Find the number of integer solutions to the equation $a+b+c=6$ where $-1 \le a < 2$ and $1\le b,\ c\le 4$.

Let positive integers $n$ and $k$ satisfy $n\ge 2k$. How many $k$-sided convex polygons are there whose vertices are those of an $n$-sided convex polygon and edges are diagonals of the same $n$-polygon.

Show that for any positive integer $n$, the value of $\frac{(n^2)!}{(n!)^{n+1}}$ is an integer.

Assuming positive integer $n$ satisfies $n\equiv 1\pmod{4}$ and $n > 1$. Let $\mathbb{P}=\{a_1,\ a_2,\ \cdots,\ a_n\}$ be a permutation of $\{1,\ 2,\ \cdots,\ n\}$. If $k_p$ denotes the largest index $k$ associated with $\mathbb{P}$ such that the following inequality holds $$a_1 + a_2 + \cdots + a_k < a_{k+1}+a_{k+2}+\cdots + a_n$$

Find the sum of $k_p$ for all possible $\mathbb{P}$.

How many ways are there to arrange $8$ girls and $25$ boys to sit around a table so that there are at least $2$ boys between any pair of girls? If a sitting plan can be simply rotated to match another one, these two are treated as the same.

Let $\mathbb{S}=\{1,\ 2,\ \cdots,\ 1000\}$ and $\mathbb{A}$ be a subset of $\mathbb{S}$. If the number of elements in $\mathbb{A}$ is $201$ and their sum is a multiple of $5$, then $\mathbb{A}$ is called $\textit{good}$. How many good $\mathbb{A}$ are there?

There are $n \ge 6$ points on a circle, every two points are connected by a line segment. No three diagonals are concurrent. How many triangles are created by these sides and diagonals?

For how many integers $n$ between $1$ and $50$, inclusive, is $$\frac{(n^2-1)!}{(n!)^n}$$

Define binary operations $\diamondsuit$ and $\heartsuit$ by $$a\diamondsuit b=a^{\log_7(b)}\qquad\text{and}\qquad a\heartsuit b=a^{\frac{1}{\log_7(b)}}$$

for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3=3\heartsuit 2$ and $$a_n=(n\heartsuit (n-1))\diamondsuit a_{n-1}$$

for all integers $n\ge 4$. To the nearest integer, what is $\log_7(a_{2019})$?

Let $\triangle{A_0B_0C_0}$ be a triangle whose angle measures are exactly $59.999^{\circ}$, $60^{\circ}$, and $60.001^{\circ}$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$& to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\triangle{A_nB_nC_n}$ is obtuse?

Dividing a circle into $n \ge 2$ sectors and coloring these sectors using $m\ge 2$ different colors. If no adjacent sectors can be colored the same, how many different color schemes are there?

$\textbf{Key Set}$

A sensitive location is protected by a door with multiple locks. This place has $11$ workers. The regulation requires that any combination of six workers can open all the locks, but any combination of five cannot. What is the minimal number of locks and how to distribute the keys?

Let $\mathbb{S}=\{a_1,\ a_2,\ \cdots,\ a_n\}$ where every element $a_i\in\{1,\ 2,\ \cdots,\ k\}$. Find the number of $\mathbb{S}$ which has an even number of $1$s.

A permutation $\{x_1,\ x_2,\ \cdots,\ x_{2n}\}$ of the set $\{1,\ 2,\ \cdots,\ 2n\}$, where $n$ is a positive integer, is said to have property $P$ if $\mid x_i − x_{i+1}\mid = n$ for at least one $i$ in $\{1,\ 2,\ \cdots,\ 2n − 1\}$. Show that, for each $n$, there are more permutations with property $P$ than without.

Let $n$ be a positive integer greater than $2$. How many non-congruent acute triangles are there whose vertices are among $n$ equally spaced points on a circle?

Let $S_n$ be the number of non-congruent triangles whose sides' lengths are all integers and circumferences equals $n$. Show that $$S_{2n-1}-S_{2n} = \left\lfloor\frac{n}{6}\right\rfloor\quad\text{or}\quad\left\lfloor\frac{n}{6}\right\rfloor +1$$

where $\lfloor{x}\rfloor$ returns the largest integer not exceeding the real number $x$.

Find the number of ways to divide a convex $n$-sided polygon into $(n-2)$ triangles using non-intersecting diagonals.

Given two distinct values $b_1$ and $b_2$, their product can be written in two ways: $b_1\times b_2$ and $b_2\times b_1$. Given three distinct values $b_1$, $b_2$, and $b_3$, their products can be expressed in $12$ ways: $b_1\times(b_2\times b_3)$, $(b_1\times b_2)\times b_3$, $b_1\times(b_3\times b_2)$, $(b_1\times b_3)\times b_2$, $b_2\times(b_3\times b_1)$, $(b_2\times b_3)\times b_1$, $b_2\times(b_1\times b_3)$, $(b_2\times b_1)\times b_3$, $b_3\times(b_1\times b_2)$, $(b_3\times b_1)\times b_2$, $b_3\times(b_2\times b_1)$, and $(b_3\times b_2)\times b_1$. Please note that in this definition, orders matter and parentheses etc cannot be simplified. The question is how many different ways to express the product of $n$ distinct values?

Let $x_i\in\{+1,\ -1\}$, $i=1,\ 2,\ \cdots,\ 2n$. If their sum equals $0$ and the following inequality holds for any positive integer $k$ satisfying $1\le k < 2n$: $$x_1+x_2+\cdots + x_k\ge 0$$

Find the number of possible ordered sequence $\{x_1,\ x_2,\ \cdots,\ x_{2n}\}$.

Solve the recursion $$a_n=\sum^{n-1}_{k=0}a_{k}a_{n-k-1}=a_0a_{n-1}+a_1a_{n-2}+\cdots+a_{n-1}a_0$$

where $a_0=a_1=1$.

Find the last $4$ digits of $2018^{2019^{2020}}$.

Determine if the following infinite series is convergent or divergent:

$$\sum_{n=2}^{\infty}\frac{1}{(\ln n)^{\ln \ln n}}$$

Evaluate $$\int_{0}^{\pi}\frac{x\sin{x}}{1+\cos^2 x}dx$$

Let $f(x)=\int_1^x\frac{\ln{x}}{1+x}dx$ for $x > 0$. Find $f(x)+f(\frac{1}{x})$.

Compute

$$\int_0^{\infty}\frac{x^2}{1+x^4}dx$$