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Find the sum of all three-digit positive integers whose cubes end with the digits 125. | open2math |
Consider the function
$$f(x)=\ds\sqrt{\f{x-2}{x^2+3x-10}}$$
defined on the real line wherever it takes real values. Find the set of all values that $f(x)$ can possibly attain. | open2math |
Given a positive integer $n$, find the sum of the squares of all positive integer divisors of $n$ if the sum of the divisors of $n$ is 78 and the sum of the divisors of $n$ excluding 1 and $n$ is 54. | open2math |
Let $x$ and $y$ be positive real numbers such that $x + y = 2$. Find the set of all possible values of $\frac{1}{x} + \frac{1}{y}$. | open2math |
Let p and q be two positive numbers such that pq=8/5.
Find 25p2q2. | open2math |
Find the least positive integer that has exactly three different prime factors and is divisible by 4. | open2math |
Let $f(x)$ be a polynomial with positive integer coefficients and degree 4 such that for any positive integer $n$, $f(n)$ can be written as a sum of distinct powers of $n$. Let $g(x)$ be the polynomial obtained by reversing the coefficients of $f(x)$. Find the sum of all positive integers $k$ for which $g(k)$ is also a... | open2math |
Find the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A(0, 3)$ and $B(5, 0)$. | open2math |
Find the value of $x$ that minimizes the expression $x^2 - 2xy + 3y^2$ given that $x$ and $y$ are positive integers. | open2math |
The following system of equations has a solution:
\begin{align*}
x+2y+3z&=6,\\
4x+5y+6z&=16,\\
7x+8y+9z&=a.
\end{align*}
Determine $a$ such that the system of equations has infinitely many solutions. | open2math |
Let's consider a unit circle $C$ centered at point $O$. Three points are chosen uniformly at random on the circle $C$. Then a point $P$ is chosen uniformly at random inside the circle $C$. What is the probability that the area of the triangle formed by $P$ and the two closest points on the circle is greater than $0.25$... | open2math |
How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1\le x\le 4$ and $1\le y\le 4$?
$\text{(A) } 496\quad \text{(B) } 500\quad \text{(C) } 512\quad \text{(D) } 516\quad \text{(E) } 560$
| open2math |
Find the value of $x$ if $x$ is positive and $\lfloor x\rfloor+x=15$.
In this problem, we need to find a positive value of $x$ such that the sum of $x$ and its floor value is equal to 15. This requires a different approach than the original problem, as we need to consider the properties of the floor function and how i... | open2math |
Find the number of lattice points that satisfy the following system of inequalities:
\[
\begin{array}{cccc}
(1) & 2x+y & > & 5 \\
(2) & x-2y & \leq & -3 \\
(3) & 3x+y & < & 11
\end{array}
\]
Graph these inequalities on a coordinate plane and find the number of lattice points (points with integer coordinates) within t... | open2math |
Find the range of values of $y$ that satisfy the equation \[\sqrt{2y-1}\cdot\sqrt{y+2}\cdot\sqrt{3y+1}=3.\] | open2math |
Find the sum of all positive integers less than 1000 that are cubes of perfect cubes minus the sum of all positive integers less than 1000 that are squares of perfect squares. | open2math |
Determine the number of integer solutions $(x, y)$ of the equation $xy-3x+4y=8$ such that $-10\leq x\leq 10$ and $-10\leq y\leq 10$. | open2math |
Find the sum of all values of $y$ such that $\sqrt{y^3+3y+4}=|y-3|$. | open2math |
For any finite sequence $A = (a_1, a_2, \ldots, a_n)$ of real numbers, define $S(A)$ to be the sum of all the elements of $A$, and $D(A)$ to be the sum of the squares of the elements of $A$. Find the smallest real number $k$ such that the inequality \[D(A) + D(B) \geq k(S(A) - S(B))^2\]holds for all pairs $(A, B)$ of s... | open2math |
There are four positive integers that are divisors of each number in the list $$36, 72, -12, 114, 96.$$Find the sum of these four positive integers.
Write another problem inspired by this one:
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | open2math |
Let $x, y, z$, and $w$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}x^2+y^2&=&z^2+w^2&=&500,\\ xy&=&zw&=&200.\end{array}\]
If $T=x+y+z+w$, determine the value of $\lceil T \rceil$. | open2math |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the maximum value of $x^3 y^2 z.$ | open2math |
Find the smallest positive integer that has exactly 5 positive integer factors, including 1 and itself, and is also a perfect square. | open2math |
If $x$ is a positive number such that \[\sqrt{8x}\cdot\sqrt{10x}\cdot\sqrt{3x}\cdot\sqrt{15x}=15,\]find all possible values for $x$. | open2math |
Find the number of positive integers less than 500 that are divisible by exactly three of the prime numbers 2, 3, 5, or 7. | open2math |
A game show contestant chooses three distinct numbers at random from the set {1, 2,..., 6}. What is the probability that the sum of the three numbers is an even number? | open2math |
Let $a$ and $b$ be integers such that $0<a<b$.
If $\cos(a \cdot \theta)=\cos(b \cdot \theta)$ for every angle $\theta$,
determine the minimum value of $b-a$ (expressed as a positive integer). | open2math |
Let $m$ be a positive integer. We define the function $\phi(m)$ to be the minimum positive integer $n$, if it exists, such that for all positive integers $a$ there exists an integer $b$ such that $ab \equiv 1 \pmod{n}$ implies $ab \equiv 1 \pmod{m}$. Otherwise we let $\phi(m) = -1$.
Prove that for every positive integ... | open2math |
The integers $G$ and $H$ are chosen such that
\[\frac{G}{x+5}+\frac{H}{x^2-4x}=\frac{x^2-2x+10}{x^3+x^2-20x}\]for all real values of $x$ except $-5$, $0$, and $4$. Find $H/G$. | open2math |
On a rectangular grid, a square is drawn so that its vertices coincide with the grid points. If two opposite vertices are located at points (5,7) and (11,13), find the side length of the square. | open2math |
Find the largest integer $n$ such that $n^2+1$ is a factor of $1156$ and $n$ is a divisor of $34$. | open2math |
In a regular octahedron, the midpoints of the 12 edges form the vertices of a smaller octahedron. The ratio of the surface area of the smaller octahedron to that of the larger is $a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. | open2math |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 100.$ Find the minimum value of $\frac{x}{y} + \frac{y}{x}.$ | open2math |
If $a$ and $b$ are positive integers such that $a^2-b^2=45$, what is the value of $a-b$? | open2math |
The quadratic equation $ax^2+bx+c=0$ has two distinct real roots. The equation $Ax^2+Bx+C=0$ is formed by reversing the coefficients of the original equation (i.e., $A=c, B=b, C=a$). Determine the sum of the values of $a$ for which the two equations have the same sum of roots. | open2math |
Count the number of ways to paint a 4x4 square grid with 4 different colors, such that no two adjacent squares have the same color. | open2math |
Find the product of all values of x for which the expression $\frac{x^2+3x-2}{x^2+5x-6}$ is not a real number. | open2math |
If $a$,$b$, and $c$ are positive real numbers such that $a(b+c) = 152$, $b(c+a) = 162$, and $c(a+b) = 170$, then find $abc.$ | open2math |
Find the value of $a$ so that the lines with the given equations are perpendicular. \begin{align*}
y &= 2x+13 \\
6y+ax &= 6.
\end{align*} | open2math |
Two fair, eight-sided dice are rolled. What is the probability that the sum of the two numbers showing is greater than 9? | open2math |
A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles? | open2math |
Determine the number of positive divisors of 3150. | open2math |
The decimal expansion of the quotient of two natural numbers is $0.\overline{ab}$. Prove that $ab$ is divisible by 99. | open2math |
Find all real values of x such that the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n(x+n)}$ converges. | open2math |
For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \le 50$.
(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions? | open2math |
What is the largest positive integer that has exactly four distinct positive factors and is a perfect square? | open2math |
Let $p(x)$ be a polynomial with integer coefficients such that $p(2007)=2007$. For how many values of $n$ is $n+p(n)+p(p(n))=2007^2$? | open2math |
Consider the function $f(x) = \frac{x^2+3x+2}{x^2+4x+4}$. Determine the sum of all values of $x$ for which $f(x)$ is undefined. | open2math |
In the construction of a rectangular solid, three identical spheres are placed on the vertices of the base so that they touch each other and the edges of the base. Find the volume of the solid in terms of the diameter of one of the spheres. | open2math |
Determine $k$ such that the equation $2x^2+kx+\frac{1}{2}=0$ has roots $x_1$ and $x_2$ satisfying the condition $x_1x_2=x_1+x_2$. | open2math |
What is the smallest whole number that must be a factor of the sum of any four consecutive positive integers? | open2math |
Find the smallest positive integer that is divisible by three distinct prime powers, where each prime power has an exponent greater than 1. | open2math |
The positive divisors of a positive integer of the form $2^i3^j$ can be paired in such a way that the sum of the divisors in each pair is a power of $2$ times a power of $3$. If the number of pairs that can be formed is $50$, what is $i + j$? | open2math |
Let $a$, $b$, $c$ be positive integers such that $a-b-c=b+c-a=(a+c)(a-b)$. Find the largest possible value of $c$ divided by $b$. | open2math |
In the triangle ABC, the lengths of the sides AB, BC, and CA are $x^2+y$, $y^2+z$, and $z^2+x$ units, respectively. Given that $x$, $y$, and $z$ are positive integers, find the minimum value of the perimeter of the triangle. | open2math |
Determine the value of $y$ such that the five-digit number $\underline{y24y2}$ is divisible by 24. | open2math |
Given that the point $(2,5)$ is on the graph of $y=g(x)$, there is one point that must be on the graph of $3y=2g(5x)-1$. What is the product of the coordinates of that point? | open2math |
Let $S$ be a set of all real numbers $x$ for which there exists a real number $y$ such that $\tan (x+y)=y$. Prove that $S$ contains a non-degenerate closed interval.
\begin{align*}
\end{align*} | open2math |
How many positive integers less than 100 can be expressed as the sum of two distinct perfect cubes? | open2math |
Let $a,$ $b,$ and $c$ be positive integers such that $abc = 720.$ Find the minimum value of $a + b + c.$ | 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 $ S $ be the set of all sides and diagonals of a regular pentagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? | open2math |
Find all possible real values of $a$ and $b$ such that the function $f(x) = ax^3 + bx^2 + 1$ satisfies the equation $f(x) + f(-x) = f(x^2)$. | open2math |
Determine the number of different arrangements of the digits in the decimal representation of 3326 that are divisible by 2. | open2math |
Determine the value of $x$ such that $(\frac{1}{2})^{\frac{x}{7}} \cdot 8^{\frac{1}{5}} = \frac{1}{16}$ | open2math |
Find the sum of all integers from 25 through 35, inclusive. | open2math |
Determine the largest positive integer $n$ such that $3^n$ divides $2^{2013}-1$. | open2math |
Determine the largest prime factor of the number obtained by multiplying all the divisors of 10080 that are not multiples of 7. | open2math |
Find the smallest integer n such that the sum of the digits of n^2 is equal to the sum of the digits of n.
In other words, we are looking for the smallest integer n such that the sum of the digits of n squared is equal to the sum of the digits of n itself.
For example, if n = 12, then n^2 = 144, and the sum of the di... | open2math |
Find the product of all integer values of $n$ such that $\frac{2n-1}{n+2}$ is a rational number.
Note: This problem requires a different approach to solve, as it involves rational numbers and a different expression. The user needs to analyze the expression and find the values of $n$ that make it a rational number. | open2math |
The vertices of a parallelogram $ABCD$ are $A(2,1)$, $B(4,3)$, $C(6,5)$ and $D(x,y)$. Given that the diagonals of the parallelogram bisect each other, find the value of $xy$. | open2math |
Determine the number of positive divisors of the number 30. | open2math |
Let $P(x)$ be a polynomial with integer coefficients such that $P(1) = 60$ and $P(x)$ is divisible by $(x-1)^2.$ Find the number of divisors of $P(1)$ that are not divisible by 3. | open2math |
Find all the real values of $x$ that satisfy the equation $f(x)=3$, given that $f(x)$ is a function that satisfies the following conditions:
- $f(x)$ is a continuous and differentiable function for all real values of $x$.
- $f(x)$ is a strictly increasing function for all real values of $x$.
- $f(2)=0$ and $f(4)=4$.
-... | open2math |
Determine the minimum value of the expression $|x^2-6x+13|+|x^2-8x+25|$ for real values of $x$. | open2math |
Find the least positive integer that can be expressed as a sum of two perfect squares in two different ways. | open2math |
Find the value of the expression $(\cos \theta + i \sin \theta)^n$ where $\theta$ is an acute angle such that $\cos \theta = \frac{3}{5}$ and $n$ is a positive integer such that $n\theta = 180^\circ$. | open2math |
Find the sum of the values of $x$ that satisfy the equation $|x-3|+|x-7|=9$. | open2math |
The vertices of a triangle are $(-2,3)$, $(5,-1)$ and $(k,7)$. Determine the value of $k$ such that the perimeter of the triangle is equal to 20 units. | open2math |
Find the smallest positive integer such that when its rightmost digit is deleted, the resulting integer is 1/13 of the original integer. | open2math |
Let $a$ and $b$ be two positive real numbers such that $a + b = 18.$ Enter the ordered pair $(a,b)$ for which $a^3 b^4$ is maximized. | open2math |
A root of the equation $x^3-21x-35=0$ is equal to $x=a+\frac{b}{\sqrt{c}}+\frac{d}{\sqrt{c^2}}$, where $a$, $b$, $c$, $d$ are positive integers. Find $a+b+c+d$. | open2math |
Let $g(x)$ be a function that satisfies $g(x)+g(y)=g(x+y)$ for all real numbers $x$ and $y$, and $g(1)=5$. Find $g(-3)$. | open2math |
Let $A_n$ be a sequence of integers such that each term is a $n$-digit integer that starts with the digit $a$ and ends with the digit $b$. The middle digits of the term can be any digits from $0$ to $9$. Let $B_n$ be a sequence of integers such that each term is a $2n$-digit integer that starts with the digit $c$ and e... | open2math |
Find the sum of the first ten terms of an arithmetic sequence, given that the sum of the first three terms is 15 and the sum of the last three terms is 105. | open2math |
Determine all real numbers $k$ such that a triangle can be constructed in the coordinate plane whose vertices are the $x$-intercept of the line $y=2x+k$, the $y$-intercept of the line $y=-x+2$, and a point $(p, q)$ lying on the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$. | open2math |
In the parallelogram $ABCD$, $A$ is at the origin, $B$ is at $(3,4)$, and $C$ is at $(2,5)$. Find the distance between the point of intersection of the diagonals of the parallelogram and one of the vertices. | open2math |
Find the number of positive integers $n$ such that $2^n$ is a divisor of $10!$ but $3^n$ is not. | open2math |
Find the number of positive integers $n$ such that the sum of the cubes of the first $n$ natural numbers is divisible by the sum of the squares of the first $n$ natural numbers. | open2math |
Let $a$ and $b$ be positive integers such that $b>1$ and $b$ is not divisible by $3$. Find the largest positive integer $k$ that divides $a^{b-1} + a^{b-2} + \ldots + a^2 + a + 1$ for any choice of $a$. | open2math |
Given a right triangle $ABC$ with legs of length 8 and 10, and hypotenuse of length 12, find the radius of the inscribed circle that is tangent to all three sides of the triangle. | open2math |
Find the greatest integer $n$ such that $n^2+3n-5$ is a factor of $n^4+2n^3-7n^2-12n+9$.
Determine $n$ by using the method of polynomial division and remainder theorem. | open2math |
The positive integers $A,$ $B,$ $C,$ and $D$ are all distinct and have the property that $A+B,$ $A+C,$ $A+D,$ $B+C,$ $B+D,$ and $C+D$ are all prime numbers. The sum of the six primes is
$\bullet$ A. even
$\bullet$ B. divisible by $3$
$\bullet$ C. divisible by $5$
$\bullet$ D. divisible by $7$
$\bullet$ E. prime | open2math |
Determine the number of complex numbers $z$ that satisfy the conditions $|z|=2$ and $|z^2-\overline{z}^2|=4$. | open2math |
Find the number of solutions of the equation $|x+5|+|x-2|=7$ and determine the distance between the two extreme solutions. | open2math |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 1.$ Find the maximum value of
\[x + y + z.\] | open2math |
Find the sum of the squares of the solutions of $x^2-13x+4=0$. | open2math |
Determine $k$ such that $\log_{3}7+\log_{3}k=1$. | open2math |
A regular decagon can be split into ten congruent sectors of equal area. A certain linear function maps the vertices of the decagon onto the center of one of the sectors. Compute the sum of the images of all the vertices, assuming the origin lies at the center of the decagon. | open2math |
Find the value of $x$ such that $(\sqrt[3]{2})^{4x}=(\sqrt[4]{2})^{3x}$. | open2math |
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