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Find a positive integer $n$ such that the equation
$$\frac{x^2-2x+n}{x^2-4x+n}=x^2$$
has a rational solution. | open2math |
Let $x$ and $y$ be two positive real numbers such that $\ds\sqrt x + \ds\sqrt y =10$ and $xy = 1600$.
Determine $x$ and $y$. | open2math |
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. | open2math |
Find the number of positive and negative integer divisors of 42. | open2math |
Find the smallest positive integer $n$ such that $n$ is divisible by 36 and $n+1$ is divisible by 132. | open2math |
How many positive two-digit integers have an odd number of positive factors? | open2math |
Find the value of $x^3$ given that there exists a real number $x$ such that $\sin (\arccos (x)) = x^2$. | open2math |
The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [asy]
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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... | open2math |
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? | open2math |
Find all real values of $x$ that satisfy $x + \frac{45}{x-4} = -10.$
Enter all the solutions, separated by commas. | open2math |
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. | open2math |
Find the largest constant $C$ so that
\[x^2 + y^2 + 1 \ge C(x + y)\]for all real numbers $x$ and $y.$ | open2math |
Find the smallest value of $c$ such that the range of $f(x)=x^2+3x+c$ includes a positive integer. | open2math |
Determine the product of the roots of the polynomial $x^4-2x^3+5x^2-7x-1=0$ that lie outside the unit circle. | open2math |
Determine the largest positive integer $n$ such that $2^n$ is a divisor of the least common multiple of the numbers $1, 2, 3, \dots, 30$. | open2math |
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. | open2math |
Find the smallest positive integer $n$ such that
$$\frac{\lfloor \sqrt{n} \rfloor}{\sqrt{n}} = \frac{3}{4}.$$
Can you solve it? | open2math |
Find the positive value of $n$ such that the equation $9x^2+nx+1=0$ has exactly one solution in $x$. | open2math |
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. | open2math |
Find $s$ such that $(s,2)$ lies on the line through $(1,6)$ and $(7,-2)$. | open2math |
Determine the sum of the positive divisors of the number $100$ except for $1$ and $100$. | open2math |
Find the value of x such that the sum of the digits of the three-digit number 245x is equal to 17. | open2math |
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 $a^3+b^3+c^3-3abc$. | open2math |
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 $n$? | open2math |
Find the number of positive integers less than 80 that have a prime factorization with an odd number of factors. | open2math |
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. | open2math |
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 $k$. | open2math |
Find the smallest positive integer that is divisible by exactly three distinct prime numbers. | open2math |
Find the maximum value of the expression $\sqrt{x+y+1}+\sqrt{4-x-y}$ for real $x$ and $y$ satisfying $x+y \leq 2$. | open2math |
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 $q/p$? | open2math |
Let $a,$ $b,$ and $c$ be real numbers such that
\[a+b+c=2,\] \[ab+bc+ca=-1,\]and \[abc=-6.\]Compute $(a+b+c)^3+(ab+bc+ca)^3+(abc)^3.$ | open2math |
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 $A+B$. | open2math |
Find the sum of all real solutions to the equation \[\frac{x-2}{x^2+4x+1} = \frac{x-5}{x^2-10x}.\] | open2math |
Find the value of $x$ such that $\sqrt[3]{x^2+27} = 3x-2$. | open2math |
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 $m+n$. | open2math |
Prove that the determinant of the matrix
\[\begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}\]
is equal to \((x+y+z)(x^2+y^2+z^2-xy-xz-yz)\). | open2math |
Determine the value of $a$ such that $a^0$ is equal to the sum of the exponents of the prime factorization of $a$. | open2math |
Given that $f(x)$ is a function such that $f(x+1) - f(x) = f(x^2)$ for all real numbers $x$, find $f(x)$. | open2math |
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 $p(x)$. | open2math |
Given that $a$, $b$, and $c$ are positive integers, simplify the expression
$\frac{a^2b^3c^4}{a^3b^2c^5}$. | open2math |
Given that the polynomial $x^2-5x+t$ has only positive integer roots, find the average of all distinct possible values of $t$. | open2math |
Consider the equation $z^6 + z^4 + z^2 + 1 = 0.$ Determine the smallest positive integer $n$ such that all the solutions of this equation are $n^{\text{th}}$ roots of unity. | open2math |
Let $z$ be a complex number such that the difference between the arguments of $z^3$ and $z$ is $2\pi/3$, and the difference between the arguments of $z^5$ and $z$ is $4\pi/5$. Find all possible values of $z$. | open2math |
Find the smallest positive integer n for which the product of two numbers of the form $2^n+1$ and $2^n-1$ is not a perfect square. | open2math |
Given that $$(a+b+c)(ab+ac+bc)=a^2+b^2+c^2$$and that $$a^3+b^3+c^3=3abc$$for real numbers $a$, $b$, and $c$, what is the value of $a+b+c$? | open2math |
Let $g$ be a function taking the positive integers to the positive integers, such that
(i) $g$ is increasing (i.e. $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m^2 + n^2 = (m + n)^2,$ then $g(m) = n$ or $g(n) = m.$
Find t... | open2math |
Let $g(x)$ be a function defined for all real numbers $x$ such that $g(x) + g(1 - x) = 2$ and $g(x) = 0$ for $x \in [-1, 0]$. Find the smallest positive integer $n$ such that $g\left(\frac{1}{n}\right) = g\left(\frac{n - 1}{n}\right)$. | open2math |
Let $\mathbf{A}$ be a $2 \times 2$ matrix, with real entries, such that $\mathbf{A}^3 = \mathbf{0}.$ Find the number of different possible matrices that $\mathbf{A}^2$ can be. If you think the answer is infinite, then enter "infinite". | open2math |
Find the maximum value of $10^x - 100^x,$ over all real numbers $x.$ | open2math |
Find the area of the rectangle with integer sides such that the
length is three times as large as the width, and the perimeter
of the rectangle is equal to the perimeter of a square with
side length 24 cm. | open2math |
Find the number of perfect squares that can be expressed as the sum of two consecutive prime numbers less than 20. | open2math |
Given that $a$, $b$, and $c$ are nonzero real numbers such that $a^3+b^3+c^3=27abc$, find the value of $\left|\frac{(a+b+c)}{(a-b)(b-c)(c-a)}\right|$. | open2math |
Let $f(x)$ be a real function such that
$f(x+1)=f(x)+f(2x)$ for all $x\neq 0$ and $f\left(\frac{1}{2}\right)=\frac{1}{2}$.
Find $f(10)$. | open2math |
Let $x$ be a positive real number such that the decimal representation of $\sqrt{x}$ starts with the digits 197. Find the smallest possible value of $x$. | 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 |
Find the minimum value of the expression $x^2+y^2-8x-6y+22$ for real $x$ and $y$. | open2math |
Given positive integers $x$ and $y$ such that $x\neq y$ and $\frac{1}{x} + \frac{1}{y} = \frac{1}{12}$, find the largest possible value of $xy$. | open2math |
A game is played with two fair dice, one with numbers 1, 2, 3, 4, 5, and 6, and the other with numbers 0, 2, 4, 6, 8, and 10. A player rolls both dice and calculates the sum of the numbers shown. What is the probability that the sum is greater than the product of the numbers shown? | open2math |
Determine the sum of the cubes of the roots of the equation
$$x^4-7x^3+5x^2+x+1=0,$$
given that the sum of the fourth powers of the roots is 1111. | open2math |
A triangle has three sides of the following side lengths: $8$, $12$, and $y^3$. 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 |
Find all real numbers $x$ such that
\[\frac{a^x + b^x}{c^x + d^x} = \frac{e}{f}.\]
Enter all the solutions, separated by commas.
Note: $a$, $b$, $c$, $d$, $e$, and $f$ are positive real numbers, and $\frac{e}{f}$ is a rational number. | open2math |
In an equilateral triangle $ABC$ with side length 6, $M$ is a point on $BC$ such that $AM=4$ and $MB=3$. Find the area of triangle $ABM$. | open2math |
Find all real numbers $x$ such that the sum of the cubes of the roots of the equation $x^3 + ax + b = 0$ is equal to $16$. Determine $a$ and $b$. | open2math |
Let $a$ be a real number such that the inequality $\left| \frac{x}{x^2 + a} \right| < 1$ has no real solutions. What is the least possible value of $a$? | open2math |
Let $a$, $b$, $c$ and $d$ be positive integers such that
$$\begin{array}{rcl}
a+b+c+d&\equiv&0\pmod{5}\\
a+b-c-d&\equiv&1\pmod{5}\\
a-b+c-d&\equiv&2\pmod{5}\\
-a+b+c-d&\equiv&3\pmod{5}.
\end{array}$$
Determine the smallest possible value of $a+b+c+d$. | open2math |
There are integers $b,c$ for which both roots of the polynomial $x^2-x-1$ are also roots of the polynomial $x^5-bx-c$. Determine the product $bc$. | open2math |
Find the largest value of m such that 7x^2+mx+90 can be factored as the product of two linear factors with integer coefficients. | open2math |
Consider the function $f(x) = \frac{x^3-2x^2-x+2}{x^4+3x^3-4x-12}$. Let $a$ be the number of local maximum points on the graph of $f$, $b$ be the number of local minimum points, $c$ be the number of inflection points, and $d$ be the number of horizontal asymptotes. Find $a+b+2c+3d$. | open2math |
Determine the value of $c$ that satisfies the equation $214_{c}+143_{c}=357_{c}$. | open2math |
The function
$$f(x)=x^4-2ax^3+3x^2+bx-1$$
has two multiple roots. Determine the sum of the possible values of $a$. | 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 |
Consider the set of all polynomials of degree 3 with real coefficients that have all roots equal to -1 or 1 and have the leading coefficient equal to 1.
Find the number of polynomials in this set, which have at least one root equal to -1. | open2math |
Let $a$ be a real number, $a > 0,$ and let $x$ and $y$ be positive real numbers such that
\[x + y + \frac{1}{x} + \frac{1}{y} = a.\]Find $xy.$ | open2math |
Find the value of $x$ in the geometric sequence $x, x^2, 3x^3$, given that the sum of the first two terms is 25. | open2math |
A positive integer is called "balanced" if the product of its digits at even positions (counting from the right) equals the product of its digits at odd positions. Find the number of balanced integers with 5 digits. | open2math |
Two positive integers, $a$ and $b$, are such that the product of their reciprocals plus the sum of their reciprocals is equal to 1. Given that both $a$ and $b$ are less than 10 and relatively prime, determine the product of $a$ and $b$. | open2math |
Find the largest positive integer $k$ such that $k^2$ is a factor of $1225 - 1$. | open2math |
Let $a,$ $b,$ and $c$ be positive integers such that $a+b+c=18$. Find the maximum value of $\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ac}{a+c}$. | open2math |
Determine the sum of all values of $x$ such that $x^3 = -8$. | open2math |
Find the integer $k$ that satisfies $0\le k<29$ and $$-500\equiv k\pmod{29}~?$$ | open2math |
A bookshelf has two types of books: fiction and non-fiction. The number of fiction books, in boxes, is at least 8 boxes more than one-third the number of non-fiction books, and is no more than three times the number of non-fiction books. Find the least number of boxes of non-fiction books that can be placed on the book... | open2math |
The integers $A$ and $B$ are chosen such that
$$\frac{A}{x+2}+\frac{B}{x^2-9x}=\frac{x^2-3x+7}{x^3-7x^2-18x}$$
for all real values of $x$ except $-2$, $0$, and $9$. Find $B/A$. | open2math |
Given a planar, closed curve $C$, define the function $f(P)$ for a point $P$ in the plane as the sum of the distances from $P$ to the two farthest points on $C$. Let $M$ be the smallest possible value of $f(P)$ over all points $P$ in the plane. Prove that if $C$ is a triangle, then $M$ is equal to the perimeter of the ... | open2math |
Let $f(x)$ be a function such that $f(0)=1$ and $f(xy)=f(x)f(y)-2f(x+y)$ for all real numbers $x$ and $y$. Find $f(1992)$. | open2math |
What is the smallest positive integer $n$ such that $n^3-13n+19$ is a prime number? | open2math |
A sequence of five consecutive positive integers has a product that is divisible by 24. What is the smallest possible value of the largest integer in this sequence? | open2math |
Find the sum of all divisors of 240 that are multiples of 3. | open2math |
Given that the point $(4,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=3f(4x)+5$. What is the sum of the coordinates of that point? | open2math |
In the equation $x^2 - 6ax + a^2 = 0$, determine the value of $a$ such that the sum of the squares of the roots is equal to 8.5. | open2math |
Call a positive integer $n$ "doubly-oddball" if there exist positive integers $a$, $b$, and $c$ such that $a^b - c^2 = n$ and both $a$ and $c$ are odd. How many doubly-oddball integers are less than $1000$? | open2math |
How many positive integers are there such that their sum is less than 20 and the sum of their squares is less than 100? | open2math |
The vertices of a cube are labeled with positive integers such that each vertex is adjacent to exactly three others with which it shares a common factor greater than 1. Find the smallest possible sum of the numbers on the vertices adjacent to the vertex labeled 15. | open2math |
Find the number of divisors of 315 that are also perfect squares. | open2math |
Let $a_1, a_2,..., a_n$ be a sequence of positive integers such that the sum of the first $k$ terms is a perfect square for each $k$. If $a_1 = 1$ and $a_2 = 8$, find $a_3$. | open2math |
Given positive integers $a$ and $b$, simplify the expression $\frac{a^{\log_{b} a}}{b^{\log_{a} b}}$. | open2math |
Determine the number of three-digit numbers that are multiples of 5 and have a tens digit that is twice the units digit. | open2math |
The legs $PQ$ and $PR$ of a right triangle $PQR$ have lengths which are positive whole numbers. Given that $PQ = k \cdot PR$ for some positive integer $k$, and that the length of the hypotenuse $QR$ is 15, what are all the possible values of $k$? | open2math |
A rectangle is inscribed in a triangle with base 12 cm and height 8 cm, such that one side of the rectangle lies on the base of the triangle and the other side is parallel to the height. If the area of the rectangle is 24 cm^2, find the perimeter of the rectangle. | open2math |
Find all real numbers $x$ such that $x$ is the geometric mean of the set $\{2x-1, x+1, 3\}$ and $x+2$ is the geometric mean of the set $\{2x+2, x+4, 4\}$. | open2math |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | open2math |
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