Given $$P(x)=(1+x+x^2)^{100}=a_0+a_1x+\cdots+a_{200}x^{200}$$

Compute the following sums:

• $S_1=a_0+a_1+a_2+a_3 +\cdots+a_{200}$
• $S_2=a_0+a_2+a_4+a_6 +\cdots+a_{200}$.

Compute: $1\times 2\times 3 + 2\times 3\times 4 + \cdots + 18\times 19\times 20$.

(Hockey Sticker Identity) Show that for any positive integer $n \ge k$, the following relationship holds: $$\binom{k}{k} +\binom{k+1}{k} + \binom{k+2}{k} + \cdots + \binom{n}{k} = \binom{n+1}{k+1} $$

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 $N$.

(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 $$\sum_{k_1+\cdots+k_p=m}\binom{n_1}{k_1}\binom{n_2}{k_2}\cdots\binom{n_p}{k_p}=\binom{n_1 + \cdots + n_p}{m}$$

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 $n$.