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

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 $m$.

 Acute scalene triangle $\triangle{ABC}$ has circumcenter $O$ and orthocenter $H$. Points $X$ and $Y$, distinct from $B$ and $C$, lie on the circumcircle of $\triangle{ABC}$ such that $\angle{BXH} = \angle{CYH} = 90^{\circ}$ . Show that if lines $XY$, $AH$, and $BC$ are concurrent, then $OH$ is parallel to $BC$.

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 $ABCD$.

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

Compute the value of $$\sum_{k=0}^{n}(-1)^k\frac{1}{k+1}\binom{n}{k}=\binom{n}{0}-\frac{1}{2}\binom{n}{1}+\frac{1}{3}\binom{n}{2} -\cdots+ (-1)^n\frac{1}{n+1}\binom{n}{n}$$

Show that $$\displaystyle\sum_{k=0}^{n}\binom{2n}{k} = 2^{2n-1}+\frac{1}{2}\binom{2n}{n}$$

Let $m$ and $n$ be two positive integers satisfying $m\le n$. Find the value of $$S_{m,n} = \displaystyle\sum_{k=0}^{m}(-1)^k\binom{m}{n}$$

Let $m$ and $n$ be two positive integers satisfying $m < n$. Show that $$S_{m,n}=\sum_{k=m}^{n}(-1)^k\binom{n}{k}\binom{k}{m}=0$$

Let $n$ be a positive integer. Show that $$\sum_{k=0}^{n}k^2\binom{n}{k}^2=n^2\binom{2n-2}{n-1}$$

Evaluate the value of $$\sum_{m=0}^{2009}\sum_{n=0}^{m}\binom{2009}{m}\binom{m}{n}$$

Find the sum of all $n$ such that $$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-\binom{n}{3}+\cdots +\binom{n}{2018} = 0$$

Let $\mathbb{S}$ be a set of integers, $\max(\mathbb{S})$ be the largest element in $\mathbb{S}$, and $\mid\mathbb{S}\mid$ be the number of elements in $\mathbb{S}$. Find the number of non-empty set $\mathbb{S}\in\{1,2,\cdots,10\}$ satisfying $\max(\mathbb{S})\le\mid\mathbb{S}\mid + 2$.

Let $m$ and $n$ be positive integers. Show that $$\frac{(m+n)!}{(m+n)^{m+n}}<\frac{m!}{m^m}\frac{n!}{n^n}$$

Show that $$\frac{1}{1-x}=1+x+x^2+x^3+x^4 + \cdots$$

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