Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.
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.
Let $p$ be an odd prime number. For positive integer $k$ satisfying $1\le k\le p-1$, the number of divisors of $k p+1$ between $k$ and $p$ exclusive is $a_k$. Find the value of $a_1+a_2+\ldots + a_{p-1}$.
Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that
$$f(yf(x)-x)=f(x)f(y)+2x$$
for all $x,\ y\in{\mathbb{R}}$.
For pairwise distinct nonnegative reals $a,b,c$, prove that
$$\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(b-a)^2}>2$$
For each integer $a_0 >$ 1, define the sequence $a_0, a_1, a_2, \cdots$ by:
$$
a_{n+1} =
\left\{
\begin{array}{ll}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer}\\
a_n + 3 & \text{otherwise}
\end{array}
\right.
$$
For all $n \ge 0$. Determine all values of $a_0$ for which there is a number $A$ such that $a_n = A$ for infinitely many values of $n$.
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all real numbers $x$ and $y$,
$$f (f(x)f(y)) + f(x + y) = f(xy)$$
An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is 1. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0$, $a_1$, $\cdots$, an such that, for each $(x, y)$ in $S$, we have:
$$a_0x^n + a_1x^{n-1}y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1$$
Prove the triple angle formulas: $$\sin 3\theta = 3\sin\theta -4\sin^3\theta$$ and $$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$$
Show that $\sin\alpha + \sin\beta + \sin\gamma - \sin(\alpha + \beta+\gamma) = 4\sin\frac{\alpha+\beta}{2}\sin\frac{\beta+\gamma}{2}\sin\frac{\gamma+\alpha}{2}$
Compute $\cot 70^\circ + 4\cos 70^\circ$
$\displaystyle\frac{2\cos 2^n A+1}{2\cos A+1}=\prod_{r=1}^{n} (2\cos 2^{r-1}A-1)$
Compute $4\cos\frac{2\pi}{7}\cos\frac{\pi}{7}-2\cos\frac{2\pi}{7}$
Let $P(x)$ be a monic cubic polynomial. The lines $y = 0$ and $y = m$ intersect $P(x)$ at points $A$, $C$, $E$ and $B$, $D$, $F$ from left to right for a positive real number $m$. If $AB = \sqrt{7}$, $CD = \sqrt{15}$, and $EF = \sqrt{10}$, what is the value of $m$?
Find an acute angle $\alpha$ such that $\sqrt{15-12\cos\alpha} + \sqrt{7-4\sqrt{3}\sin\alpha}=4$. (Find at least two different solutions.)
Show that $\frac{1}{k+1}\binom{n}{k}=\frac{1}{n+1}\binom{n+1}{k+1}$.