instruction_seed stringlengths 30 7.99k | source stringclasses 1
value |
|---|---|
In a library, there are $n$ shelves, each containing a certain number of books. When the total number of books on all the shelves was counted, it was found that the number of books on one of the shelves was mistakenly counted twice, resulting in an incorrect total of $2500$ books. However, it is known that the number o... | nvidia |
Find the maximum value of $a^2+b^2$ given that $a$ and $b$ are positive integers such that $a^2b^2=100$. | nvidia |
Problem:
Three positive integers $x$, $y$, and $z$ satisfy the equations $x+y+z=12$ and $x^2+y^2+z^2=144$. Find the value of $xyz$. | nvidia |
A rectangle with a given perimeter of 20 cm is to be inscribed in a circle of radius 5 cm. Determine the value of x for which the area of the rectangle is maximized, where x is the distance from the center of the circle to one of the vertices of the rectangle. | nvidia |
Find the number of positive integers $n$ such that the number of digits in $n$ is equal to the number of digits in $n^2$. | nvidia |
What is the sum of the areas of all rectangles with whole number dimensions that can be inscribed in a circle of radius $3\sqrt{2}$? | nvidia |
Count the number of three-digit integers that have exactly one digit equal to 5, and the remaining two digits are distinct and add up to 10. | nvidia |
Find the only positive real number $x$ such that $x^2$ is equal to the sum of the roots of the quadratic equation $\frac{x-1}{2x} = \frac{1}{x-2}$. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 1$ and
\[f(x^2 - y^2) = (x - y) (f(x) + f(y))\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$ | nvidia |
Determine the last three digits of the number $1993^3$. | nvidia |
Charles has two six-sided die. One of the die is fair, and the other die is biased so that it comes up six with probability $\frac{2}{3}$ and each of the other five sides has probability $\frac{1}{15}$. Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes... | nvidia |
Determine the largest positive integer $n$ such that $\frac{n}{10n-4}$ is an integer. | nvidia |
The product of two positive integers is 150. The sum of their squares is 3269. What is the sum of the two integers? | nvidia |
If the point $(3,6)$ is on the graph of $y=g(x)$, and $h(x)=(g(x))^2$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? | nvidia |
Given a positive integer $n$, find the number of possible graphs on $n$ vertices such that the graph is symmetric about a fixed line of reflection and has no isolated vertices. | nvidia |
Find all values of x for which the expression $$\frac{x+1}{\log_2{(x+2)}-\log_2{5}}+\frac{2x-1}{\log_2{(x+2)}-\log_2{7}}+\frac{x-1}{\log_2{(x+2)}+\log_2{7}}+\frac{3x-2}{\log_2{(x+2)}+\log_2{5}}$$ is equal to zero. | nvidia |
Let $a$ and $b$ be real numbers. If the polynomial $x^2+ax+b$ has exactly one real root and $a=b-2$, find the value of the sum of all possible values of $b$. | nvidia |
How many positive integers less than 60 have an even number of positive divisors? | nvidia |
Let $a$, $b$, and $c$ be three positive real numbers such that
$$a^3+b^3+c^3+3abc=64$$
and
$$a^2(b+c)+b^2(a+c)+c^2(a+b)=128$$
Find $ab+ac+bc$. | nvidia |
Find the largest positive integer $m$ that has exactly $6$ positive integral divisors and is a divisor of $10!$. Determine the number of different prime divisors of $m$. | nvidia |
Let $f(z)$ be a polynomial such that $f(z)=f(z^{-1})$. If there are exactly four distinct roots that lie on the unit circle, what is the maximum number of real roots that $f(z)$ can have? | nvidia |
Find the number of positive integers between 1 and 500, inclusive, that can be expressed as the sum of the cubes of two nonnegative integers. | nvidia |
Two positive integers, $x$ and $y$, are said to be "good" if both of the following conditions are met:
1. The graphs of the two functions $f(t) = t^x$ and $g(t) = t^y$ intersect at exactly one point.
2. The number of real solutions to the equation $f(t) = g(t)$ is equal to the number of real solutions to the equation ... | nvidia |
Find the value of $x$ such that $\left|\frac{x}{2}\right|+\left|\frac{2}{x}\right|=3$. | nvidia |
If the quadratic $2x^2-5nx+n$ has exactly one real root, find the positive value of $n$. | nvidia |
A right circular cylinder has a base with radius $300$ and height $150\sqrt{3}.$ A bug starts at a point on the surface of the cylinder whose distance from the bottom base of the cylinder is $50$, and crawls along the surface of the cylinder to a point on the exact opposite side of the cylinder whose distance from the ... | nvidia |
Let $x$ and $y$ be positive integers such that $x > y$ and $xy + \frac{x}{y} + \frac{y}{x} = 99$. Determine the value of $x$. | nvidia |
Find the product of all positive integers less than 200 that are cubes of perfect cubes. | nvidia |
Find the least positive integer $x$ that satisfies $x+4609 \equiv 2104 \pmod{12}$. | nvidia |
Find the number of positive integers $n$ such that $3.6n+1.2<8.4$ and $n$ is a multiple of $3$. | nvidia |
Find the sum of all solutions to the equation $(x-6)^2=25$. | nvidia |
What is the largest prime number that can be represented as the sum of three consecutive positive integers? | nvidia |
Find the number of solutions of the system of equations
$$x^{2^k}=y^{3^k}, \text{ for } k=1,2,3,\ldots,n,$$
where $x,y$ are integers and $n$ is a fixed positive integer. | nvidia |
Given $a,$ $b,$ $c,$ $d$ are positive real numbers such that $\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}=6$ and
\[a+b+c+d=26.\]
Find the largest possible value of $abcd.$ | nvidia |
A portion of the graph of a quadratic function $f(x)$ is shown below.
Let $g(x)=-f(x-2)$ and $h(x)=f(x+1)$. If $a$ is the number of points where the graphs of $y=f(x)$ and $y=g(x)$ intersect, and $b$ is the number of points where the graphs of $y=f(x)$ and $y=h(x)$ intersect, then what is $5a+2b$?
[asy]
size(150);
re... | nvidia |
In a regular octahedron, the midpoints of the edges are connected to form a smaller polyhedron. The ratio of the surface area of the smaller polyhedron to that of the larger octahedron is $a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. | nvidia |
Given a fixed positive real number $\alpha,$ let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = \alpha.$ Find the maximum value of $\frac{x^2 + y^2}{xy(x + y + z)}$. | nvidia |
Let $\mathbf{B}$ be a $3 \times 3$ matrix, with real entries, such that $\mathbf{B}^2 = \mathbf{0}.$ Find the number of different possible matrices that $\mathbf{B}^3$ can be. | nvidia |
Find the sum of the values of $x$ that satisfy the equation $|2x|+5=21$. | nvidia |
A square is drawn such that one of its sides coincides with the line $y = 5$, and so that the endpoints of this side lie on the parabola $y = x^2 + 3x + 2$. What is the area of the square? | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) - y) = g(x) + g(g(y) - g(-x)) + |x|\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(0),$ and let $t$ be the sum of all possible values of $g(0).$ Find $m \times t.$ | nvidia |
Two numbers are relatively prime if their greatest common divisor is 1. Prove that if two integers $m$ and $n$ are relatively prime, then there exist integers $a$ and $b$ such that $m(a+b)=n(a-b)$. | nvidia |
Find the number of five-digit numbers $ABCDE$ with no leading zeroes, where the three-digit number $ABC$ and the two-digit number $DE$ are both prime, and each of the digits $A$, $B$, $C$, $D$, and $E$ are not necessarily distinct. | nvidia |
Let $Q(x) = x^3 + dx^2 + ex + f$ be a cubic polynomial with integral coefficients such that $x = 2$ is a double root, and the sum of the squares of its zeros is equal to 12. Given that the product of the zeros of $Q(x)$ is equal to 16, determine $d$. | nvidia |
Let $P_n$ denote the set of polynomials of degree $n$ with coefficients $0$ or $1$, let $Q_n$ denote the number of polynomials in $P_n$ with distinct real roots and let $R_n$ denote the number of polynomials in $P_n$ whose roots are all rational.
Compute $\sum_{n=1}^{20} (Q_n-R_n)$. | nvidia |
Let $A_0=(0,0)$. Distinct points $A_1,A_2,\dots$ lie on the $x$-axis, and distinct points $B_1,B_2,\dots$ lie on the graph of $y=\sqrt{x}$. For every positive integer $n,\ A_{n-1}B_nA_n$ is an equilateral triangle. What is the least $n$ for which the length $A_0A_n\geq100$?
$\textbf{(A)}\ 13\qquad \textbf{(B)}\ 15\qqua... | nvidia |
Let $x_1, x_2, \dots, x_n$ be positive real numbers such that $x_1 \cdot x_2 \cdots x_n = 1.$ Find the minimum value of $x_1^2 + x_2^2 + \dots + x_n^2.$ | nvidia |
For each integer $n\geq 4$, let $a_n$ denote the base-$n$ number $0.\overline{133}_n$. The product $a_4a_5 \dotsm a_{99}$ can be expressed as $\frac{m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$? | nvidia |
What is the sum of the digits of the sum of all the even positive integers between 20 and 120? | nvidia |
Find the largest three-digit number that is not divisible by any of the first five prime numbers. | nvidia |
Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$. Find $g^{-1}(f(-7))$. | nvidia |
Let $P(x)$ be a polynomial of degree $3$ with real coefficients satisfying the following conditions:
(i) There exist real numbers $r$ and $s$ such that $P(x) = (x-r)^2(x-s)$;
(ii) There exist positive integers $m$ and $n$ such that $P(m) = 0$ and $P(n) = 0$;
(iii) For every positive integer $k$ such that $P(k) \neq ... | nvidia |
A tetrahedron with four equilateral triangular faces has vertices $A$, $B$, $C$, and $D$, and the side length of the tetrahedron is 2. The tetrahedron is placed inside a cube so that the three vertices $A$, $B$, and $C$ lie on the vertices of one face of the cube. Find the length of the diagonal of the cube that passes... | nvidia |
Find the largest possible value of x such that 18x and 24x are both factors of 432. | nvidia |
Find the smallest possible value of the radius of a circle that circumscribes a triangle whose sides are 3, 4, and 5. | nvidia |
In the polynomial $P(x) = (x-1)^n$, there exist three consecutive terms such that their coefficients form an arithmetic progression with common difference 2. Determine the minimum value of $n$ that satisfies this condition. | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $a<b<c$ and $\text{lcm}(a,b,c)=abc$. Determine the smallest possible value of $a+b+c$. | nvidia |
Find the length of the segment between the center of the ellipse with equation $\frac{x^2}{9}+\frac{y^2}{4}=1$ and the point $(-7,-2)$, given that this point lies on the line tangent to the ellipse at $(3\cos t, 2\sin t)$. | nvidia |
Find the median of the set of positive integers that are less than 30 and are not divisible by 2 or 3. | nvidia |
Find all real numbers $x$ such that $\sin x = 1 + x^2$. | nvidia |
Let $a$ and $b$ be two relatively prime positive integers such that the sum of the first 1000 terms of the sequence
$$a, a^2, a^3, a^4, \dots$$
is divisible by $b$. Find the smallest possible value of $b$. | nvidia |
For all odd integers $m$, what is the smallest integer that always divides into the sum of $m$ and the square of $m$? | nvidia |
Let $x$ and $y$ be two positive numbers such that their geometric mean is equal to $\sqrt{2}$. Suppose that $z$ is a number such that $x$, $y$, and $z$ are in an arithmetic progression. Determine the maximum possible value of $z$. | nvidia |
Let $C$ be the number of three-digit numbers that are divisible by neither 5 nor 7, and let $D$ be the number of three-digit numbers that are divisible by 35. Find $C-D$. | nvidia |
Find the value of $a$ that satisfies the equation $293_{a}+468_{a}=73B_{a}$, where $B_{a}=11_{10}$. | nvidia |
Find the number of integers $n$ such that $n$ and $2n$ have the same remainder when divided by $13$, and $n$ and $3n$ have the same remainder when divided by $17$. | nvidia |
Consider the matrix
\[\begin{pmatrix} x & 4 \\ -7 & y \end{pmatrix}\]
such that its transpose is its own inverse. Determine the number of possible pairs (x, y) that satisfy this condition. | nvidia |
Let $n$ be an integer such that $2^{31} \equiv n \pmod{15}$. Determine the value of $n$ that satisfies this congruence relation. | nvidia |
The lengths of the sides of a triangle with positive area are $\log_{10}12$, $\log_{10}75$, and $\log_{10}n$, where $n$ is a positive integer. Find the number of possible values for $n$. | nvidia |
Compute $(a+b)^3+(a^3+b^3)$, given that $a$ and $b$ are the roots of the equation $x^2+3x+2=0$. | nvidia |
The graph of the parabola described by the equation $y=-3x^2-30x-81$ is translated horizontally by $k$ units such that its vertex lies on the line $y=x$. Determine the value of $k$. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a^2 + b^2 = 100$. Find the maximum value of $ab + 2a + 2b$. | nvidia |
For positive real numbers $a_1,a_2,\ldots,a_n,$ find the minimum value of
\[\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+\cdots+\frac{a_n^2}{a_n+a_1}\]
as $n$ varies over all positive integers. | nvidia |
Find the sum of all $x$ and $y$ that satisfy the system of equations
\begin{align*}
x^2 + y^2 &= 25, \\
(x-3)^2 + (y-4)^2 &= 100.
\end{align*} | nvidia |
Find the real number $a$ for which the equation
\[3^x + 3^{-x} = 2a\]
has exactly three solutions. | nvidia |
Evaluate the product \[ (a-10) \cdot (a-9) \cdot \dotsm \cdot (a-1) \cdot a, \] where $a=2$. | nvidia |
Let $m$ and $n$ be positive integers such that $m+n=32$ and $\gcd(m,n)=8$. Find the smallest possible value of $\mathrm{lcm}(m,n)$. | nvidia |
In a regular hexagon $ABCDEF$, three congruent isosceles triangles are drawn such that each triangle shares a side with the hexagon and has its third vertex inside the hexagon. The maximum possible total area of these triangles can be written in the form $a\sqrt{b}-c$, where $a$, $b$, and $c$ are positive integers. Fin... | nvidia |
A curve is defined by the equation
$$xy=k,$$
where $k$ is a positive number.
Point $M$ with coordinates $(12,14)$ is located on the curve.
The tangent line to the curve at point $M$ has the slope of $-\frac{1}{10}$.
Find the largest possible value of $k$. | 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 |
The equation $x^2-kx-12=0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$? | nvidia |
Two circles are tangent to each other and have radii that differ by 5. The area of the larger circle is 4 times the area of the smaller circle. What is the sum of the radii of the two circles? | nvidia |
The sequence $a_n$ is defined recursively as follows:
$a_1 = 1$
$a_n = a_{n-1} + n$ for $n > 1$
Prove that for every positive integer $n$, $a_n$ is divisible by $n$. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(xf(y) + z) = xf(z) + zf(y)\]for all real numbers $x,$ $y,$ and $z.$
Let $m$ be the number of possible values of $f(4),$ and let $t$ be the product of all possible values of $f(4).$ Find $m \times t.$ | nvidia |
Consider the function $g : \mathbb{R}^+ \to \mathbb{R}^+$ that satisfies the following properties:
(i) $g(x^2) = x \cdot g(x)$ for all $x \in \mathbb{R}^+$.
(ii) $g(x) + g(y) = 2 + g(xy)$ for all $x, y \in \mathbb{R}^+$.
Find the number of possible values of $g(1)$ and the sum of all these values, then calculate the... | nvidia |
A sequence of real numbers $a_1,a_2,\ldots$ is defined such that the set $\{a_1,a_2,\ldots,a_n\}$ has exactly $n$ distinct elements for every positive integer $n$. Suppose that the sequence satisfies $a_{n+1}+a_{n-1}=2a_n$ for all $n\geq 2$. If $a_1+a_2=6$, what is the maximum number of distinct elements in the set $\{... | nvidia |
Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$. Find $g^{-1}(f(-7))$. | nvidia |
A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order. | nvidia |
What is the product of all values of $a$ such that the equation $x^3-ax^2+2a=0$ has three distinct real roots that form an arithmetic progression? | nvidia |
Given a 2x2 matrix
$$A=\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}$$
where a, b, c, d are real numbers and $ad-bc\neq 0$.
Find the equation of the set of vectors that remain unchanged under the linear transformation defined by this matrix. | nvidia |
Find the sum of the reciprocals of the roots of $x^2-13x+4=0$. | nvidia |
Compute the value of $x$ such that $\dbinom{x}{x} = \dbinom{133}{133}$. | nvidia |
Find all positive integers n such that 2n is a divisor of n^2 + 1. | nvidia |
Point $Q$ lies on the line $y=4$ and is 8 units from the point $(2,-3)$. Find the sum of all possible $x$-coordinates that satisfy the given conditions. | nvidia |
Let $a$, $b$ and $c$ be positive real numbers satisfying the condition $ab=12$, $ac=7$, and the triangle inequality $a+b>c$. How many integer values of $a$ satisfy these conditions? | nvidia |
If the quadratic $x^2+4mx+m$ has exactly one real root, find the positive value of $m$.
Write another problem inspired by this one:
The quadratic $x^2+4mx+m$ has exactly one real root. Find the range of values of $m$ such that the quadratic has a real root and the product of its roots is greater than $-1$. | nvidia |
The arithmetic mean of four numbers is 15. Two of the numbers are 10 and 18 and the other two are equal. What is the product of the two equal numbers? | nvidia |
The two sides of a right triangle are $4\sqrt 3$ and 4 units, respectively. A circle with center at the triangle's right angle has radius $r$. Given that the circle is tangent to the hypotenuse, find the area of the circle. | nvidia |
Find the smallest positive integer that can be subtracted from 2023 so that the result is divisible by 7. | nvidia |
Find a positive integer that is divisible by 20 and has a cube root between 12 and 12.5. | nvidia |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.