instruction_seed stringlengths 30 7.99k | source stringclasses 1
value |
|---|---|
In a triangle $ABC$, a circle with radius $r$ is inscribed inside the triangle, touching sides $AB$, $BC$, and $CA$ at points $D$, $E$, and $F$, respectively. The perimeter of the triangle is equal to $3p$, and the area of the triangle is $4r^2$. If $p=15$ cm, find the length of the shortest side of the triangle. | nvidia |
Let $S$ be the set of all integers $n$ such that $1000 < n < 9999$, and the four-digit number $n$ is divisible by the integer formed by its digits in reverse order. How many integers are in $S$? | nvidia |
Determine the smallest integer value of $y$ such that the solution set of the inequality $3-2y<11$ contains at least one integer. | nvidia |
Find the number of positive integers $m \le 2000$ such that $24m$ is a perfect cube. | nvidia |
A positive integer is said to be square-free if it is not divisible by the square of any integer greater than 1. Which number in the set {16, 18, 20, 21, 22, 24} has the smallest number of positive divisors among all its positive divisors that are square-free? | nvidia |
The sum of the squares of two consecutive positive integers is equal to the square of a third integer that is 10 more than the smaller of the two consecutive integers. Find the three integers. | nvidia |
What is the largest possible area of a triangle that can be inscribed in a circle of radius $4$ cm, such that the triangle is formed by the endpoints of a diameter and a point on the circle that forms a given angle with the diameter? | nvidia |
The decimal expansion of $\frac{1}{9}$ is $0.\overline{1}$. Find the value of $k$ such that the decimal expansion of $\frac{1}{3^k}$ has a repeating block of length $3$. | 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 |
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 $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 10.$ Find the minimum value of
\[\frac{1}{a} + \frac{1}{b} + \frac{1}{c}.\] | nvidia |
If $\alpha$, $\beta$ and $\gamma$ are positive integers such that $\sqrt[3]{\alpha x} \cdot \sqrt[3]{\beta x} \cdot \sqrt[3]{\gamma x}=6x$, where $x$ is a positive real number, determine $\alpha \beta \gamma$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x + y = 1$. Find the minimum value of $x^2 + y^2 + \frac{1}{x^2} + \frac{1}{y^2}$. | nvidia |
Let $m$ and $n$ be positive integers such that
$$\frac{3}{5} < \frac{m}{n} < \frac{2}{3}$$
and $n$ is as small as possible. What is $n-m$? | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x + f(y)) = x + y\]for all real numbers $x$ and $y.$ | nvidia |
What is the largest positive integer $n$ such that $n!$ has exactly 3 different prime factors? | nvidia |
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $\displaystyle {{m+n\pi}\over
p}$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m+n+p$. | nvidia |
Find the sum of all integers $k$ such that $\binom{23}{4} + \binom{23}{5} = \binom{24}{k}$. | nvidia |
Find the largest four-digit integer that can be expressed as the sum of three distinct perfect squares, such that the digits of the integer form an arithmetic sequence. | nvidia |
Find the value of $x$ that maximizes the expression $\frac{x^2+15x-54}{x^2-6x+1}$. | nvidia |
Find the least positive integer $n$ such that the product of its positive whole number divisors is greater than 1000. | nvidia |
A triangle with vertices at points A(1,3), B(5,2), and C(7,1) is rotated 180° around the midpoint of segment AB. Find the sum of the coordinates of the image of point C. | nvidia |
Count the number of positive integers less than 1000 that are multiples of 9 and have exactly two digits equal to 7. | nvidia |
Solve for $y$ in the equation $\sqrt{y^2+4y+4} = \sqrt{2y+5}$, given that $y$ is a positive real number. | nvidia |
Let $C$ equal the number of three-digit integers that are multiples of 3. Let $D$ equal the number of three-digit integers that are multiples of 2. Find $C+D$. | nvidia |
Given that $n$ is a positive integer and $p$ is the product of all the distinct prime factors of $10^n-1$, determine the smallest value of $n$ for which $p>10$. | nvidia |
Let $g(n)$ be the product of all the proper divisors of a positive integer $n$. If $g(g(n)) = n$, then call $n$ self-similar. Determine the smallest self-similar positive integer greater than 1. | nvidia |
What is the smallest positive integer $k$ such that $k(n-1)(n-2)(n-3)(n-4)$ is a multiple of $24$ for all positive integers $n$? | nvidia |
Find the number of solutions to the equation $|z|^2 + az + b = 0$ that lie inside the unit circle, where $a$ and $b$ are real numbers and $|a| + |b| = 1$. | nvidia |
Given the convex quadrilateral $ABCD$, let $O$ be its circumcenter and $r$ be the radius of the circumcircle.
Prove that
\[|OA|\cdot |OB|\cdot |OC|\cdot |OD| \geq 16 r^4.\] | nvidia |
Find the number of ways to arrange the digits of the number 12345 such that the sum of the digits in each arrangement is a multiple of 5. | nvidia |
Compute the positive integer $x$ such that $7^{3x+1}$ has exactly 5 digits. | nvidia |
Compute the number of 3-digit positive integers divisible by 7. | nvidia |
Find the sum of all the perfect squares between 10 and 100 that are also multiples of 4. | nvidia |
If $a$ and $b$ are integers such that $a>1$ and $b>1$, compute $a\left(\frac{1}{b}\right)^4$, given that $a^2 + 2b^2 = 15$. | nvidia |
How many three-digit positive integers have the property that the product of their digits is equal to 36? | nvidia |
Let $a$, $b$, and $c$ be the roots of the cubic equation $x^3 + mx^2 + nx + p = 0$. Find the number of cubic equations for which the sum of the cubes of their roots is equal to the sum of the squares of their roots multiplied by a root.
That is, find the number of triples $(m,n,p)$ such that $$a^3+b^3+c^3=(a^2+b^2+c^2... | nvidia |
Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?
$\textbf{(A)}\ \sqrt{15} \qquad \textbf{(B)}\ \sqrt{21} \qquad \textbf{(C)}\ 2\sqrt{6} \qquad \textbf{(D)}\ 5 \qquad \textbf{(... | nvidia |
Find the number of points with integer coordinates that satisfy the inequality $|x^2-3x+2|+|y^2-3y+2|\le 1$. | nvidia |
Find the sum of all the positive integers less than or equal to 100 that have exactly three distinct factors. | nvidia |
Let's consider the product of all the divisors of a positive integer $n$ which are not multiples of 3. For example, for $n=7$, we get $1\cdot7=7$, and for $n=9$, we get $1$. Determine the largest value of $n$ for which the product is 1. | nvidia |
Find the maximum value of
\[|x| + |y| + |x - y|\]
over all real numbers $x$ and $y$ such that $x^2 + y^2 \leq 1.$ | nvidia |
A function machine has three possible outputs: $A,B,C.$ The input is a two-digit number. If the sum of the digits of the input number is odd, the output is $A.$ If the sum of the digits of the input number is even, but the number itself is odd, then the output is $B.$ In all other cases, the output is $C.$ What is the ... | nvidia |
What is the sum of the squares of all the positive integer divisors of 24 that are also perfect squares? | nvidia |
Find the sum of squares of all real solutions to the equation $\tan(x) = x^2$ in the interval $[-1,1]$. | nvidia |
Consider a family of real functions
$$\gamma_s(x)=\cos x+s(x+1)+\ln (x+s),$$
where $s$ is a real parameter.
The set of $x$ such that $f(x)$ is defined will be denoted by $A$.
Find the sum of all $s$ such that the inverse function $\gamma_s^{-1}(x)$ is defined on the entire range $\gamma_s(A)$ of $\gamma_s(x)$. | nvidia |
Find a positive integer $n$ such that the equation
$$\frac{x^2-2x+n}{x^2-4x+n}=x^2$$
has a rational solution. | nvidia |
What are the last two digits in the sum of factorials of the first 100 positive integers? | nvidia |
Let $x$ and $y$ be two positive real numbers such that $\ds\sqrt x + \ds\sqrt y =10$ and $xy = 1600$.
Determine $x$ and $y$. | nvidia |
Determine the number of four-digit positive integers with non-repeated digits that are divisible by 5 and have digits that form an increasing sequence. | nvidia |
Find the number of positive and negative integer divisors of 42. | nvidia |
The product of all the positive divisors of 1, each taken once, is equal to 1. The product of all the positive divisors of 2, each taken once, is equal to 2. The product of all the positive divisors of 3, each taken once, is equal to 3. The product of all the positive divisors of 4, each taken once, is equal to 8. Cont... | nvidia |
Find the set of all x-values such that the function
\[h(x) = \frac{3x^2+6x+8}{x^3 - 2x + 1}\]
is bounded. | nvidia |
Find the smallest positive integer $n$ such that $n$ is divisible by 36 and $n+1$ is divisible by 132. | nvidia |
How many positive two-digit integers have an odd number of positive factors? | nvidia |
Find the value of $x^3$ given that there exists a real number $x$ such that $\sin (\arccos (x)) = x^2$. | 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 |
What is the smallest positive integer $n$ such that the product of the digits of $n$ is equal to the difference between $n$ and the sum of its digits? | nvidia |
Find all real values of $x$ that satisfy $x + \frac{45}{x-4} = -10.$
Enter all the solutions, separated by commas. | nvidia |
Find the value of $x$ such that $\lfloor x \rfloor + \lceil 2x \rceil = 7$, given that $x$ is a real number between 2 and 3. | nvidia |
Find the largest constant $C$ so that
\[x^2 + y^2 + 1 \ge C(x + y)\]for all real numbers $x$ and $y.$ | nvidia |
Find the smallest value of $c$ such that the range of $f(x)=x^2+3x+c$ includes a positive integer. | nvidia |
Determine the product of the roots of the polynomial $x^4-2x^3+5x^2-7x-1=0$ that lie outside the unit circle. | nvidia |
A real number $b$ is chosen randomly and uniformly from the interval $[-15, 20]$. Find the probability that the roots of the polynomial
\[x^3 - 3bx^2 + (3b + 1)x - 1\]are all real. | nvidia |
Determine the largest positive integer $n$ such that $2^n$ is a divisor of the least common multiple of the numbers $1, 2, 3, \dots, 30$. | nvidia |
In a triangle, there are three integers that are common factors of the lengths of all three sides. Given that the sides of the triangle are 12, 15, and 18, find the sum of these three integers. | nvidia |
Find the smallest positive integer $n$ such that
$$\frac{\lfloor \sqrt{n} \rfloor}{\sqrt{n}} = \frac{3}{4}.$$
Can you solve it? | nvidia |
A deck of 52 cards is shuffled and two cards are drawn at random. What is the probability that the sum of the ranks of the two cards is a multiple of 3? | nvidia |
Find the positive value of $n$ such that the equation $9x^2+nx+1=0$ has exactly one solution in $x$. | nvidia |
Consider a 4x4 grid of squares. We want to color some of the squares black such that each row, column, and diagonal contains at most two black squares, and no two black squares share an edge. Determine the maximum number of black squares that can be colored under these conditions. | nvidia |
Find $s$ such that $(s,2)$ lies on the line through $(1,6)$ and $(7,-2)$. | nvidia |
Determine the sum of the positive divisors of the number $100$ except for $1$ and $100$. | nvidia |
A polynomial with real coefficients $P(x)$ has a complex root $1+i\sqrt{3}$ and satisfies $P(-1)=0$. If $P(x)$ can be written in the form $(x^2+cx+d)(x+1)$, where $c$ and $d$ are real numbers, find the value of $c+d$. | nvidia |
Find the value of x such that the sum of the digits of the three-digit number 245x is equal to 17. | nvidia |
Let $a$, $b$, $c$ be positive real numbers such that $a^2+b^2+c^2+a+b+c=abc$.
Find the maximum value of $a^3+b^3+c^3-3abc$. | nvidia |
A set of $n$ different positive integers has the following property: for any three distinct elements $a$, $b$, and $c$, the product $abc$ is divisible by the sum $a+b+c$. What is the largest possible value of $n$? | nvidia |
Find the number of positive integers less than 80 that have a prime factorization with an odd number of factors. | nvidia |
Compute the largest positive integer $n$ such that the graph of the polynomial $p(x) = x^3 + nx^2 + 1$ has at least three distinct real roots and the sum of the pairwise distances between the $x$-intercepts of the graph is less than 2015. | nvidia |
Let $f(x)$ be a quadratic function such that its graph is symmetric with respect to the line $y=-2x+3$. Given that the point $(2,0)$ is on the graph of $f(x)$ and the vertex of the parabola is at the point $(h,k)$, where $h$ and $k$ are both integers, find the value of $k$. | nvidia |
Find the smallest positive integer that is divisible by exactly three distinct prime numbers. | nvidia |
Find the minimum possible value of
\[\sum_{1 \le i < j \le 100} a_i a_j,\]
given that each of the numbers $a_1,$ $a_2,$ $\dots,$ $a_{100}$ is a complex number with absolute value 1. | nvidia |
A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the square to the area of the circle? | nvidia |
Find the maximum value of the expression $\sqrt{x+y+1}+\sqrt{4-x-y}$ for real $x$ and $y$ satisfying $x+y \leq 2$. | nvidia |
A die is rolled repeatedly until the same number appears three consecutive times. Let $p$ be the probability that all three consecutive numbers are the same. Let $q$ be the probability that the first two consecutive numbers are the same and the third number is different. What is the value of $q/p$? | nvidia |
Let $a,$ $b,$ and $c$ be real numbers such that
\[a+b+c=2,\] \[ab+bc+ca=-1,\]and \[abc=-6.\]Compute $(a+b+c)^3+(ab+bc+ca)^3+(abc)^3.$ | nvidia |
A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls is at least 8, given that the first roll is even? | nvidia |
Simplify the expression $\sqrt[3]{x^6 \cdot \sqrt{x^2 + 1}}$ where $x$ is a positive integer. The answer can be written as $\sqrt[3]{Ax^B}$, where $A$ and $B$ are integers, $A$ is positive, and $B$ is greater than or equal to 1. Find the minimum possible value of $A+B$. | nvidia |
What is the degree measure of the smaller angle between the hour hand and the minute hand of a clock at exactly 2:30 p.m. on a 12-hour analog clock?
A right triangle is inscribed in a circle of radius 1, with one vertex at the center of the circle and the other two vertices on the circle. The hypotenuse of the triangl... | nvidia |
Let $g(x)$ be a polynomial function such that $g(g(x)) = x^4 + 2x^2 - 8$ for all real numbers $x$. Find the smallest possible degree of $g(x)$. | nvidia |
Find the sum of all real solutions to the equation \[\frac{x-2}{x^2+4x+1} = \frac{x-5}{x^2-10x}.\] | nvidia |
Find the value of $x$ such that $\sqrt[3]{x^2+27} = 3x-2$. | nvidia |
The diagonal of a cube with twice the surface area of a cube with edge length 8 inches can be expressed in the form $m\sqrt[2]{n}$ where $m$ and $n$ are positive integers and $n$ contains no perfect square factors. Compute $m+n$. | nvidia |
Prove that the determinant of the matrix
\[\begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}\]
is equal to \((x+y+z)(x^2+y^2+z^2-xy-xz-yz)\). | nvidia |
Determine the value of $a$ such that $a^0$ is equal to the sum of the exponents of the prime factorization of $a$. | nvidia |
A positive integer is chosen randomly between 1 and 100 (inclusive). What is the probability that the number is composite and has exactly 3 positive divisors? | nvidia |
Given that $f(x)$ is a function such that $f(x+1) - f(x) = f(x^2)$ for all real numbers $x$, find $f(x)$. | nvidia |
Given that the graphs of $y=h(x)$ and $y=j(x)$ intersect at $(2,2),$ $(4,6),$ $(6,12),$ and $(8,12),$ there is one point where the graphs of $y=h(2x)$ and $y=2j(x)$ must intersect. What is the sum of the coordinates of that point? | nvidia |
A polynomial $p(x)$ has the property that its $n$ roots are the numbers $x_1, x_2, \dots, x_n$ with the property that $x_2 = x_1^2, x_3 = x_2^2, \dots, x_n = x_{n-1}^2$. Find all such polynomials $p(x)$. | nvidia |
Given that $a$, $b$, and $c$ are positive integers, simplify the expression
$\frac{a^2b^3c^4}{a^3b^2c^5}$. | nvidia |
Given that the polynomial $x^2-5x+t$ has only positive integer roots, find the average of all distinct possible values of $t$. | nvidia |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.