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?

What is the value of \[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?\]

What is the hundreds digit of $(20! - 15!)$?

A child builds towers using identically shaped cubes of different colors. How many different towers with a height of $8$ cubes can the child build with $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.)

For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k=0.232323\cdots_k$. What is $k$?

Positive real numbers $x\ne 1$ and $y\ne 1$ satisfy $\log_2x=\log_y16$ and $xy=64$. What is $\left(\log_2\frac{x}{y}\right)^2$?

For a certain complex number $c$, the polynomial $$P(x)=(x^2-2x+2)(x^2-cx+4)(x^2-4x+8)$$

has exactly $4$ distinct roots. What is $\mid c\mid$?

Positive real numbers $a$ and $b$ have the property that $$\sqrt{\log a}+\sqrt{\log b} +\log\sqrt{a} + \log\sqrt{b}=100$$

and all four terms on the left are positive integers, where $\log$ denotes the base-$10$ logarithm. What is $ab$?

Let $$z=\frac{1+i}{\sqrt{2}}$$

What is $$\left(z^{1^2}+z^{2^2}+z^{3^2}+\cdots+z^{12^2}\right)\left(\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\cdots+\frac{1}{z^{12^2}}\right)$$

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 $\log_7(a_{2019})$?

Equally divide each side of a triangle into $n$ parts and then connect these points to draw lines which are parallel to one of the triangle's sides. Find the number of parallelograms created by these lines.

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?

Find the number of non-negative integer solutions to the equation $$2x_1+x_2+x_3+\cdots+x_9+x_{10}=3$$

Let $\mathbb{S}=\{1,\ 2,\ 3,\ \cdots,\ n\}$ and positive integer $m$ satisfying $n + 1\ge 2m$. Find the number of subsets of $\mathbb{S}$ which has $m$ elements and no two elements are consecutive.

How many ordered integers $(x_1,\ x_2,\ x_3,\ x_4)$ are there such that $0 < x_1 \le x_2\le x_3\le x_4 < 7$?

How many different ways to write a positive integer $n$ as a sum of $m$ different positive integers? Different sequences are treated as distinct.

$\textbf{Key Set}$

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 $\mathbb{S} =\{a_1,\ a_2,\ \cdots,\ a_n\}$ be a permutation of $\{1,\ 2,\ \cdots,\ n\}$ which satisfies the condition that for every $a_i$, $(i=1$, $2$, $\cdots$, $n)$, there exists an $a_j$ where $i< j \le n$ such that $a_j=a_i+1$ or $a_j=a_i-1$. Find the number of such $\mathbb{S}$.

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.

Let $n$ be an even integer. Find the number of ways to select four distinct integers $a$, $b$, $c$, $d$ between $1$ and $n$, inclusive, satisfying $a+c=b+d$. Order of these four numbers does not matter.

In the Banana Country, only Mr Decent always tells the truth and only Mr Joke always tells lies. Everyone else has a probability of $p$ to tell a lie. One day, Mr Decent has decided to run for the President and told his decision to the first person who in turn told this to the second person. The second person then told this to the third person, and so on, till the $n^{th}$ person who told this news to Mr Joke. No one has been told this news twice in this process. Finally, Mr Joke announced Mr Decent's decision to everyone. What is the probability that Mr Joke's statement agrees with Mr Decent's intention?

Let $a$ and $b$ be two positive real numbers satisfying $(a-b)^2=4(ab)^3$. Find the minimal value of $\frac{1}{a}+\frac{a}{b}$.

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.