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

Let $n$ be a positive integer. Find the number $a_n$ of polynomials $f(x)$ with coefficients in $\{0, 1, 2, 3\}$ such that $f(2)=n$.

Let $a_0$, $a_1$, $a_2$, $\cdots$ be an increasing sequence of non-negative integers such that every non-negative integer can be expressed uniquely in the form of $(a_i + 2a_j+4a_k)$ where $i$, $j$, and $k$ are not necessarily distinct. Determine $a_{1998}$.

Let $\mathbb{N}$ be the set containing all positive integers. Is it possible to partition $\mathbb{N}$ to more than one but still a finite number of arithmetic sequences with no two having the same common difference?

Let $p$ be an odd prime number. Find the number of subsets $\mathbb{A}$ of the set $\{1, 2, \cdots, 2p\}$ such that $\mathbb{A}$ has exactly $p$ elements and the sum of all elements in $\mathbb{A}$ is divisible by $p$.

Let $a$, $b$, $p$, and $q$ be fixed positive integers. If an $a\times b$ grid can be tiled using some $1\times p$ and $q\times 1$ pieces, show that either $a$ is divisible by $p$ or $b$ is divisible by $q$. Here, a $1\times k$ and $k\times 1$ grids are treated as different.

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

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

(right shifting) Find the generating function for the sequence: $$\underbrace{0, 0, \cdots, 0}_{k}, 1, 1, 1, 1, \cdots$$

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

Find the generating function for the sequence $1$, $2$, $3$, $4$, $\cdots$.

Find the generating function for the sequence $0$, $1$, $2$, $3$, $\cdots$.

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

Find the generating function for the sequence $2,\ 3,\ 4,\ 5,\ \cdots$.

Find the number of subsets of $\{1,\ 2,\ ,3\ ,\cdots,\ 2021\}$ the sum of whose elements is divisible by $101$. (Empty subset is permitted).

Seven students took a math exam. Every problem was solved at at most $3$ students. For every pair of students, there is at least one problem which was solved by both of these two students. What is the minimal number of problem this exam contains?

A partition of a positive integer $n$ is to write $n$ as a sum of some positive integers. Let $k$ be a positive integer. Show that the number of partitions of $n$ with exactly $k$ parts equals the number of partitions of $n$ whose largest part is exactly $k$.

A partition of a positive integer $n$ is to write $n$ as a sum of some positive integers.. Show that the number of partitions of a positive integer $n$ into distinct parts is equal to the number of partitions of $n$ where all parts are odd integers.

Find the number of parallelograms in the following equilateral triangle of side length $n$ which is made of some smaller unit equilateral triangles.

Let $n$ be a positive integer. Find the number of paths from $(0,0)$ to $(n,n)$ on a $n\times n$ grid using only unit up and right steps so that the path stays in the region $x\ge y$.

Find the number of expressions containing $n$ pairs of parentheses which are correctly matched. For example, when $n=3$, the answer is $5$ because all the valid expressions are $$((())),\ (()()),\ (())(),\ ()(()),\ ()()()$$

A tree is a structure which grows from a single root node. By convention, the root node is usually placed at the top. Then, new nodes, referred as children nodes, can be attached to an existing node, referred as parent node, by edges. The root node has no parent and all other nodes have exactly one parent. A node may have any number of children nodes, including no child at all. No looping chain of nodes is permitted in this structure. For example, there are exactly $5$ different types of trees with $4$ nodes. They are shown below. The goal is to find the number of different types of tree with $n$ nodes where $n$ is a positive integer greater than $1$.

Find the generating function for the sequence $\ 0,\ 1,\ -\frac{1}{2},\ \frac{1}{3},\ -\frac{1}{4},\ \cdots$

There are $30$ identical souvenirs to be distributed among $50$ students. Each student may receive more than one souvenir. How many different distribution plans are there?