Show that the number $(2n^{3k}+4n^{k}+10)$ cannot be a product of consecutive integers for any positive integers $n$ and $k$.

Let $a$, $b$, and $x_0$ all be positive integers. Sequence $\{x_n\}$ is defined as $x_{n+1}=ax_n + b$ where $n \ge 1$. Show that $x_1$, $x_2$, $\cdots$ cannot be all prime.

Determine all positive integer $n$ such that the following equation is solvable in integers: $$x^n + (2+x)^n + (2-x)^n = 0$$

Let integers $l > m > n$ be the side lengths of a triangle satisfying $\left\{\frac{3^l}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$ where function $\{x\}$ returns the decimal part of real number $x$. Find the least possible value of this triangle's perimeter.

Let $n$ be an integer which is divisible by neither $2$ nor $5$. Show that $n$ must be divisible by a number whose digits are all $1$.

Let $N=4568^{7777}$, $a$ be the sum of digits in $N$, $b$ be the sum of digits in $a$, and $c$ be the sum of digits in $b$. Find $c$.

Show that for any integer $x$, the number $\left(\frac{x^5}{5}+\frac{x^3}{3}+\frac{7x}{15}\right)$ is an integer.

In the following $(8\times 5)$ grid, how many shortest routes are there from point $A$ to point $B$?

In the following $5\times 4\times 3$ grid system, how many shortest routes are there from point $A$ to point $B$?

There are $5$ red balls and $4$ green balls in a bag. One ball is retrieved a time until all the balls are taken out. How many possible ways are there such that all the red balls are taken out before all the green balls are taken out?

There are $5$ red balls, $4$ green balls and $3$ yellow balls in a bag. One ball is retrieved a time until all the balls are taken out. What is the probability that all the red balls are retrieved before all the green or yellow balls are retrieved?

Let $m$ and $n$ be positive integers, $m$ be odd, and $(m, 2^{n} - 1)=1$. Show that $\displaystyle\sum_{k=1}^{m}k^n$ is a multiple of $m$.

Find, with proof, all ordered pairs of positive integers $(a, b)$ with the following property: there exist positive integers $r$, $s$, and $t$ such that for all $n$ for which both sides are defined, $$\binom{\binom{n}{a}}{b}=r\binom{n+s}{t}$$

Let $O$ and $H$ be the circumcenter and orthocenter of $\triangle{ABC}$ respectively. Show that $OH\parallel BC$ if and only if $\tan{B}\tan{C}=3$.

(Vandermonde's Identity) Show that $$\displaystyle\sum_{k=0}^r\binom{m}{k}\binom{n}{r-k}=\binom{m+n}{r}$$

(Generalized Vandermonde's Identity) Show that $$\sum_{k_1+\cdots+k_p=m}\binom{n_1}{k_1}\binom{n_2}{k_2}\cdots\binom{n_p}{k_p}=\binom{n_1 + \cdots + n_p}{m}$$

Find the number of ordered quadruples of integer $(a, b, c, d)$ satisfying $1\le a < b < c < d \le 10$.

How many different strings of length $10$ which contains only letter $A$ or $B$ contains no two consecutive $A$s are there?

Let $N$ be the number of possible ways to pick up two adjacent squares in a $(n\times m)$ grid. Find $N$.

Lizzie writes a list of fractions as follows. First, she writes $\frac{1}{1}$ , the only fraction whose numerator and denominator add to $2$. Then she writes the two fractions whose numerator and denominator add to $3$, in increasing order of denominator. Then she writes the three fractions whose numerator and denominator sum to 4 in increasing order of denominator. She continues in this way until she has written all the fractions whose numerator and denominator sum to at most $1000$. So Lizzie’s list looks like: $$\frac{1}{1}, \frac{2}{1} , \frac{1}{2} , \frac{3}{1} , \frac{2}{2}, \frac{1}{3}, \frac{4}{1}, \frac{3}{2} , \frac{2}{3}, \frac{1}{4} ,\cdots, \frac{1}{999}$$

Let $p_k$ be the product of the first $k$ fractions in Lizzie’s list. Find, with proof, the value of $p_1 + p_2 +\cdots+ p_{499500}$.

Cyclic quadrilateral $ABCD$ has $AC\perp BD$, $AB + CD = 12$, and $BC + AD = 13$. Find the greatest possible area for $ABCD$.

$\textbf{Eel}$

An eel is a polyomino formed by a path of unit squares that makes two turns in opposite directions. (Note that this means the smallest eel has four cells.) For example, the polyomino shown below is an eel. What is the maximum area of a $1000\times 1000$ grid of unit squares that can be covered by eels without overlap? 

Let $n$ and $k$ be two positive integers. Show that $$\frac{1}{\binom{n}{k}}=\frac{k}{k-1}\left(\frac{1}{\binom{n-1}{k-1}}-\frac{1}{\binom{n}{k-1}}\right)$$

Let $f(x)=a_0+a_1x+a_2x^2+\cdots +a_nx^n$ be a $n$-degree polynomial and all its coefficients $a_i$ $(0\le i\le n)$ be either $1$ or $-1$. If $f(x)$ has only real roots, what is the maximum value of $n$?