instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Find the sum of all three-digit positive integers whose cubes end with the digits 125. | nvidia |
Consider the function
$$f(x)=\ds\sqrt{\f{x-2}{x^2+3x-10}}$$
defined on the real line wherever it takes real values. Find the set of all values that $f(x)$ can possibly attain. | nvidia |
Given a positive integer $n$, find the sum of the squares of all positive integer divisors of $n$ if the sum of the divisors of $n$ is 78 and the sum of the divisors of $n$ excluding 1 and $n$ is 54. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x + y = 2$. Find the set of all possible values of $\frac{1}{x} + \frac{1}{y}$. | nvidia |
Let p and q be two positive numbers such that pq=8/5.
Find 25p2q2. | nvidia |
Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a + b + c = 21.$ Find the largest possible value of $b$. | nvidia |
Find the least positive integer that has exactly three different prime factors and is divisible by 4. | nvidia |
Let $f(x)$ be a polynomial with positive integer coefficients and degree 4 such that for any positive integer $n$, $f(n)$ can be written as a sum of distinct powers of $n$. Let $g(x)$ be the polynomial obtained by reversing the coefficients of $f(x)$. Find the sum of all positive integers $k$ for which $g(k)$ is also a... | nvidia |
Find the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A(0, 3)$ and $B(5, 0)$. | nvidia |
Find the value of $x$ that minimizes the expression $x^2 - 2xy + 3y^2$ given that $x$ and $y$ are positive integers. | nvidia |
The following system of equations has a solution:
\begin{align*}
x+2y+3z&=6,\\
4x+5y+6z&=16,\\
7x+8y+9z&=a.
\end{align*}
Determine $a$ such that the system of equations has infinitely many solutions. | nvidia |
Let's consider a unit circle $C$ centered at point $O$. Three points are chosen uniformly at random on the circle $C$. Then a point $P$ is chosen uniformly at random inside the circle $C$. What is the probability that the area of the triangle formed by $P$ and the two closest points on the circle is greater than $0.25$... | nvidia |
How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1\le x\le 4$ and $1\le y\le 4$?
$\text{(A) } 496\quad \text{(B) } 500\quad \text{(C) } 512\quad \text{(D) } 516\quad \text{(E) } 560$
| nvidia |
What is the least number of colors needed to shade the following pattern, such that no two adjacent regions have the same color?
[asy]
for(int i=0;i<5;++i){
for(int j=0;j<5;++j){
filldraw(circle((i,j),0.4),white);
if(i<4 && j<4){
draw((i+0.4,j+0.4)--(i+1-0.4,j+0.4)--(i+1-0.4,j+1-0.4)--(i+0.4,j+1-0.4)--cycle);
}
}
}
[/... | nvidia |
Find the value of $x$ if $x$ is positive and $\lfloor x\rfloor+x=15$.
In this problem, we need to find a positive value of $x$ such that the sum of $x$ and its floor value is equal to 15. This requires a different approach than the original problem, as we need to consider the properties of the floor function and how i... | nvidia |
Find the number of lattice points that satisfy the following system of inequalities:
\[
\begin{array}{cccc}
(1) & 2x+y & > & 5 \\
(2) & x-2y & \leq & -3 \\
(3) & 3x+y & < & 11
\end{array}
\]
Graph these inequalities on a coordinate plane and find the number of lattice points (points with integer coordinates) within t... | nvidia |
Find the range of values of $y$ that satisfy the equation \[\sqrt{2y-1}\cdot\sqrt{y+2}\cdot\sqrt{3y+1}=3.\] | nvidia |
Find the sum of all positive integers less than 1000 that are cubes of perfect cubes minus the sum of all positive integers less than 1000 that are squares of perfect squares. | nvidia |
Determine the number of integer solutions $(x, y)$ of the equation $xy-3x+4y=8$ such that $-10\leq x\leq 10$ and $-10\leq y\leq 10$. | nvidia |
Find the sum of all values of $y$ such that $\sqrt{y^3+3y+4}=|y-3|$. | nvidia |
For any finite sequence $A = (a_1, a_2, \ldots, a_n)$ of real numbers, define $S(A)$ to be the sum of all the elements of $A$, and $D(A)$ to be the sum of the squares of the elements of $A$. Find the smallest real number $k$ such that the inequality \[D(A) + D(B) \geq k(S(A) - S(B))^2\]holds for all pairs $(A, B)$ of s... | nvidia |
There are four positive integers that are divisors of each number in the list $$36, 72, -12, 114, 96.$$Find the sum of these four positive integers.
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 $x, y, z$, and $w$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}x^2+y^2&=&z^2+w^2&=&500,\\ xy&=&zw&=&200.\end{array}\]
If $T=x+y+z+w$, determine the value of $\lceil T \rceil$. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the maximum value of $x^3 y^2 z.$ | nvidia |
Find the smallest positive integer that has exactly 5 positive integer factors, including 1 and itself, and is also a perfect square. | nvidia |
If $x$ is a positive number such that \[\sqrt{8x}\cdot\sqrt{10x}\cdot\sqrt{3x}\cdot\sqrt{15x}=15,\]find all possible values for $x$. | nvidia |
Find the number of positive integers less than 500 that are divisible by exactly three of the prime numbers 2, 3, 5, or 7. | nvidia |
Triangle $PQR$ has $PQ=25$, $PR=26$ and $QR=24$. Points $S$ and $T$ are located on $\overline{PQ}$ and $\overline{PR}$, respectively, such that $\overline{ST}$ is parallel to $\overline{QR}$ and contains the center of the inscribed circle of triangle $PQR$. Then $ST=a/b$, where $a$ and $b$ are relatively prime positive... | nvidia |
A game show contestant chooses three distinct numbers at random from the set {1, 2,..., 6}. What is the probability that the sum of the three numbers is an even number? | nvidia |
Let $a$ and $b$ be integers such that $0<a<b$.
If $\cos(a \cdot \theta)=\cos(b \cdot \theta)$ for every angle $\theta$,
determine the minimum value of $b-a$ (expressed as a positive integer). | nvidia |
Let $a$ be the product of all positive integers less than $8!$ which are not coprime with $8!$. Find the remainder when $a$ is divided by $8!$.
(Here $n!$ denotes $1\times\cdots\times n$ for each positive integer $n$.) | nvidia |
Let $m$ be a positive integer. We define the function $\phi(m)$ to be the minimum positive integer $n$, if it exists, such that for all positive integers $a$ there exists an integer $b$ such that $ab \equiv 1 \pmod{n}$ implies $ab \equiv 1 \pmod{m}$. Otherwise we let $\phi(m) = -1$.
Prove that for every positive integ... | nvidia |
The integers $G$ and $H$ are chosen such that
\[\frac{G}{x+5}+\frac{H}{x^2-4x}=\frac{x^2-2x+10}{x^3+x^2-20x}\]for all real values of $x$ except $-5$, $0$, and $4$. Find $H/G$. | nvidia |
On a rectangular grid, a square is drawn so that its vertices coincide with the grid points. If two opposite vertices are located at points (5,7) and (11,13), find the side length of the square. | nvidia |
Find the largest integer $n$ such that $n^2+1$ is a factor of $1156$ and $n$ is a divisor of $34$. | nvidia |
In a regular octahedron, the midpoints of the 12 edges form the vertices of a smaller octahedron. The ratio of the surface area of the smaller octahedron to that of the larger is $a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. | nvidia |
Let $C_1$ and $C_2$ be two circles with radii $r_1$ and $r_2$ respectively, such that $r_1+r_2=12$. Let $T$ be a circle that is externally tangent to $C_1$ and internally tangent to $C_2$. Find the radius of the largest circle that can be inscribed in the region bounded by $C_1$, $C_2$, and $T$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 100.$ Find the minimum value of $\frac{x}{y} + \frac{y}{x}.$ | nvidia |
If $a$ and $b$ are positive integers such that $a^2-b^2=45$, what is the value of $a-b$? | nvidia |
The quadratic equation $ax^2+bx+c=0$ has two distinct real roots. The equation $Ax^2+Bx+C=0$ is formed by reversing the coefficients of the original equation (i.e., $A=c, B=b, C=a$). Determine the sum of the values of $a$ for which the two equations have the same sum of roots. | nvidia |
Count the number of ways to paint a 4x4 square grid with 4 different colors, such that no two adjacent squares have the same color. | nvidia |
Find the product of all values of x for which the expression $\frac{x^2+3x-2}{x^2+5x-6}$ is not a real number. | nvidia |
How many ordered pairs of positive integers $(x,y)$ with $x$ and $y$ less than or equal to $100$ satisfy the equation $\cos(xy)=-0.8$? | nvidia |
Let $f(x) = x^3 + ax^2 + bx + c$ be a cubic polynomial with real coefficients. It is known that the roots of $f(x)$ are not all real. Suppose that $g(x) = x^3 + px^2 + qx + r$ is another cubic polynomial with real coefficients such that the roots of $f(x) + g(x)$ are all real. Find the minimum value of $|c - r|$. | nvidia |
In a right triangle, the sum of the squares of the two legs is 1000, and the product of the two legs is 1440. What is the length of the hypotenuse? | nvidia |
The sum of the proper divisors of a positive integer $N$ is equal to $N$ itself. What is the largest such integer less than 500? | 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 |
Consider a $5\times5$ grid of points, where each point is $1$ unit away from its nearest neighbors. A "decreasing path" is a sequence of distinct points in the grid such that the distance between consecutive points in the sequence is strictly decreasing. Let $M$ be the maximum possible number of points in a decreasing ... | nvidia |
Find the value of $a$ so that the lines with the given equations are perpendicular. \begin{align*}
y &= 2x+13 \\
6y+ax &= 6.
\end{align*} | nvidia |
Two fair, eight-sided dice are rolled. What is the probability that the sum of the two numbers showing is greater than 9? | nvidia |
Let $M$ be a matrix such that $M^3=0$ and $M^2 \neq 0$.
Suppose that $M$ has real entries and that $M$ can be written in the form $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ for some real numbers $a$, $b$, $c$, and $d$.
How many such matrices are there? | nvidia |
A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles? | nvidia |
Determine the number of positive divisors of 3150. | nvidia |
The decimal expansion of the quotient of two natural numbers is $0.\overline{ab}$. Prove that $ab$ is divisible by 99. | nvidia |
Find all real values of x such that the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n(x+n)}$ converges. | nvidia |
For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \le 50$.
(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions? | nvidia |
What is the largest positive integer that has exactly four distinct positive factors and is a perfect square? | nvidia |
Given that the determinant of a 2x2 matrix is 14, find the determinant of the matrix that results when the rows and columns of the original matrix are swapped, and then the matrix is multiplied by a scalar value of 3. | nvidia |
Let $p(x)$ be a polynomial with integer coefficients such that $p(2007)=2007$. For how many values of $n$ is $n+p(n)+p(p(n))=2007^2$? | nvidia |
Let $x,$ $y,$ $z$ be positive real numbers such that
\[\frac{x}{y + z} = \frac{y}{z + x} = \frac{z}{x + y} = p.\]
Find the value of $p$ that maximizes the product $xyz.$ | nvidia |
Consider the function $f(x) = \frac{x^2+3x+2}{x^2+4x+4}$. Determine the sum of all values of $x$ for which $f(x)$ is undefined. | nvidia |
In the construction of a rectangular solid, three identical spheres are placed on the vertices of the base so that they touch each other and the edges of the base. Find the volume of the solid in terms of the diameter of one of the spheres. | nvidia |
Determine $k$ such that the equation $2x^2+kx+\frac{1}{2}=0$ has roots $x_1$ and $x_2$ satisfying the condition $x_1x_2=x_1+x_2$. | nvidia |
Given a polynomial $p(x)$ of degree $n$, determine the number of possible values that the polynomial $q(x) = p(p(x))$ can take when $p(x) = 0$ for exactly one real value of $x$. | nvidia |
What is the smallest whole number that must be a factor of the sum of any four consecutive positive integers? | nvidia |
Find the smallest positive integer that is divisible by three distinct prime powers, where each prime power has an exponent greater than 1. | nvidia |
The positive divisors of a positive integer of the form $2^i3^j$ can be paired in such a way that the sum of the divisors in each pair is a power of $2$ times a power of $3$. If the number of pairs that can be formed is $50$, what is $i + j$? | nvidia |
Let $a$, $b$, $c$ be positive integers such that $a-b-c=b+c-a=(a+c)(a-b)$. Find the largest possible value of $c$ divided by $b$. | nvidia |
In the triangle ABC, the lengths of the sides AB, BC, and CA are $x^2+y$, $y^2+z$, and $z^2+x$ units, respectively. Given that $x$, $y$, and $z$ are positive integers, find the minimum value of the perimeter of the triangle. | nvidia |
Determine the value of $y$ such that the five-digit number $\underline{y24y2}$ is divisible by 24. | nvidia |
Given that the point $(2,5)$ is on the graph of $y=g(x)$, there is one point that must be on the graph of $3y=2g(5x)-1$. What is the product of the coordinates of that point? | nvidia |
Let $S$ be a set of all real numbers $x$ for which there exists a real number $y$ such that $\tan (x+y)=y$. Prove that $S$ contains a non-degenerate closed interval.
\begin{align*}
\end{align*} | nvidia |
How many positive integers less than 100 can be expressed as the sum of two distinct perfect cubes? | nvidia |
Find all possible values of $k$ such that the equation
$$(x^2 + kx + 1)^2 + x^2 = (x^2 + x + k)^2$$
has integer solutions. | nvidia |
Let $a,$ $b,$ and $c$ be positive integers such that $abc = 720.$ Find the minimum value of $a + b + c.$ | 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 |
Let $ S $ be the set of all sides and diagonals of a regular pentagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? | 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 |
Determine the number of different arrangements of the digits in the decimal representation of 3326 that are divisible by 2. | nvidia |
Determine the value of $x$ such that $(\frac{1}{2})^{\frac{x}{7}} \cdot 8^{\frac{1}{5}} = \frac{1}{16}$ | nvidia |
Find the sum of all integers from 25 through 35, inclusive. | nvidia |
Determine the largest positive integer $n$ such that $3^n$ divides $2^{2013}-1$. | nvidia |
Determine the largest prime factor of the number obtained by multiplying all the divisors of 10080 that are not multiples of 7. | nvidia |
Find the smallest integer n such that the sum of the digits of n^2 is equal to the sum of the digits of n.
In other words, we are looking for the smallest integer n such that the sum of the digits of n squared is equal to the sum of the digits of n itself.
For example, if n = 12, then n^2 = 144, and the sum of the di... | nvidia |
Find the product of all integer values of $n$ such that $\frac{2n-1}{n+2}$ is a rational number.
Note: This problem requires a different approach to solve, as it involves rational numbers and a different expression. The user needs to analyze the expression and find the values of $n$ that make it a rational number. | nvidia |
The vertices of a parallelogram $ABCD$ are $A(2,1)$, $B(4,3)$, $C(6,5)$ and $D(x,y)$. Given that the diagonals of the parallelogram bisect each other, find the value of $xy$. | nvidia |
Determine the number of positive divisors of the number 30. | nvidia |
The average of two 3-digit positive integers is equal to the decimal number obtained by writing one of the three-digit integers before the decimal point and the other three-digit integer after the decimal point. What is the smaller of the two integers? | 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 the real values of $x$ that satisfy the equation $f(x)=3$, given that $f(x)$ is a function that satisfies the following conditions:
- $f(x)$ is a continuous and differentiable function for all real values of $x$.
- $f(x)$ is a strictly increasing function for all real values of $x$.
- $f(2)=0$ and $f(4)=4$.
-... | nvidia |
Determine the minimum value of the expression $|x^2-6x+13|+|x^2-8x+25|$ for real values of $x$. | nvidia |
Find the least positive integer that can be expressed as a sum of two perfect squares in two different ways. | nvidia |
The sum of three different numbers is 67. The two larger numbers differ by 7 and the two smaller numbers differ by 3. What is the value of the largest number? | nvidia |
Find the value of the expression $(\cos \theta + i \sin \theta)^n$ where $\theta$ is an acute angle such that $\cos \theta = \frac{3}{5}$ and $n$ is a positive integer such that $n\theta = 180^\circ$. | nvidia |
Find the sum of the values of $x$ that satisfy the equation $|x-3|+|x-7|=9$. | nvidia |
Find the value of $x$ that satisfies the equation $\sin x + \sin 2x + \sin 3x = 0$, where $0 < x < \frac{\pi}{2}$. | nvidia |
The vertices of a triangle are $(-2,3)$, $(5,-1)$ and $(k,7)$. Determine the value of $k$ such that the perimeter of the triangle is equal to 20 units. | nvidia |
Find the smallest positive integer such that when its rightmost digit is deleted, the resulting integer is 1/13 of the original integer. | nvidia |
Let $a$ and $b$ be two positive real numbers such that $a + b = 18.$ Enter the ordered pair $(a,b)$ for which $a^3 b^4$ is maximized. | nvidia |
My co-worker Erich is very odd. He only likes numbers that are divisible by 5. How many different last digits are possible in numbers that Erich likes? | nvidia |
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