Each point of a circle is colored either red or blue.
(a) Prove that there always exists an isosceles triangle inscribed in this circle such that all its vertices are colored the same.
(b) Does there always exist an equilateral triangle inscribed in this circle such that all its vertices are colored the same?
Numbers $1,2,\cdots, 1974$ are written on a board. You are allowed to replace any two of these numbers by one number which is either the sum or the difference of these numbers. Show that after $1973$ times performing this operation, the only number left on the board cannot be $0$.
On an $8\times 8$ chess board, there are $32$ white pieces and $32$ black pieces, one piece in each square. If a player can change all the white pieces to black and all the black pieces to white in any row or column in a single move, then is it possible that after finitely many movies, there will be exactly one black piece left on the board?
Four $x$'s and five $o$'s are written around the circle in an arbitrary order. If two consecutive symbols are the same, then a new $x$ is inserted in between. Otherwise, a new $o$ is inserted. After nine new symbols are inserted, the previous 9 old ones are erased. Is it possible to get nine $o$'s after having repeated this operation for a finite time?
There are three piles which contain $8$, $9$, and $19$ stones, respectively. You are allowed to choose two piles and transfer one stone from each of them to the third pile. Is it possible to make all piles all contain exactly $12$ stones after several such operations?
Let the lengths of five line segments be $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$, respectively, where $a_1 \ge a_2\ge a_3\ge a_4\ge a_5$. If any three of these five line segments can form a triangle, then prove at least one of such triangle is acute.
Given $n > 2$ points on a plane. Prove if any straight line passing two of these points, it must pass another one among these points, then all these $n$ points must be collinear.
If all sides of a convex pentagon $ABCDE$ are equal in length and $\angle{A}\ge\angle{B}\ge\angle{C}\ge\angle{D}\ge\angle{E}$, show that $ABCDE$ is a regular pentagon.
Let $n$ be an integer greater than $2$, prove $n^{n+1} > (n+1)^n$.
Let $\{a_n\}$ be a sequence defined as $a_1=1$ and $a_n=\frac{a_{n-1}}{1+a_{n-1}}$ when $n\ge 2$. Find the general formula of $a_n$.
Prove a positive proper fraction $\frac{m}{n}$ must be a sum of some reciprocals of distinct integers.
$\displaystyle\frac{2\cos 2^n A+1}{2\cos A+1}=\prod_{r=1}^{n} (2\cos 2^{r-1}A-1)$
For $m=4k+1$ where $k$ is a positive integer. Show that $$\frac{1}{\sqrt{m}}\Big(\Big(\frac{1+\sqrt{m}}{2}\Big)^n-\Big(\frac{1-\sqrt{m}}{2}\Big)^n\Big)$$ must be an integer for any positive integer $n$.
(Bezout's theorem) Show that two positive integers $a$ and $b$ are co-prime if there exist integer $x$ and $y$ satisfying $ax+by=1$.
Show there exist infinite many primes in the form of $(4k+1)$ where $k$ is a positive integer.
Given any five real numbers, show that at least two of them $x$ and $y$ satisfy the condition $|xy+1|>|x-y|$.
Prove that $\cos 1^\circ$ is irrational.
$\textbf{Flip the Grid}$
Given two grids shown below, is it possible to transform ($a$) to ($b$) after a series of operations? In each operation, one can change all the signs in either one entire row or one entire column.
Is it possible to use twenty seven $1\times 2\times 4$ blocks to construct a $6\times 6\times 6$ cube?
Find all pairs of positive integers $(a, b)$ satisfying $a! + b! = a^b + b^a$.
Take a list of positive integers $1$, $2$, $3$, $\cdots$, $2017$. At each step, pick up two of the numbers on the list, say $a$ and $b$, cross them out and replace them by the single number $(ab+a+b)$. Keep doing this until only a single number is left. What is (are) the possible value(s) of this last number?
Find the minimal value of $\sqrt{x^2 - 4x + 5} + \sqrt{x^2 +4x +8}$.
There are $100$ lights lined up in a long room. Each light has its own switch and is currently off. The room has an entry door and an exit door. There are $100$ people lined up outside the entry door. Each light is numbered consecutively from $1$ to $100$. So is each person.
Person No. $1$ enters the room, switches on every light, and exits. Person No. $2$ enters and flips the switch on every second light (i.e. turn off lights $2$, $4$, $6$...). Person No. $3$ enters and flips the switch on every third light (i.e. toggle lights $3$, $6$, $9$...). This continues until all $100$ people have passed through the room. How many of the lights are on at the end?
(Thue's theorem) Let $p$ be a prime. Show that for any integer $a$ such that $p\not\mid a$, there exist positive integers $x$, $y$ not exceeding $\lfloor{\sqrt{p}}\rfloor$ satisfying $ax\equiv y\pmod{p}$ or $ax\equiv -y\pmod{p}$.
Let $n$ be a positive odd integer. Show that at least one of the following numbers is a multiple of $n$. $$2-1, 2^2 -1, \cdots, 2^{n-1} -1$$