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?

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 $\{x_1,\ x_2,\ \cdots,\ x_{2n}\}$.

(Hanoi Tower) There are $3$ identical rods labeled as $A$, $B$, $C$; and $n$ disks of different sizes which can be slide onto any of these three rods. Initially, the $n$ disks are stacked in ascending order of their sizes on $A$. What is the minimal number of moves in order to transfer all the disks to $B$ providing that each move can only transfer one disk to another rod's topmost position and at no time, a bigger disk can be placed on top of a smaller one.

Find the total number of sequences of length $n$ containing only letters $A$ and $B$ such that no two $A$s are next to each other. For example, for $n = 2$, there are $3$ possible sequences: $AB$, $BA$, and $BB$.

How many distinct permutations of the letters of the word REDDER are there that do not contain a palindromic substring of length at least two? (A substring is a contiguous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.)

Reimu and Sanae play a game using $4$ fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the 4 coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then neither of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins? 

Yannick is playing a game with $100$ rounds, starting with $1$ coin. During each round, there is a $n\%$ chance that he gains an extra coin, where $n$ is the number of coins he has at the beginning of the round. What is the expected number of coins he will have at the end of the game?

Contessa is taking a random lattice walk in the plane, starting at $(1, 1)$. (In a random lattice walk, one moves up, down, left, or right $1$ unit with equal probability at each step.) If she lands on a point of the form $(6m, 6n)$ for $m$, $n\in\mathbb{Z}$, she ascends to heaven, but if she lands on a point of the form $(6m+ 3, 6n+ 3)$ for $m,\ n\in\mathbb{Z}$, she descends to hell. What is the probability that she ascends to heaven? 

A point $P$ lies at the center of square $ABCD$. A sequence of points $\{P_n\}$ is determined by $P_0 = P$, and given point $P_i$ , point $P_{i+1}$ is obtained by reflecting $P_i$ over one of the four lines $AB$, $BC$, $CD$, $DA$, chosen uniformly at random and independently for each $i$. What is the probability that $P_8 = P$? 

In an election for the Peer Pressure High School student council president, there are $2019$ voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice’s boyfriend Bob votes for Alice as well. Then one by one, each of the remaining $2016$ voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate’s votes. For example, the first undecided voter David has a $2/3$ probability of voting for Alice and a $1/3$ probability of voting for Celia. What is the probability that Alice wins the election (by having more votes than Celia)?

For a positive integer $N$, we color the positive divisors of $N$ (including $1$ and $N$) with four colors. A coloring is called multichromatic if whenever $a$, $b$ and $\gcd(a, b)$ are pairwise distinct divisors of $N$, then they have pairwise distinct colors. What is the maximum possible number of multichromatic colorings a positive integer can have if it is not the power of any prime? 

How many ways can one fill a $3\times 3$ square grid with nonnegative integers such that no nonzero integer appears more than once in the same row or column and the sum of the numbers in every row and column equals $7$?

Fred the Four-Dimensional Fluffy Sheep is walking in $4$-dimensional space. He starts at the origin. Each minute, he walks from his current position $(a_1,\ a_2,\ a_3,\ a_4)$ to some position $(x_1,\ x_2,\ x_3,\ x_4)$ with integer coordinates satisfying $$(x_1−a_1)^2+(x_2−a_2)^2+(x_3−a_3)^2+(x_4−a_4)^2 = 4$$

and $\mid (x_1 + x_2 + x_3 + x_4) − (a_1 + a_2 + a_3 + a_4)\mid = 2$. In how many ways can Fred reach $(10, 10, 10, 10)$ after exactly $40$ minutes, if he is allowed to pass through this point during his walk?

Consider a $2\times 3$ grid where each entry is one of $0$, $1$, and $2$. For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$? One valid grid is shown below: $$\begin{bmatrix}  1& 2 & 0\\ 2& 1 &0 \end{bmatrix}$$

Let $a$ and $b$ be five-digit palindromes (without leading zeroes) such that $a < b$ and there are no other five-digit palindromes strictly between $a$ and $b$. What are all possible values of $b - a$? (A number is a palindrome if it reads the same forwards and backwards in base $10$.) 

A $4\times 4$ window is made out of $16$ square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors? Two different windowpanes are neighbors if they share a side.