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Find the number of positive integers less than or equal to $24$ that are relatively prime to $24$. | open2math |
Suppose that $a$ and $b$ are integers such that $$3b = 8 - 2a.$$How many of the first six positive integers must be divisors of $2b + 12$? | open2math |
How many different integers can be expressed as the sum of two or more consecutive positive integers? | open2math |
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. | open2math |
Find the interval of all $x$ such that both $5x$ and $7x$ are in the interval $(3,4)$. | open2math |
Determine the value of $a$ such that the expression $ax^2+4x+a^2$ has a minimum value of $-1$ for real $x$. | open2math |
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 $f(-1).$ | open2math |
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 $p$. | open2math |
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 $\frac{1}{a+bi}$. | open2math |
Let $n$ be an odd integer with exactly 11 positive divisors. Find the number of positive divisors of $8n^3$. | open2math |
Determine a positive integer $M$ with three or fewer digits such that the first three digits of $M^2$ are 123. | open2math |
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. | open2math |
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. | open2math |
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 $xyz$. | open2math |
Let $a, b,$ and $c$ be positive real numbers such that $a + b + c = 1.$ Find set of all possible values of $\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}.$ | open2math |
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. | open2math |
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. | open2math |
Find the number of positive integers less than 1000 that are not divisible by any of the prime numbers 2, 3, or 5. | open2math |
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. | open2math |
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. | open2math |
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? | open2math |
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 $(a,b)$. | open2math |
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. | open2math |
Let $a$ and $b$ be positive real numbers such that $a+b = \sqrt{16+a^2+b^2}$. Find the product $ab$. | open2math |
When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \neq 0$$ | open2math |
Find the sum of the values of $x$ that satisfy the equation $\sqrt{x}+\sqrt{x+1}=3$. | open2math |
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? | open2math |
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 $S$. | open2math |
What is the sum of all positive integer divisors of 77? | open2math |
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 $ABC$. | open2math |
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 $m+n$? | open2math |
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... | open2math |
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. | open2math |
Find the smallest digit $B$ in the five-digit integer $27B,\!418$ such that the entire number is divisible by 9. | open2math |
Find the largest positive integer $n$ such that $2n \equiv 578 \pmod{19}.$ | open2math |
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. | open2math |
Find the number of positive integers less than or equal to 30 that are not divisible by any perfect square other than 1. | open2math |
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 $a-b$. | open2math |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}^n = \mathbf{I}.\] | open2math |
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. | open2math |
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... | open2math |
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 $x$. | open2math |
A function $g$ has the property that $g(2x^2+1)=\frac{1}{x^2+1}$ for all real numbers $x$. What is $g(5)$? | open2math |
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. | open2math |
Find the positive value of x that satisfies the equation
$$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+x}}}}= \sqrt{x}+1$$ | 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 |
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. | open2math |
Find the number of positive integer divisors of $M$ if $M = 3^4 \cdot 5^3 \cdot 7^2 \cdot 11^2$. | open2math |
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. | open2math |
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? | open2math |
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. | open2math |
Find the sum of all positive integers less than 15 that are not multiples of 3 or 5. | open2math |
Find the largest value of $c$ such that the function $f(x) = x^3 - 2x^2 + cx + 1$ has at least one real root. | open2math |
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 $12^{12}$. | open2math |
What is the least positive integer $n$ such that $4125$ is a factor of $n!$? | open2math |
Compute the value of the infinite series \[
\sum_{n=3}^{\infty} \frac{n^5+2n^3+12n^2+15n+12}{3^n \cdot \left(n^5+8\right)}
\] | open2math |
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. | open2math |
Find the value of $y$ such that the three-digit number $\underline{2y6}$ is divisible by 9. | open2math |
Determine all real values of $x$ such that $\log_3(x-1) + \log_3(x-2) = \log_3(2x)$. | open2math |
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. | open2math |
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. | open2math |
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 $S$? | open2math |
Determine the smallest integer value of $y$ such that the solution set of the inequality $3-2y<11$ contains at least one integer. | open2math |
Find the number of positive integers $m \le 2000$ such that $24m$ is a perfect cube. | open2math |
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? | open2math |
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. | open2math |
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? | open2math |
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. | open2math |
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? | open2math |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 10.$ Find the minimum value of
\[\frac{1}{a} + \frac{1}{b} + \frac{1}{c}.\] | open2math |
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 $\alpha \beta \gamma$. | open2math |
Let $x$ and $y$ be positive real numbers such that $x + y = 1$. Find the minimum value of $x^2 + y^2 + \frac{1}{x^2} + \frac{1}{y^2}$. | open2math |
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 $n-m$? | open2math |
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 $y.$ | open2math |
What is the largest positive integer $n$ such that $n!$ has exactly 3 different prime factors? | open2math |
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 $m+n+p$. | open2math |
Find the sum of all integers $k$ such that $\binom{23}{4} + \binom{23}{5} = \binom{24}{k}$. | open2math |
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. | open2math |
Find the value of $x$ that maximizes the expression $\frac{x^2+15x-54}{x^2-6x+1}$. | open2math |
Find the least positive integer $n$ such that the product of its positive whole number divisors is greater than 1000. | open2math |
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. | open2math |
Count the number of positive integers less than 1000 that are multiples of 9 and have exactly two digits equal to 7. | open2math |
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 $C+D$. | open2math |
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 $p>10$. | open2math |
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. | open2math |
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 $n$? | open2math |
Given the convex quadrilateral $ABCD$, let $O$ be its circumcenter and $r$ be the radius of the circumcircle.
Prove that
\[|OA|\cdot |OB|\cdot |OC|\cdot |OD| \geq 16 r^4.\] | open2math |
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. | open2math |
Compute the positive integer $x$ such that $7^{3x+1}$ has exactly 5 digits. | open2math |
Compute the number of 3-digit positive integers divisible by 7. | open2math |
Find the sum of all the perfect squares between 10 and 100 that are also multiples of 4. | open2math |
If $a$ and $b$ are integers such that $a>1$ and $b>1$, compute $a\left(\frac{1}{b}\right)^4$, given that $a^2 + 2b^2 = 15$. | open2math |
How many three-digit positive integers have the property that the product of their digits is equal to 36? | open2math |
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{(... | open2math |
Find the number of points with integer coordinates that satisfy the inequality $|x^2-3x+2|+|y^2-3y+2|\le 1$. | open2math |
Find the sum of all the positive integers less than or equal to 100 that have exactly three distinct factors. | open2math |
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. | open2math |
Find the maximum value of
\[|x| + |y| + |x - y|\]
over all real numbers $x$ and $y$ such that $x^2 + y^2 \leq 1.$ | open2math |
What is the sum of the squares of all the positive integer divisors of 24 that are also perfect squares? | open2math |
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 $\gamma_s(x)$. | open2math |
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