nvidia |
Compute the number of pairs of distinct positive integers whose sum is equal to 51. | nvidia |
Find the smallest positive integer $z$ such that the expression $xy+4x-12$ is divisible by $15$, where $x$ and $y$ are positive integers with a difference of $2$. | nvidia |
Find the number of positive integers less than 60 that can be expressed as the sum of two perfect squares in more than one way. | nvidia |
Given a positive real number $a$, let $x$ be a solution to the equation
$$\sqrt[3]{1-x^3} + \sqrt[3]{1+a^3x^3} = 1.$$Find the value of $a$ that maximizes the product $ax^3$. | nvidia |
If $x$ and $y$ are positive integers such that $x^a = y^b$, where $a$ and $b$ are also positive integers, find the value of $x^{1/a} + y^{1/b}$. | nvidia |
Determine the maximum number of real solutions to the equation
$$\sum_{k=0}^{n} k!x^k=0$$
where $n$ is a positive integer. | nvidia |
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