Let $a, b, c, d, e$ be distinct positive integers such that $a^4 + b^4 = c^4 + d^4 = e^5$. Show that $ac + bd$ is a composite number.

There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.

Let $S$ be the set of all ordered triple of integers $(a_1,a_2,a_3)$ with $1 \le a_1,a_2,a_3 \le 10$. Each ordered triple in $S$ generates a sequence according to the rule $a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |$ for all $n \ge 4$. Find the number of such sequences for which $a_n=0$ for some $n$.

Let $f(x)$ be a third-degree polynomial with real coefficients satisfying $$|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.$$ Find $|f(0)|$.

For each integer $n \ge 2$, let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\le x \le n$ and $0\le y \le x \left\lfloor \sqrt x \right\rfloor$, where $\left\lfloor \sqrt x \right\rfloor$ is the greatest integer not exceeding $\sqrt x$. Find the number of values of $n$ with $2\le n \le 1000$ for which $A(n)$ is an integer.

Steve says to Jon, 'I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $$P(x) = 2x^3-2ax^2+(a^2-81)x-c$$

for some positive integers $a$ and $c$. Can you tell me the values of $a$ and $c$?' After some calculations, Jon says, 'There is more than one such polynomial.' Steve says, 'You're right. Here is the value of $a$.' He writes down a positive integer and asks, 'Can you tell me the value of $c$?' Jon says, 'There are still two possible values of $c$.' Find the sum of the two possible values of $c$.

Let $x$ and $y$ be real numbers satisfying $x^4y^5+y^4x^5=810$ and $x^3y^6+y^3x^6=945$. Evaluate $2x^3+(xy)^3+2y^3$.

Let $x_1< x_2 < x_3$ be the three real roots of the equation $\sqrt{2014} x^3 - 4029x^2 + 2 = 0$. Find $x_2(x_1+x_3)$.

Let $m$ be the largest real solution to the equation$$\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}=x^2-11x-4$$There are positive integers $a$, $b$, and $c$ such that $m=a+\sqrt{b+\sqrt{c}}$. Find $a+b+c$.

The repeating decimals $0.abab\overline{ab}$ and $0.abcabc\overline{abc}$ satisfy
\[0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},\]
where $a$, $b$, and $c$ are (not necessarily distinct) digits. Find the three digit number $abc$.

Let $f(x)=(x^2+3x+2)^{cos(\pi x)}$. Find the sum of all positive integers $n$ for which \[\left |\sum_{k=1}^nlog_{10}f(k)\right|=1.\]

Let $f(x) = x^4 + ax^3 + bx^2 + cx + d$. If $f(-1) = -1$, $f(2)=-4$, $f(-3) = -9$, and $f(4) = -16$. Find $f(1)$.

Solve in positive integers $x^2 - 4xy + 5y^2 = 169$.

Solve in integers the question $x+y=x^2 -xy + y^2$.

Solve in integers $\frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$

Prove the product of 4 consecutive positive integers is a perfect square plus 1.

For any arithmetic sequence whose terms are all positive integers, show that if one term is a perfect square, this sequence must have infinite number of terms which are perfect squares.

Prove there exist infinite number of positive integer $a$ such that for any positive integer $n$, $n^4 + a$ is not a prime number.

Find all positive integer $n$ such that $(3^{2n+1} -2^{2n+1}- 6^n)$ is a composite number.

The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.

The domain of the function $f(x) = \arcsin(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the remainder when the smallest possible sum $m+n$ is divided by 1000.

Positive integers $a$ and $b$ satisfy the condition \[\log_2(\log_{2^a}(\log_{2^b}(2^{1000}))) = 0.\] Find the sum of all possible values of $a+b$.

A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$-th centimeter. Suppose it takes $T$ seconds for the candle to burn down completely. Then $\tfrac{T}{2}$ seconds after it is lit, the candle's height in centimeters will be $h$. Find $10h$.

Let $A = \{1, 2, 3, 4, 5, 6, 7\}$, and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$.

The terms of an arithmetic sequence add to $715$. The first term of the sequence is increased by $1$, the second term is increased by $3$, the third term is increased by $5$, and in general, the $k$th term is increased by the $k$th odd positive integer. The terms of the new sequence add to $836$. Find the sum of the first, last, and middle terms of the original sequence.

Algabra Polynomial Equation Root VietaTheorem AlgebraFundamentalTheorem Delta RationalRootTheorem SpecialEquation CeilingAndFloor AbsoluteValue ExpAndLog Sequence BasicSequence LinearRecursion TelescopingSeries InfiniteRepitition SpecialSequence Function FunctionProperty FunctionEquation AlgebraMisc DifferentBase LogAndExp