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

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 $n$?

Show that $$\frac{1}{(1-x)^n}=\sum_{k=0}^{\infty}\binom{n-1+k}{n-1}x^k$$

Let $n$ be an odd positive integer, and $$N=6^n + \binom{n}{1}\cdot 6^{n-1}+\cdots + \binom{n}{n-1}\cdot 6-1$$

Find the remainder when $N$ is being divided by $8$.

Assuming that $$(1-2x)^7=a_0 + a_1x+a_2x^2+\cdots+a_7x^7$$

Find the value of

• $S_1 = a_1+a_2+\cdots + a_7$
• $S_2 = a_1+a_3+a_5+a_7$
• $S_3 = a_0+a_2+a_4+a_6$
• $S_4 = \mid a_0\mid + \mid a_1\mid +\cdots + \mid a_7\mid$

Let $m$ and $n$ be positive integers satisfying $1 < m < n$. Show that $(1+m)^n > (1+n)^m$.

Show that the following relation holds for any positive integers $1 < k \le m < n$: $$\binom{n}{k}m^k > \binom{m}{k}n^k$$

Let $\{a_n\}$ be a geometric sequence whose initial term is $a_1$ and common ratio is $q$. Show that $$a_1\binom{n}{0}-a_2\binom{n}{1}+a_3\binom{n}{2}-a_4\binom{n}{3}+\cdots+(-1)^na_{n+1}\binom{n}{n}=a_1(1-q)^n$$

where $n$ is a positive integer.

Find the constant term in the expansion of $\left(\frac{x}{2}+\frac{1}{x}+\sqrt{2}\right)^5$.

Let $n$ be a positive integer and the coefficient of the $x^3$ term in the expansion of $(1+\frac{x}{n})^n$ be $\frac{1}{16}$. Find $n$.