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?

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.)

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?

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 $x$.

How many $3\times 3$ matrices of non-negative integers are there such that the sum of every row and every column equals $n$?

Let $n$, $m$ and $k$ be three positive integers satisfying $m(k-1) < n$. Find the number of ways to select $k$ items from $\{1,\ 2,\ \cdots,\ n\}$ for form a strict increasing sequence and the difference between adjacent terms is no more than $m$.

As shown, an isosceles trapezoid is obtained by removing the top part of an equilateral triangle. The lengths of its two bases are $a$ and $b$, respectively, which are both integers. Both bases and sides are equally divided into unit-length parts. Their ending points are then connected to create several segments which are parallel to either two bases or one side. Find the number of equilateral triangles in this diagram.

How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bc+c, a\ne 0$, and the roots are $r$ and $s$, then the requirement is that $\{a,\ b,\ c\}=\{r,\ s\}$.)

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

Real numbers x, y, and z are chosen from the interval $[−1, 1]$ independently and uniformly at random. What is the probability that $$|x| + |y| + |z| + |x + y + z| = |x + y| + |y + z| + |z + x|$$

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?