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

Compute the following sums:

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

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 $N$.

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 $(1+m)^n > (1+n)^m$.

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 $n$.

Calculate the value of $$\displaystyle\sum_{k=0}^{n}\frac{1}{2^k}\binom{n}{k}$$

Let the binary representation of positive integer $n$ be $b_tb_{t-1}\cdots b_1b_0$. Show that $$\binom{n}{2^j} \equiv b_j \pmod{2}$$

where $j$ is a non-negative integer. Note that $\binom{n}{m} = 0$ if $m > n$.

Show that $$\sqrt{1+x}=1+\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n\cdot 2^{2n-1}}\binom{2n-2}{n-1}x^n$$

(Generalized binomial expansion) If $a$, $b$, and $r$ are some real or complex numbers, then $$(a+b)^r = \sum_{k=0}^{\infty}\binom{r}{k}a^{r-k}b^k$$

Here, the following definition still holds when $r$ is a real or complex number: $$\binom{r}{k}=\frac{r(r-1)\cdots(r-k+1)}{1\cdot 2\cdots k}$$

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 the limit of $f(n)=\left(1+\frac{1}{n}\right)^n$ exits when $n$ becomes infinitely large.