instruction_seed stringlengths 30 7.97k | source stringclasses 1
value |
|---|---|
Find the positive integer $n$ for which $\gcd(n, n+1) + \text{lcm}(n, n+1) = 88$. | open2math |
A positive integer is a divisor of 135 that is a palindrome. What is the sum of all of its digits? | open2math |
Suppose that $x$ and $y$ are integers such that $$2y = 9 - 3x.$$How many of the first five positive integers must be divisors of $3y + 15$? | open2math |
Given a positive integer $n$, determine the largest possible value of $k$ such that $n^3+2n^2+3n+k$ is divisible by 4 for all values of $n$. | open2math |
Compute the number of subsets of {1,2,...,15} that have 3 elements and contain at least two even numbers. | open2math |
Let $\tau (n)$ denote the number of positive integer divisors of $n$ (including $1$ and $n$). Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$.
| open2math |
Given that the sum of the squares of the roots of a quadratic equation is 52 and the sum of the roots is -8, find the product of the roots. | open2math |
The blue parabola shown is the graph of the equation $y = ax^2 + bx + c$. Find $c$. Assume that the graph has the standard unit scale.
[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 tick... | open2math |
Find the sum of all real solutions to the equation:
$$\frac{x-1}{x^2+3x+2} = \frac{x-4}{x^2-7x+1}$$ | open2math |
Determine all real numbers $x$ such that $\sin (2x+30^\circ) = \cos (3x-20^\circ)$. | open2math |
Find the number of positive three-digit integers that can be formed using only the digits 1, 2, 3, 5, and 7, such that the sum of the digits is a multiple of 5. | open2math |
What is the greatest integer $n$ such that the number $n!$ contains the same number of zeros at the end as it contains nonzero digits? | open2math |
Let $f(x)$ be a polynomial with integer coefficients. If $f(17)=13$, $f(13)=17$ and for every positive integer $k$ not equal to $13$ or $17$, there is a positive integer $n$ such that $n$ and $k+n$ are roots of $f(x)$, find $f(x)$. | open2math |
Let $k$ be a real number such that $k > 1$ and
\[\sum_{n=1}^{\infty} \frac{5n-1}{k^n} = \frac{13}{4}.\]Find $k.$ | open2math |
A $\textit{composite number}$ is a number that has two or more prime factors. The number 145 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers? | open2math |
An equilateral triangle is circumscribed around a circle. One of the sides of the triangle is $12$, and the circle passes through the midpoint of this side. Find the area of the triangle. | open2math |
Consider a set of $n$ complex numbers, each with magnitude $1$. These numbers are equally spaced around the unit circle, with the first number at position $(1,0)$. We want to assign a sign ($+1$ or $-1$) to each number so that the real part of the sum of the signed numbers is maximized.
What is the largest possible va... | open2math |
Let $f(x)=6x^2+x-1$. Prove that for all positive integers $n$ we have
$$\binom{n}{2} f(1) + \binom{n}{3} f(2) + \dots + \binom{n}{n} f(n-1) = f(n).$$ | open2math |
Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. | open2math |
Let $A$ be a $3 \times 3$ matrix with integer entries such that $A^3 = I$, where $I$ is the $3 \times 3$ identity matrix. Find the smallest possible value of the sum of the absolute values of the entries of $A$. | open2math |
Let $m$ be a positive integer. In the set of ordered pairs of positive integers $(x,y)$, where $\gcd(x,y)=m$, it is known that the pair with the greatest sum $x+y$ has $x=30m$ and $y=45m$. Find $m$. | open2math |
Determine the number of real solutions to the equation $$\log_{\sqrt{x}}\left(\sin\left(\arctan\left(\frac{1}{\sqrt{x}}\right)\right)\right)=\frac{1}{2}.$$ | open2math |
A box contains 18 marbles, some of which are blue and the rest of which are green. When two marbles are drawn simultaneously at random, the probability that both marbles are green is $\frac{1}{8}$. How many marbles in the box are green? | open2math |
When the digits in the number $2005$ are reversed we obtain the number $5002,$ and $5002 = a \cdot b \cdot c$, such that $a$, $b$ and $c$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$ and $p_3$ such that $p_1 + p_2 + p_3 = a+b+c$? | open2math |
In a regular polygon, the number of diagonals that can be drawn from a single vertex is one less than the number of sides. If the number of diagonals from one vertex is multiplied by the number of sides, the product is 165. Find the number of sides of the polygon. | open2math |
How many five-digit positive integers have the property that the sum of their digits is a multiple of 5? | open2math |
Let's determine if there exists a positive real number $x$ such that for any complex number $z$, if the point $x$ lies on the segment connecting $z$ and $\frac{1}{z}$ in the complex plane, then
$$|z| + \left|\frac{1}{z}\right| \geq 2\sqrt{2}.$$ | open2math |
The numbers $p$, $q$, $r$ are consecutive terms of a geometric sequence, where $p$ and $r$ are positive integers. Given that $q$ is a rational number and $pr = 100$, determine the value of $q$. | open2math |
Determine the value of $k$ such that $\cos 10^\circ = \sin (2k-5)^\circ.$ | open2math |
Find the positive real number $y$ such that $\lceil y \rceil \cdot y = 135$. Express $y$ as a decimal. | open2math |
Let $P$ be the product of the first $100$ positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k .$
| open2math |
Find all real numbers $x$ that satisfy
\[\frac{x}{\sqrt{x^2 + 1}} + \frac{\sqrt{x^2 - 1}}{x} \ge 2.\] | open2math |
Find the smallest positive integer $n$ such that the equation $x^3 - 2x^2 - nx + 8 = 0$ has at least one rational root. | open2math |
Find all real values of $x$ that satisfy $x + \frac{45}{x-4} = -10.$
Enter all the solutions, separated by commas. | open2math |
What is the sum of all possible values of the coefficient of $x$ in the equation $x^2 + mx + 20 = 0$ if the equation has two distinct positive integer solutions? | open2math |
Determine the smallest positive integer that is divisible by the sum of its digits, the product of its digits, and each of its distinct digits, with no repeated digits allowed. | open2math |
Find the largest value of $d$ such that $3$ is in the range of $g(x)=x^2-5x+d$. | open2math |
Find all possible values of the base $b$ if $271_b \cdot 60_b \cdot 2002_b = 77_b \cdot 21_b \cdot 1700_b$ | open2math |
What is the smallest positive integer $n$ such that $\left(\dfrac{3}{4}\right)^n$ is less than 0.01? | open2math |
Let $f(x)$ be a function such that $f(x+1)-f(x) = 2x+1$ for all real numbers $x$. If $f(0) = 1$, find $f(-1)+f(1)$. | open2math |
Find the integer $k$ such that the polynomial $x^3 - 2x + k$ divides the polynomial $(x-1)^{10} + (x+1)^{10}$. | open2math |
Two integers, $m$ and $n$, are randomly selected from the set of positive integers less than 100, with replacement. If the probability that the sum of the two integers is even is $\frac{3}{5}$, what is the probability that the sum of the two integers is a multiple of 3? | open2math |
Let $\mathbf{v}$ be a non-zero vector such that
$\mathbf{v}\cdot\begin{pmatrix}1\\-1\\1\end{pmatrix}=0$. Find $\mathbf{v}$ such that
\[\frac{\mathbf{v}\cdot\begin{pmatrix}1\\2\\1\end{pmatrix}}{\mathbf{v}\cdot\begin{pmatrix}1\\0\\1\end{pmatrix}}=\frac{5}{3}.\] | open2math |
What is the largest positive integer $n$ such that every interval of length $n$ contains a prime or a number that can be written as the sum of two primes? | open2math |
Find the smallest integer $n$ such that $2^n > \pi \cdot 10^{12}$. | open2math |
Let's consider a function with a similar trigonometric theme and domain restriction.
Find the smallest value of the positive integer $k$ such that the function $g(x) = \arccos(\sqrt{kx})$ has a domain that includes the interval $[0, \frac{1}{2}]$. | open2math |
Find the least positive integer $y$ that satisfies $y+2743 \equiv 1876 \pmod{8}$. | open2math |
A certain type of tile is used to create patterns on a wall. Each tile has a square shape with an area of 1 square foot. The pattern requires that the tiles be arranged in a way that each row has the same number of tiles, and each column has the same number of tiles. In how many ways can a rectangular wall with an area... | open2math |
Let $x$ be the number of positive multiples of $12$ that are less than $40$. Let $y$ be the number of positive integers that are less than $40$, and a multiple of $4$ and a multiple of $3$. Compute $(x - y)^3$. | open2math |
Find the greatest integer value of $b$ for which the expression $\frac{9x^3+4x^2+11x+7}{x^2+bx+8}$ has a domain of all real numbers. | open2math |
A five-digit number $abcde$ is divisible by 99 if and only if the sum of its digits $a+b+c+d+e$ is divisible by 9 and the number formed by the last two digits $de$ is divisible by 11. How many different five-digit numbers $abcde$ can be formed such that $a+b+c+d+e=27$ and $de$ is divisible by 11? | open2math |
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)? | open2math |
Given that $f(x) = \frac{x^3 - 1}{2}$, find the value of $x$ such that $f^{-1}(x) = -\frac{5}{6}$. | open2math |
Given that the point $(2,5)$ is on the graph of $y=g(x)$, and $g(x)$ is an odd function, there is one point that must be on the graph of $y=g(-2x)+2$. What is the product of the coordinates of that point? | open2math |
Find the largest of four consecutive odd integers whose sum is equal to the sum of the first $15$ positive odd integers. | open2math |
Circle $A$ has its center at $(3, -8)$ and a radius of $7$ units. What is the sum of the $x$-coordinates of the two points on circle $A$ that are also on the $x$-axis? | open2math |
Find the product of all positive integral values of $n$ such that $n^2-35n+306= p$ for some prime number $p$. Note that there is at least one such $n$. | open2math |
Given that
$$A=(-5+x)(-2+x)(-4+x)(1+x)(-1+x)(-3+x)$$
and that $A$ can be written in the form
$$B(x^2+ax+b)(x^2+cx+d),$$
determine $B$. | open2math |
Find all real numbers $a$ such that for every real polynomial $p(x)$ of degree $n \ge 2$, the following equation holds:
$$p(x - a) + p(x + a) = p(x) + p(x + 2a)$$
for all real numbers $x$. | open2math |
Let a, b, c be distinct real numbers. For how many real numbers d do there exist non-zero real numbers x, y, z such that all the lines
y=a(x-d)
y=b(x-d)
y=c(x-d)
are tangent to the parabola y=x^2? | open2math |
What is the largest integer n such that the product of any five consecutive integers is divisible by n? | open2math |
Let \(P(x)\) be a polynomial with real coefficients such that \(P(\cos \theta) = \frac{1}{\sin \theta}\) for all \(\theta \in (0, \pi)\). If \(Q(x) = xP(x)\), find the sum of all real roots of \(Q(x) = 0\). | open2math |
A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater than the number of the green point, but les... | open2math |
Find the smallest positive integer that can be expressed as the sum of three distinct prime numbers, each greater than 20. | open2math |
Let $n$ be a positive integer. A function $f:\mathbb{Z}\to\mathbb{Z}$ satisfies the following two properties:
(i) $f(m+n)=f(m)+f(n)+3mn$ for all integers $m$ and $n$.
(ii) $f(mn)=f(m)f(n)+m^2n^2$ for all integers $m$ and $n$.
Determine the number of possible values of $f(3)$. | open2math |
A parabola is graphed below.
[asy]
import graph;
size(180);
real lsf=0.5;
pen dps=linewidth(0.7)+fontsize(10);
defaultpen(dps); pen ds=black;
real xmin=-3.25,xmax=4.25,ymin=-7.25,ymax=6.25;
Label laxis; laxis.p=fontsize(10);
xaxis("$x$",xmin,xmax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),abo... | open2math |
The function $f(x)$ satisfies
\[f(x + y) = f(x) f(y)\]for all real numbers $x$ and $y.$ Find all possible values of $f(0).$
Enter all the possible values, separated by commas. | open2math |
Find all real values of $a$ such that the equation $ax^2 + (a + 3)x + 4 = 0$ has at least one root that is a root of the equation $2x^2 + 3x + 1 = 0$. | open2math |
Find the smallest value of $b$ such that the quadratic expression $2x^2 + bx + 32$ can be expressed as the difference of two perfect squares with integer coefficients. | open2math |
Let $S$ be a set of positive integers such that for each element $n \in S$, the base-$n$ representation of the integer $2022$ is a palindrome. Determine the number of elements in $S$. | open2math |
Let $a$, $b$, and $c$ be positive integers such that $\sqrt{a+b\sqrt{c}}=m+n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers. Find the smallest possible value of $m+n+p$. | open2math |
A circle is inscribed in a triangle with sides $3x$, $4x$, and $5x$. Find the area of the triangle when the radius of the circle is 2. | open2math |
The greatest common divisor of two positive integers $m$ and $n$ is 17. Determine the smallest positive integer $a$ so that $4m + 7n + a$ is divisible by 17. | open2math |
Determine all positive integers $n\geq 2$ such that $n^2+1$ is a perfect square. | open2math |
Find all positive three-digit integers such that the sum of the squares of their digits is equal to the product of their digits. | open2math |
If $a$, $b$ are nonzero real numbers such that $a^2+b^2=8ab$, find the value of $\left|\frac{a+b}{a-b}\right|$. | open2math |
In a sequence of real numbers, the sum of any three consecutive terms forms an arithmetic progression with a common difference of $d$. If the sum of the first three terms is 15 and the sum of the next three terms is 18, determine the value of the 100th term. | open2math |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x+y) = g(x)g(y) + xg(y) + yg(x) + xy\]
for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(0),$ and let $t$ be the product of all possible values of $g(0).$ Find $m \times t.$ | open2math |
Find the smallest positive integer $x$ such that $7x \equiv 25 \pmod{13}.$ | open2math |
Let $n$ be a positive number such that when $237$ is divided by $n$, the remainder is $8.$ Compute the sum of all possible three-digit values of $n$. | open2math |
Compute the number of distinct subsets of size $3$ that can be formed from the elements $\left\{-\frac{1}{4}, \frac{7}{10}, \pi,\frac{5}{6},-11,49,-99,72,\sqrt{8}, 45\right\}$. | open2math |
What is the sum of all real numbers that are tripled when multiplied by their squares? | open2math |
The area of the smallest circle that will contain a square of side length 6 is what? | open2math |
Let x be a real number such that $\lfloor x\rfloor+\lceil x\rceil=13$.
Determine the value of x. | open2math |
Find the largest positive integer with only odd digits that is less than $100000$ and is a multiple of $3$. | open2math |
Let $g(x)=x^2-5x$. What is the smallest value of $n$ such that the $n$th iterate of $g$, $g^n(x)$, is equal to $g^n(0)$ for all real numbers $x$? | open2math |
Find the point on the plane defined by
$$3x-2y+z=5$$
that is farthest from the origin $(0,0,0)$. | open2math |
Find the area of the triangle with vertices $(6,5,3),$ $(3,3,1),$ and $(15,11,9).$ | open2math |
What is the largest positive integer $n$ such that $n!$ can be expressed as a product of $n$ consecutive integers? | open2math |
The equation $x^4-4x^2+4=0$ has two real solutions and two non-real solutions. What is the positive difference between the two real solutions? | open2math |
Find the sum of all positive integer divisors of 100 that are not multiples of 3. | open2math |
Let $a$ and $b$ be real numbers. If $z$ is a complex number such that $|z-a|=2$ and $|z-b|=4$, determine $|a-b|$. | open2math |
Find the number of positive integers $n$ such that $n^3+2n^2-3n+1$ is between $\sqrt{50}$ and $\sqrt{500}$. | open2math |
Find the product of all constants $m$ such that the quadratic $x^2 + mx + 6$ can be factored in the form $(x+c)(x+d)$, where $c$ and $d$ are integers. | open2math |
Among all pairs of real numbers $(x, y)$ such that $\cos \cos x = \cos \cos y$ with $-5\pi \le x, y \le 5\pi$, Alex randomly selects a pair $(X, Y)$. Compute the probability that $X = Y$. | open2math |
Let $p$ and $q$ be positive real numbers such that $\sqrt{p} + \sqrt{q} = 5$ and $\frac{1}{\sqrt{p}} + \frac{1}{\sqrt{q}} = 2.$ Find $\frac{p+q}{pq}$. | open2math |
Let $a$, $b$, and $c$ be positive integers such that $\gcd(a,b)=1$ and $\gcd(b,c)=1$. Find the largest possible value of $\gcd(a,c)$, if $ab+bc+ca=1509$. | open2math |
Let $a$, $b$, and $c$ be three positive integers such that $\gcd(a,b)=12$, $\gcd(a,c)=42$, and $a$, $b$, and $c$ are pairwise relatively prime. Determine the smallest possible value of $a+b+c$. | open2math |
Find the sum of the digits of the base $5$ representation of $625_{10}$. | open2math |
In triangle $ABC$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC$, and $AB$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the circumcenter $O$, and that $\frac{AC'}{C'B} \cdot \frac{BA'}{A'C} \cdot \frac{CB'}{B'A} = \frac{11}{13}$, find $\frac{AO}{OC}$. | open2math |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.