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?

In a sports contest, there were $m$ medals awarded on $n$ successive days ($n > 1$). On the first day, one medal and $1/7$ of the remaining $(m − 1)$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining medals were awarded; and so on. On the $n^{th}$ and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?

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.

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.

Alicia had two containers. The first was $\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\frac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container?

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement.

$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$

In a high school wit $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?

All lines with equation $ax+by=c$ such that $a,b,c$ form an arithmetic progression pass through a common point. What are the coordinates of that point?

$\textbf{(A) } (-1,2) \qquad\textbf{(B) } (0,1) \qquad\textbf{(C) } (1,-2) \qquad\textbf{(D) } (1,0) \qquad\textbf{(E) } (1,2)$

Triangle $ABC$ lies in the first quadrant. Points& $A$, $B$, and $C$ are reflected across the line $y=x$ to points $A'$, $B'$, and $C'$, respectively. Assume that none of the vertices of the triangle lie on the line $y=x$. Which of the following statements is not always true?

$\textbf{(A) }$ Triangle $A'B'C'$" lies in the first quadrant.

$\textbf{(B) }$ Triangles $ABC$ and $A'B'C'$ have the same area.

$\textbf{(C) }$ The slope of line $AA'$ is $-1$.

$\textbf{(D) }$ The slopes of lines $AA'$ and $CC'$ are the same.

$\textbf{(E) }$ Lines $AB$ and $A'B'$ are perpendicular to each other.

There is a real $n$ such that $(n+1)! + (n+2)! = n! \cdot 440$. What is the sum of the digits of $n$?