instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Find the largest value of n such that the product of the first n terms of the sequence 2!, 3!, 4!,... is less than or equal to 10^6. | nvidia |
Find $x^2+y^2$ if $x$ and $y$ are positive integers such that \[\begin{aligned} xy+x+y&=71 \\ x^2y+xy^2 &= 880.\end{aligned}\] | nvidia |
$\sqrt{53+20\sqrt{7}}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$. | nvidia |
Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$. | nvidia |
Find the smallest positive integer $m$ such that $2^m \equiv 1 \pmod{17}$. | nvidia |
The polynomial $Q(x) = x^4 + dx^3 + ex^2 + fx + g$ has the property that the sum of its zeros, the product of its zeros taken three at a time, and the sum of the squares of its coefficients are all equal. If the value of $Q(-1)$ is 7, what is $e$? | nvidia |
A $99$-vertex polygon $P_1$ is drawn in the Cartesian plane. The sum of the squares of the $y$-coordinates of the $99$ vertices equals $121$. The midpoints of the sides of $P_1$ form a second $99$-vertex polygon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $99$-vertex polygon, $P_3$. Find the s... | nvidia |
Let $$x$$ be a real number such that $$\arctan x = 3\arctan \frac{x}{3},$$
then compute $$\cos(\arctan x).$$ | nvidia |
An isosceles triangle is inscribed in a circle, with the base being one of its diameters. Given that the height of the triangle is 3 cm less than the radius of the circle, find the length of the base. | nvidia |
Find the value(s) of $x$ such that $\frac{2x+3y}{x+4y}$ is equal to $\frac{x-2y}{2x+5y}$ for all values of $y$. | nvidia |
A polynomial function $P(x)$ is defined such that $P(x) = P(2-x)$ for all real numbers $x.$ If $P(x) = 0$ has exactly $3$ real roots, what is the sum of all the real roots of $P(x) = 0$? | nvidia |
Point $P$ lies on the line $x= -3$ and is 10 units from the point $(5,2)$. Find the product of all possible $y$-coordinates that satisfy the given conditions. | nvidia |
A convex polygon has all its sides and diagonals of different integer lengths. What is the minimum number of sides of such a polygon that has a side of length at least 100 units? | nvidia |
Let $G$ be the set of all positive integers. Let $g : G \to G$ be a function such that
\[g(x) + g(y) = g(\gcd(x,y)g(\text{lcm}(x,y)))\]for all $x,$ $y \in G.$
Let $m$ be the number of possible values of $g(12),$ and let $t$ be the sum of all possible values of $g(12).$ Find $m \times t.$ | nvidia |
Let $A$ be a skew-symmetric matrix and $B$ be a symmetric matrix. Prove that the matrix product $AB$ is also skew-symmetric. | nvidia |
Find the sum of all values of $x$ such that $x$ is a solution to the equation $|x+3|=|x-2|$. | nvidia |
Given that the greatest common divisor (gcd) of two positive integers $m$ and $n$ is $d$, how many values of $a$ can be chosen such that the gcd of $m^a$ and $n^a$ is $d^2$? | nvidia |
Given a positive real number $x$, find the value of the parameter $a$ in the expression $2x^4+6x^3-12x^2+ax+b$, such that when divided by $x^2+2x-3$, the remainder is a perfect square trinomial. | nvidia |
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 |
Three non-negative real numbers $a,$ $b,$ and $c$ satisfy $a+b+c=100$ and $a^2+b^2+c^2=2500.$ Find the maximum value of the expression $(a+b+c)^2-a^2-b^2-c^2.$ | nvidia |
The sum of the lengths of all the edges of a rectangular prism is 84 cm. Find the number of cubic centimeters in the volume of the prism. | nvidia |
Determine the number of different prime divisors of the number
$$N=2^{40}+3^{40}$$ | nvidia |
Determine the product of all $y$ that satisfy the equation $\frac{4y^2-1}{y^2+2y-3} = \frac{y+1}{y-1} + \frac{3}{y+3}.$ | nvidia |
Given that $x$ is real and $x^3+\frac{1}{x^3}=52$, find $x+\frac{1}{x}$. | nvidia |
Of the numbers 1, 2, 3, ..., 15, which number has the greatest number of divisors (the dots mean that we are including all the integers between 1 and 15)? | nvidia |
For how many positive integers $n$ less than or equal to 1000 is $$(\sin t+i\cos t)^n=\sin nt+i\cos nt$$true for all real $t$? | nvidia |
In the diagram shown, each side is perpendicular to its adjacent sides, and all 32 of the sides are congruent. The perimeter of the polygon is 80. Find the area of the polygon.
[asy]
unitsize(0.5 cm);
draw((4,0)--(5,0)--(5,1)--(6,1)--(6,2)--(7,2)--(7,3)--(8,3)--(8,4)--(7,4)--(7,5)--(6,5)--(6,6)--(5,6)--(5,7)--(4,7)--... | nvidia |
The sum of the squares of two consecutive integers is 121. What is the smaller of the two integers? | nvidia |
Let $g$ be a function which maps real numbers to real numbers, such that
\[g(x)+2g\left(\frac{1}{x}\right) = 3x,\]
for all $x\neq 0.$ Find all possible values of $g(-5).$ | nvidia |
[asy] draw(circle((0,0),10),black+linewidth(.75)); draw((-10,0)--(10,0),black+linewidth(.75)); draw((-10,0)--(8.5,5.2),black+linewidth(.75)); draw((-10,0)--(8.5,-5.2),black+linewidth(.75)); draw((4,0)--(8.5,5.2),black+linewidth(.75)); draw((4,0)--(8.5,-5.2),black+linewidth(.75)); MP("X",(4,0),N);MP("A",(-10,0),W);MP("D... | nvidia |
Find all pairs $(a,b)$ of real numbers such that for any real number $x$, the equality $\frac{(ax+b)}{(bx+a)}= \frac{(a+b)}{(b+a)}$ holds. | nvidia |
Find the sum of all positive integers $n$ such that $1.2n-4.45$ is a perfect square. | nvidia |
Determine all values of $n$ for which $2^n$ can be expressed in the form $3m+1$, where $m$ is an integer. | nvidia |
For a positive integer $m$ and distinct digits $x$, $y$, and $z$, let $X_m$ be the $m$-digit integer each of whose digits is equal to $x$; let $Y_m$ be the $m$-digit integer each of whose digits is equal to $y$, and let $Z_m$ be the $3m$-digit (not $m$-digit) integer each of whose digits is equal to $z$. What is the gr... | nvidia |
Find all real numbers $x$ such that
\[\frac{3^x + 5^x}{6^x + 10^x} = \frac{11}{8}.\] | nvidia |
The sum of all the three-digit positive integers is:
$$S = \sum_{n=100}^{999} n$$
Find the value of $S$ using the formula for the sum of an arithmetic series, where $a$ is the first term, $l$ is the last term, and $n$ is the number of terms:
$$S = \frac{n(a+l)}{2}$$
Now, let's create a new problem inspired by this ... | nvidia |
Parallelogram $A$ has its vertices at $(-4, 5)$, $(-2, 4)$, $(6, 7)$, and $(4, 8)$. What is the sum of the $x$-coordinates of the two points on the diagonal of parallelogram $A$ that are also on the line $y = 6$? | nvidia |
Determine the complex number $z$ satisfying the equation $2z-3i\bar{z}=-7+3i$. Note that $\bar{z}$ denotes the conjugate of $z$. | nvidia |
Determine the largest positive integer $m$ such that there exist positive integers $a, b, c$ so that \[
m^3 = a^3+b^3+c^3+3a^2b+3a^2c+3ab^2+3ac^2+3b^2c+3bc^2+6abc+2a+2b+2c-12
\] | nvidia |
Let $x$ be a real number such that
$\left(x^2-1\right)\left(x^2-4\right)=\left(x^2-2\right)^2$
and $\left(x^2-4\right)\left(x^2-9\right)=\left(x^2-6\right)^2$.
Find $x$. | nvidia |
The product of two consecutive positive integers is 650. What is the sum of the two integers? | nvidia |
Find the number of real solutions of the equation
\[\frac{x}{100} = \cos^2 x.\] | nvidia |
Find the smallest positive value of $x$ such that the expression $x^3-4x^2-7x+11$ changes sign. | nvidia |
Point $Q$ lies on the line $y = 2$ and is 15 units from the point $(-2, 4)$. Find the sum of all possible $x$-coordinates that satisfy the given conditions. | nvidia |
Determine the number of combinations of two letters from the English alphabet that form a palindrome. | nvidia |
Consider the function $f(x)$ that maps positive integers to real numbers, such that for all positive integers $x$ and $y$ it holds that
\[xf(y) + yf(x) = f(xy).\]
Find all possible values of $f(20).$ | nvidia |
Find the sum of the squares of the solutions to the equation $x^4-17x^3+26x^2-24x-60=0$. | nvidia |
Let a sequence $a_n$ be defined by $a_1=1$ and for all positive integers $n\ge 2$, $a_n=a_{n-1}+\frac{1}{n}$. Find the least $n$ such that $a_n>2$. | nvidia |
Find all values of $a$ for which the system of equations
$$\frac{1}{x} + \frac{1}{y} = a$$
$$\frac{1}{x^2} + \frac{1}{y^2} = 2a^2$$
has a solution in positive integers $x$ and $y$. | nvidia |
Let $S$ be the set of all positive integer divisors of 16!. A number $x$ is chosen at random from the set $S$. What is the probability that $x$ is a perfect square? | nvidia |
Let $g$ be a function defined on the integers as follows:
\[g(n) = \left\{
\begin{array}{cl}
2n & \text{if $n$ is odd}, \\
n - 3 & \text{if $n$ is even}.
\end{array}
\right.\]Given that the range of $g(g(g(n)))$ is $\{-3,-6\}$, find the smallest positive integer $k$ that is in the domain of $g$. | nvidia |
Let $S$ be the set of all points in the plane that can be reached by a sequence of rotations (about any point) and translations (by any distance in any direction), starting from a given point $P$.
Determine the subset of points $T$ in the plane that cannot be reached from $P$ by this process if there are two points $A... | nvidia |
The sum of the fourth powers of the solutions of $x^2+2x-6=0$ equals $a$. Determine the value of $2a+15$. | nvidia |
In the set of positive integers, let $m$ be the smallest number that is a multiple of the sum of its digits, and let $n$ be the largest number that is a factor of the sum of its digits. What is the value of $m + n$? | nvidia |
Find the smallest positive integer $n$ such that the product of its digits is equal to the number of distinct positive factors of $n$. | nvidia |
In the set of all convex quadrilaterals with integer side lengths, find the smallest perimeter that guarantees the existence of at least 100 different quadrilaterals (considering two quadrilaterals as the same if one can be obtained from the other by a rotation and a translation). | nvidia |
Given two spheres with centers at $(0, 0, 3)$ and $(0, 0, -3),$ and radii of 6 and $\sqrt{26},$ respectively, find the shortest distance between a point on the first sphere and a point on the second sphere, such that the points are connected by a segment that passes through the origin. | nvidia |
Find the sum of all two-digit integers that have a digits sum of 7. | nvidia |
Find the sum of all positive integers $m$ for which $m^2-22m+144$ is a perfect square. | nvidia |
Let $f(n)$ return the number of distinct ordered pairs of positive integers $(a, b)$ such that for each ordered pair, $a^2 + b^2 = n$. Note that when $a \neq b$, $(a, b)$ and $(b, a)$ are distinct. What is the smallest positive integer $n$ for which $f(n) = 3$? | nvidia |
Find the number of positive integers less than or equal to 100 that are not divisible by 2 or 5. | nvidia |
The sum of the squares of two positive integers is 90. The product of the two integers is 27. What is the sum of the two integers? | nvidia |
Find the sum of all odd numbers between 100 and 200 that are divisible by 5. | nvidia |
In the triangle ABC, the lengths of the sides AB, BC, and CA are 3x, 4x, and 5x respectively. The point D is on the side BC such that the ratio of BD to DC is 2:3. If the area of the triangle ABD is 24 square units, find the length of CD. | nvidia |
Consider a three-digit number $ABC$ where $A$, $B$, and $C$ are distinct digits. If the digits are rotated to form two other three-digit numbers, and the smallest of the three numbers is subtracted from the largest, what prime number must be a factor of the difference? | nvidia |
In the triangle ABC, the cevian AD divides the triangle into two triangles of areas $60 \text{ and } 30$. The cevian BE divides the triangle into two triangles of areas $45 \text{ and } 45$, while the cevian CF divides the triangle into two triangles of areas $36 \text{ and } 54$. If these cevians are concurrent, find ... | nvidia |
Find the largest positive integer $n$ such that $n$ divides $40304$ and $30203$ and also satisfies the condition that $\frac{40304}{n} + \frac{30203}{n}$ is a perfect square. | nvidia |
Simplify $\displaystyle\frac{2+2i}{-3+4i}$.
Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers.
If $a$ and $b$ are real numbers, find the values of $a$ and $b$ such that $\displaystyle\frac{a+bi}{2+2i}$ is a real number. | nvidia |
Let f(x) be a continuous function on [0,∞) such that
∫20f(x)dx=5 and f(x)≥0.
Find the minimum value of the integral ∫20(f(x))2dx. | nvidia |
Let $\mathcal{S}$ be the set of positive integers such that for any $n\in\mathcal{S}$, the decimal representation of $n!$ contains all the digits from $1$ to $9$. Find the smallest positive integer $k$ such that $|\mathcal{S}\cap\{1,2,3,...,k\}|\geq 10$. | nvidia |
Suppose that $a$ and $b$ are integers such that $$3b = 8 - 2a.$$How many of the first six positive integers must be divisors of $2b + 12$? | nvidia |
Find the sum of the solutions of the equation $-32x^2 + 84x + 135=0$. | nvidia |
Six 6-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal? | nvidia |
Determine all real numbers x that satisfy the equation
$${{\left( \frac{1-x}{1+x} \right)}^{2017}}+{{\left( \frac{1+x}{1-x} \right)}^{2017}}=2.$$ | nvidia |
How many four-digit numbers are there such that the sum of their digits is equal to the sum of the digits of their reverse? | nvidia |
The polynomial $f(x) = x^3 + x^2 + 2x + 3$ has three distinct roots. Let $g(x) = x^3+bx^2+cx+d$ be a cubic polynomial with leading coefficient $1$ such that the roots of $g(x)$ are the squares of the roots of $f(x)$. Find the ordered triple $(b,c,d)$. | nvidia |
Find the largest positive integer $a$ such that $x^2 - ax + 1$ is not composite for any integer $x$. | nvidia |
Three fair six-sided dice are rolled. What is the probability that the sum of the numbers shown on the first two dice is equal to the number shown on the third die? | nvidia |
In triangle $DEF$, $D'$, $E'$, and $F'$ are on the sides $EF$, $FD$, and $DE$, respectively. Given that $DD'$, $EE'$, and $FF'$ are concurrent at the point $P$, and that $\frac{DP}{PD'}+\frac{EP}{PE'}+\frac{FP}{PF'}=43$, find $\frac{DP}{PD'}\cdot \frac{EP}{PE'}\cdot \frac{FP}{PF'}$. | nvidia |
Determine all real values $a$ such that the expression $(a + i)^4 - (a - i)^4$ is a real number. | nvidia |
A right-angled triangle has an altitude to the hypotenuse that divides the triangle into two smaller right-angled triangles. The length of the altitude is equal to the geometric mean of the two segments into which the hypotenuse is divided. If the hypotenuse is equal to 15 units, and one of the segments is equal to 6 u... | nvidia |
The complex numbers $a+bi$ and $c+di$ are such that $\left|a+bi\right|=\left|c+di\right|$ and $\left|a+c\right|=\left|b+d\right|$.
Find the number of real values $k$ for which $\left|a+bi+k\right|=\left|c+di+k\right|$. | nvidia |
Let $f(x)$ be a polynomial of degree $n$ with positive coefficients such that $f(x) + f(-x) \geq 2$ for all real numbers $x$. Find the largest possible value of $n$.
(Try to solve it yourself or ask for help if needed) | nvidia |
Determine the smallest positive integer $n$ such that $n^2$ is a multiple of 18 and $n^3$ is a multiple of 42. | nvidia |
If the four solutions of $z^4=81$ are expressed in the form $c+di$, where $c$ and $d$ are real, find the sum of those solutions with $c<0$. | 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 |
The function $f(x)$ satisfies
\[f(2^x) + xf(2^{-x}) = 1\]for all real numbers $x.$ Find $f(2).$ | nvidia |
The sum of the squares of two consecutive positive integers is 365. Find the difference between the larger integer and the smaller one. | nvidia |
What is the smallest positive integer $n$ such that $3n \equiv 1356 \pmod{22}?$ | 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 |
There are three real numbers $x$ that are not in the domain of $$f(x) = \frac{1}{1+\frac{1}{1+\frac 1x}}.$$ What is the sum of those three numbers? | nvidia |
What is the product of all positive integers less than 200 that are cubes of perfect cubes? | nvidia |
Find the largest positive integer $a$ for which $x^2+ax-2010$ has no integer roots, but the roots of $x^2+(a+1)x-2010$ are both integers. | nvidia |
Let $x$ be a real number such that $x + \frac{1}{x} = 2.$
Define the sequence $T_n = x^n + \frac{1}{x^n}.$
Find the value of $T_9$. | nvidia |
Determine the smallest value of $a$ such that the equation $2x^3-4x^2+ax+1=0$ has at least one root in the interval $(-1, 1)$. | nvidia |
Find the number of positive integers $m \le 1000$ such that $3^m - 1$ is a multiple of $7$. | nvidia |
Find the number of ordered triples $(x,y,z)$ of real numbers such that $x + y = 2$ and $xy - z^2 = 1.$ | nvidia |
Find a prime number that is a factor of every sum of four consecutive integers. | nvidia |
A regular octahedron has eight faces and six vertices. Eight positive integers are written on the faces of the octahedron. Each vertex is labeled with the sum of the four numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to 186, then what is the sum of the numbers written o... | nvidia |
Compute the value of the infinite series \[
\sum_{n=2}^{\infty} \frac{n^4+3n^2+10n+10}{2^n \cdot \left(n^4+4\right)}
\]
Another problem:
Find the sum of the infinite series:
$$\sum_{n=1}^{\infty} \frac{1}{n^2+2n+1} \cdot \left(\frac{1}{n} - \frac{1}{n+1}\right)$$ | nvidia |
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