Find the number of non-decrease sequences of length $n$ and each element is a non-negative integer not exceeding $d$.

Find, with proof, all pairs of positive integers $(n, d)$ with the following property: for every integer $S$, there exists a unique non-decreasing sequence of n integers $a_1$, $a_2$, $\cdots$, $a_n$ such that $a_1 + a_2 + \cdots + a_n = S$ and $a_n-a_1 = d$.

Let $p$ be an odd prime. Show that $$\sum_{j=0}^p\binom{p}{j}\binom{p+j}{j}\equiv 2^p +1 \pmod{p^2}$$

Let $p$ be a prime and $k$ be a positive integer less than $p$. Show that $\binom{p}{k} \equiv 0 \pmod{p}$.

Let integer $N=\left\lfloor{(\sqrt{29}+\sqrt{21})^{2020}}\right\rfloor$ where $\lfloor{x}\rfloor$ denotes the largest integer not exceeding $x$. Find the last two digits of $N$.

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 $B$?

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?

(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 $p_1 + p_2 +\cdots+ p_{499500}$.

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

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 $n$.