instruction_seed stringlengths 30 7.99k | source stringclasses 1
value |
|---|---|
Compute the sum of all the solutions to the equation $z^6 = -1$, where $z$ is a complex number. | nvidia |
A circle of radius $1$ is drawn inside a regular hexagon. A point is chosen at random on the boundary of the hexagon, and another point is chosen at random on the boundary of the circle. What is the probability that the distance between the two points is greater than $2$? | nvidia |
Find the number of all integer powers of 2 that lie between $\sqrt{200}$ and $\sqrt{3500}$. | nvidia |
Let $n$ be the smallest positive integer that is a multiple of $75$ and has exactly $75$ positive integral divisors, including $1$ and itself. Find $\frac{n}{75}$.
| nvidia |
Find the number of positive divisors of 8100 that are not multiples of 5. | nvidia |
Consider the function $g(x)$ that satisfies the equation:
\[g(3^x) - xg(3^{-x}) = 2\]
for all real numbers $x.$ Find $g(3).$ | nvidia |
Consider a cube with dimensions 3 x 3 x 3, where each vertex is assigned a unique integer value from 1 to 27, inclusive, such that the sum of the values of each pair of adjacent vertices (connected by an edge) is always even. What is the smallest possible sum of the values of all the vertices that have odd coordinates ... | nvidia |
What is the largest possible value of the sum of the coefficients of a polynomial with integer coefficients if the polynomial has degree 5, the leading coefficient is positive, and the sum of the roots is -3? | nvidia |
If $A$, $B$, $C$, and $D$ are four complex numbers such that $A+B+C+D=i$, $A-B=C-D=2i$, and $A^2+C^2=10$, find $B^2+D^2$. | nvidia |
In triangle $ABC$, $AC=BC$, and $D$ is a point on $\overline{AB}$ such that $AD=CD$. If $\angle CAD=12^\circ$, find the measure of $\angle ACD$. | nvidia |
Let $f$ be a function with the property that for each real number $x$ in its domain, $1/x$ is also in its domain and
\[
f(x) + f\left(\frac{1}{x}\right) = x.
\]
Suppose we are given that $f(x)$ is an even function, i.e. $f(x) = f(-x)$ for all $x$ in its domain. What is the largest set of real numbers that can be in the... | nvidia |
Find the number of positive integers $n$ such that $n^3+1000$ is a perfect square. | nvidia |
The sum of the squares of two positive integers is 200. Find the greatest common divisor of these two integers. | nvidia |
What is the least positive integer that must divide the sum of the first fifteen terms of any arithmetic sequence whose terms are positive integers? | nvidia |
An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free? | nvidia |
Find the smallest prime number that can be expressed as the difference of two consecutive cubes. | nvidia |
A circle with center $(3,2)$ passes through the point $(5,4)$. Find the equation of the line that is tangent to the circle at the point $(5,4)$. | nvidia |
Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that
the points $P_1, P_2,\ldots, P_n$ are coplanar and no three of them are collinear,
each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint,
all of t... | nvidia |
Let $S$ be the set of all positive real numbers $x$ such that $\lfloor x \rfloor \geq \lfloor 2x \rfloor$. Find the number of positive integers $N$ such that the equation $2^x = N$ has a solution for $x$ in $S$. | nvidia |
A positive real number x is randomly selected such that x < 1. What is the probability that the integer nearest to x is odd? | nvidia |
The sum of the squares of two consecutive positive integers can be represented as $(n+1)^2+n^2=2n^2+2n+1$. Show that this expression can also be written as $N^2+(N+1)^2$ for some integer $N$. What is the smallest value of $N$? | nvidia |
What is the smallest positive integer n such that the product of all positive integers less than or equal to n is divisible by 2007? | nvidia |
Find the number of positive integers less than or equal to $24$ that are relatively prime to $24$. | 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 |
A triangle is formed with one vertex at the vertex of the parabola $y=x^2-1$ and the other two vertices at the intersections of the line $y=r$ and the parabola. If the area of the triangle is between $8$ and $64$ inclusive, find all possible values of $r$. Express your answer in interval notation. | nvidia |
How many different integers can be expressed as the sum of two or more consecutive positive integers? | nvidia |
A sphere is inscribed in a right circular cylinder such that the diameter of the sphere is 8 and the radius of the cylinder is 5.
The ratio of the volume of the cylinder to the volume of the sphere is $k^3$, where $k$ is an integer.
Find the radius of the cylinder. | nvidia |
Let $f(x) = \left\{\begin{array}{cl}
1 & \text{if } x > 0 \\
-1 & \text{if } x \le 0
\end{array}\right.$. Find all real $a$ such that the function $g(x) = \int_{-1}^{x} f(t) \;dt$ satisfies $g(x) + g^{-1}(x) = 0$ for all $x \in (-1, 1)$. | nvidia |
Let $f(x)$ be a quadratic function such that $-50 \leq f(x) \leq 50$ for all values of $x$ in the range $x_1 \leq x \leq x_2$. Find the minimum possible difference between $x_2$ and $x_1$. | nvidia |
Find the interval of all $x$ such that both $5x$ and $7x$ are in the interval $(3,4)$. | nvidia |
Find the largest possible area of a triangle inscribed in a unit circle.
(Note: A triangle is inscribed in a circle if all its vertices lie on the circle.) | nvidia |
A circle with center $O$ passes through the points $A(4, 6)$ and $B(0, 0)$. A line with slope $2$ intersects the circle at the points $A$ and $C$. Find the coordinates of the point $C$. | nvidia |
Determine the value of $a$ such that the expression $ax^2+4x+a^2$ has a minimum value of $-1$ for real $x$. | nvidia |
Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2+px+q=0$. If $g(x)=x^2+px+q$ and $h(x)=x^2+(1-p)x+(1-q)$, find the value of $x$ such that $g(x)=h^{-1}(x)$. | nvidia |
The function $f(x)$ satisfies
\[f(x^2) = f(x)^3\]for all real numbers $x,$ and $f(x) \neq 0$ for all real numbers $x.$ Find $f(-1).$ | nvidia |
Given that $x^2+px+q=0$ and the roots of the equation are in the ratio of 2:3, if the larger root is 3 more than the smaller root, find the sum of the possible values of $p$. | nvidia |
Let $a$ and $b$ be real numbers such that $|a+bi|=1$ and $a\neq 0$. Find the value of the expression $\frac{a^2-b^2}{a}$, given that $b$ is the imaginary part of $\frac{1}{a+bi}$. | nvidia |
Given that $d = c-5$, $c = b+3$, $b = a-1$, and $a = 2$, evaluate the expression \[ \frac{a+3}{a+2} \cdot \frac{b+2}{b+1} \cdot \frac{c-2}{c-3} \cdot \frac{d+1}{d-1}, \] provided that none of the denominators are zero. | nvidia |
Let $n$ be an odd integer with exactly 11 positive divisors. Find the number of positive divisors of $8n^3$. | nvidia |
Let $T$ be the set of all real values of $x$ with $0 < x < \frac{\pi}{4}$ such that $\sin x$, $\cos x$, and $\sin x + \cos x$ form the side lengths (in some order) of an isosceles triangle. Compute the sum of $\sin^2 x + \cos^2 x$ over all $x$ in $T$. | nvidia |
Determine a positive integer $M$ with three or fewer digits such that the first three digits of $M^2$ are 123. | nvidia |
Find the smallest possible value of the radius of a circle that circumscribes a triangle whose sides are 3, 4, and 5. | nvidia |
A triangle has three sides of the following side lengths: $9$, $15$, and $y^2$. What are all of the positive integer values of $y$ such that the triangle exists? Separate your answers using commas and express them in increasing order. | nvidia |
A number is divisible by 9 if the sum of its digits is divisible by 9.
Find the largest digit $d$ such that the sum of the digits in the number $23d456$ is divisible by 9, and $d$ is the units digit of the sum of the other digits. | nvidia |
If $x$,$y$, and $z$ are positive real numbers such that $x^2(y+z) = 100$, $y^2(x+z) = 125$, and $z^2(x+y) = 144$, then find $xyz$. | nvidia |
Let $a, b,$ and $c$ be positive real numbers such that $a + b + c = 1.$ Find set of all possible values of $\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}.$ | nvidia |
A line with a slope of -3 intersects the positive x-axis at A and the positive y-axis at B. A second line intersects the x-axis at C(6,0) and the y-axis at D. The lines intersect at E(3,3). What is the area of the shaded quadrilateral OBEC? | nvidia |
Given a regular hexagon, find the length of its side if its vertices are located on the lattice points of a coordinate plane with integer coordinates and all the coordinates of the vertices are between 0 and 10. | nvidia |
Find all real numbers $c$ such that the solutions of the equation
$$2x^3-5x^2+4cx-3=0$$
can be the side lengths of a triangle. Express your answer as a common fraction. | nvidia |
Determine the smallest digit $e$ so that the number formed by erasing one digit from $413{,}924$ is divisible by 3. | nvidia |
Find the number of positive integers less than 1000 that are not divisible by any of the prime numbers 2, 3, or 5. | nvidia |
Find the smallest whole number that must be a divisor of the product of any four consecutive positive integers and is also a power of 2. | nvidia |
Find the greatest integer that can be formed by the digits of the number 123456789, such that it is divisible by 9 and the sum of its digits is equal to 45. | nvidia |
If $z$ is a complex number such that
$z+\ds\f{1}{z}=2\cos 3^\circ$,
determine
$z^2+\ds\f{1}{z^2}$. | nvidia |
Let $f(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point. How many positive integer divisors does $f(2025)$ have? | nvidia |
Let $f(x)=e^{2x}+ax+b$ be a function defined for all real numbers $x$. The function $f$ has the property that $f(f(x))=x$ for all $x$, and $f(0)=1$. Find the ordered pair $(a,b)$. | nvidia |
Consider the sequence of numbers obtained by adding the squares of the positive integers with even exponents and subtracting the squares of the positive integers with odd exponents: $2^2-1^2+4^2-3^2+6^2-5^2+8^2-7^2+...$. Find the sum of the first 100 terms of this sequence. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a+b = \sqrt{16+a^2+b^2}$. Find the product $ab$. | nvidia |
Find the integer $n$, $10 \le n \le 15$, such that
$n \equiv 2311 \pmod{7}$. | nvidia |
When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \neq 0$$ | nvidia |
Find the sum of the values of $x$ that satisfy the equation $\sqrt{x}+\sqrt{x+1}=3$. | nvidia |
Two distinct vertices of a square are chosen at random. What is the probability that the area of the triangle formed by the square's center and these two vertices is less than half the area of the square? | nvidia |
The sum $S$ of four positive integers is $3$ times their product, and two of the integers are the sum of the other two. Find the product of all possible values of $S$. | nvidia |
What is the sum of all positive integer divisors of 77? | nvidia |
In a triangle $ABC$, the lengths of the sides $AB$ and $AC$ are $8$ cm and $10$ cm, respectively. The angle $A$ is equal to $60^{\circ}$. Find the area of the triangle $ABC$. | nvidia |
Two positive integers $m$ and $n$ are chosen such that $m$ is the smallest positive integer with only two positive divisors and $n$ is the largest integer less than $100$ with exactly three positive divisors. What is $m+n$? | nvidia |
A regular hexagon is inscribed in a circle of radius $3$. Let $A$ be one of the vertices of the hexagon. Through $A$, draw a line $l$ that is tangent to the circle and intersects the opposite side of the hexagon at point $B$. Let $C$ be the foot of the perpendicular from $B$ to the line passing through $A$ and the cent... | nvidia |
The Fibonacci sequence is defined by the recurrence relation $F_n=F_{n-1}+F_{n-2}$, where $F_1=1$ and $F_2=1$. Find the least positive integer $k$ such that $F_k$ is divisible by 17. | nvidia |
Find the smallest digit $B$ in the five-digit integer $27B,\!418$ such that the entire number is divisible by 9. | nvidia |
Find the smallest possible degree of the numerator of a rational function, given that the graph of the function has a slant asymptote. | nvidia |
Find the largest positive integer $n$ such that $2n \equiv 578 \pmod{19}.$ | nvidia |
In the system of equations
$$\begin{cases}
u+v-2w=-5,\\
u-w+2v=-11,
\end{cases}$$
find the minimal value of $|w|$ provided that $u$, $v$, and $w$ are positive integers. | nvidia |
Given a point $P$ on the unit circle that is $\arctan\left(\frac{3}{4}\right)$ radians from the point $(1,0)$ in the counterclockwise direction, compute the coordinates of $P$. | nvidia |
Find the number of positive integers less than or equal to 30 that are not divisible by any perfect square other than 1. | nvidia |
Let $p^a$ be the greatest power of a prime $p$ that divides $100!$ (the product of all positive integers up to 100), and let $q^b$ be the greatest power of a prime $q$ that divides $100!$. If $p$ and $q$ are distinct primes such that $p+q=10$, find the value of $a-b$. | 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 set of 16 distinct points on a circle, determine the maximum number of non-overlapping isosceles right triangles that can be formed using these points as vertices. | nvidia |
Find the number of ordered pairs of integers $(c,d)$ with $1 \leq c \leq 50$ and $d \geq 0$ such that the polynomial $x^2+cx+d$ can be expressed as the sum of two perfect squares with integer coefficients. | nvidia |
Problem:
In a round-robin tournament with $n$ teams, each team plays every other team exactly once, and the probability of a tie is zero. Instead of winning or losing, each team earns a score based on the outcome of each game. The team with the highest score at the end of the tournament is declared the winner.
Each t... | nvidia |
In a quadrilateral, the angles are $80^{\circ}$, $60^{\circ}$, $80^{\circ}$, and $x$. If the sum of the measures of the interior angles is equal to the product of the number of sides and the average of the measures of the angles, find the value of $x$. | nvidia |
A function $g$ has the property that $g(2x^2+1)=\frac{1}{x^2+1}$ for all real numbers $x$. What is $g(5)$? | nvidia |
Find all possible values of the expression
\[\frac{a^2+b^2+c^2}{|a^2+b^2+c^2|} + \frac{ab+bc+ca}{|ab+bc+ca|}.\]Enter all possible values, separated by commas. | nvidia |
Find the positive value of x that satisfies the equation
$$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+x}}}}= \sqrt{x}+1$$ | nvidia |
If $a$, $b$ are nonzero real numbers such that $a^2+b^2=8ab$, find the value of $\left|\frac{a+b}{a-b}\right|$. | nvidia |
Find the sum of the solutions of the equation $-32x^2 + 84x + 135=0$. | nvidia |
Find the number of ordered pairs $(A, B)$ of subsets of a set with $n$ elements, such that the symmetric difference of $A$ and $B$ has exactly $k$ elements, and the intersection of $A$ and $B$ has exactly $m$ elements. | nvidia |
Find the number of positive integer divisors of $M$ if $M = 3^4 \cdot 5^3 \cdot 7^2 \cdot 11^2$. | nvidia |
Find the value of $x$ such that the fourth degree polynomial $f(x) = 2x^4-17x^3+26x^2-24x-60$ has a maximum value. | nvidia |
Let $g(x)$ be a function piecewise defined as $g(x) = c \log_b (|x|)$ for all real $x \neq 0$, and $g(0) = k$. What are all possible values of $c$ and $k$ that satisfy the equation
\[g(a^x) - xg(a^{-x}) = d\]
for all real numbers $x$, given constants $a, b, d$ where $a$ and $b$ are positive integers? | nvidia |
Find all positive values of $d$ so that the inequality $x^2-8x+d<0$ has real solutions for $x$. Express your answer in interval notation. | nvidia |
Find the sum of all positive integers less than 15 that are not multiples of 3 or 5. | nvidia |
Find the largest value of $c$ such that the function $f(x) = x^3 - 2x^2 + cx + 1$ has at least one real root. | nvidia |
Find the number of positive integers $k$ such that the greatest common divisor of $6^6$, $8^8$, and $k$ is $2^2 \cdot 3^3$ and $k$ is a divisor of $12^{12}$. | nvidia |
What is the least positive integer $n$ such that $4125$ is a factor of $n!$? | nvidia |
Compute the value of the infinite series \[
\sum_{n=3}^{\infty} \frac{n^5+2n^3+12n^2+15n+12}{3^n \cdot \left(n^5+8\right)}
\] | nvidia |
Find all real numbers $b$ such that the roots of the polynomial
$$x^3 - 6x^2 + 21x + b$$
have the same absolute value and are not all real. | nvidia |
Find the value of $y$ such that the three-digit number $\underline{2y6}$ is divisible by 9. | nvidia |
Determine all real values of $x$ such that $\log_3(x-1) + \log_3(x-2) = \log_3(2x)$. | nvidia |
Find the number of different 6-letter strings that can be formed using only the letters A, B, and C, with the condition that no two adjacent letters are the same. | nvidia |
Determine the least positive integer $n$ such that the remainders of $n^2$, $2n$, and $3n$ upon division by 7, 11, and 13 respectively are all the same. | nvidia |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.