Suppose that $f : \mathbb R^+ \to \mathbb R^+$ is a decreasing function such that \[f(x+y)+f(f(x)+f(y))=f(f(x+f(y))+f(y+f(x)), \quad \forall x,y \in \mathbb R^+.\] Prove that $f(x) = f^{-1}(x).$

How many ways are there to remove an $11\times11$ square from a $2011\times2011$ square so that the remaining part can be tiled with dominoes ($1\times 2$ rectangles)?

The acute triangle $ABC$ satisfies $\overline {AB}<\overline {BC}<\overline {CA}$. Denote the foot of perpendicular from $A,B,C$ to opposing sides as $D,E,F$. Let $P$ a foot of perpendicular from $F$ to $DE$, and $Q(\neq F)$ a intersection point of line $FP$ and circumcircle of $BDF$. Prove that $\angle PBQ=\angle PAD$.

Let us call a set of positive integers [i]nice[/i], if its number of elements is equal to the average of all its elements. Call a number $n$ [i]amazing[/i], if one can partition the set $\{1,2,\ldots,n\}$ into nice subsets. a) Prove that any perfect square is amazing. b) Prove that there exist infinitely many positive integers which are not amazing.

Which triangles satisfy the equation $\frac{c^2-a^2}{b}+\frac{b^2-c^2}{a}=b-a$ when $a, b$ and $c$ are sides of a triangle?

There are $ 3n, n \in \mathbb{Z}^\plus{}$ girl students who took part in a summer camp. There were three girl students to be on duty every day. When the summer camp ended, it was found that any two of the $ 3n$ students had just one time to be on duty on the same day. (1) When $ n\equal{}3,$ is there any arrangement satisfying the requirement above. Prove yor conclusion. (2) Prove that $ n$ is an odd number.

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

$7$ boys each went to a shop $3$ times. Each pair met at the shop. Show that $3$ must have been in the shop at the same time.

Let the sequence $(a_{n})$ be defined by $a_{1} = t$ and $a_{n+1} = 4a_{n}(1 - a_{n})$ for $n \geq 1$. How many possible values of t are there, if $a_{1998} = 0$?

Find all $4$-tuples $(a,b,c,d)$ of natural numbers with $a \le b \le c$ and $a!+b!+c!=3^d$

Let $ABCD$ be a square with side $4$. Find, with proof, the biggest $k$ such that no matter how we place $k$ points into $ABCD$, such that they are on the interior but not on the sides, we always have a square with sidr length $1$, which is inside the square $ABCD$, such that it contains no points in its interior(they can be on the sides).

The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$. [hide="comment"] Hi guys , Here is a nice problem: Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that : 1) $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u [i]Edited by Orl.[/i] [/hide]

Let $x$, $y$, and $z$ be real numbers such that $x<y<z<6$. Solve the system of inequalities: \[\left\{\begin{array}{cc} \dfrac{1}{y-x}+\dfrac{1}{z-y}\le 2 \\ \dfrac{1}{6-z}+2\le x \\ \end{array}\right.\]

Let $n\ge 2$ be an integer. Solve in reals: \[|a_1-a_2|=2|a_2-a_3|=3|a_3-a_4|=\cdots=n|a_n-a_1|.\]

Find all pairs $(m, n)$ of positive integers such that $m^2 + n^2 =(m + 1)(n + 1).$

Show that an arbitary triangle can be dissected by straight line segments into three parts in three different ways so that each part has a line of symmetry.

Let $ABC$ be a triangle. Construct rectangles $BA_1A_2C$, $CB_1B_2A$, and $AC_1C_2B$ outside $ABC$ such that $\angle BCA_1=\angle CAB_1=\angle ABC_1$. Let $A_1B_2$ and $A_2C_1$ intersect at $A'$ and define $B',C'$ similarly. Prove that line $AA'$ bisects $B'C'$. [i]Linus Tang[/i]

Consider the natural number prime $p, p> 5$. From the decimal number $\frac1p$, randomly remove $2012$ numbers, after the comma. Show that the remaining number can be represented as $\frac{a}{b}$ , where $a$ and $b$ are coprime numbers , and $b$ is multiple of $p$.

What is the biggest positive integer which divides $p^4 - q^4$ for all primes $p$ and $q$ greater than $10$?

Theseus starts at the point $(0, 0)$ in the plane. If Theseus is standing at the point $(x, y)$ in the plane, he can step one unit to the north to point $(x, y+1)$, one unit to the west to point $(x-1, y)$, one unit to the south to point $(x, y-1)$, or one unit to the east to point $(x+1, y)$. After a sequence of more than two such moves, starting with a step one unit to the south (to point $(0, -1)$), Theseus finds himself back at the point $(0, 0)$. He never visited any point other than $(0, 0)$ more than once, and never visited the point $(0, 0)$ except at the start and end of this sequence of moves. Let $X$ be the number of times that Theseus took a step one unit to the north, and then a step one unit to the west immediately afterward. Let $Y$ be the number of times that Theseus took a step one unit to the west, and then a step one unit to the north immediately afterward. Prove that $|X - Y| = 1$. [i]Mitchell Lee[/i]

Given an $\triangle ABC$ isoceles with base $BC$ we note with $M$ the midpoint of said base. Let $X$ be any point on the shortest arc $AM$ of the circumcircle of $\triangle ABM$ and let $T$ be a point on the inside $\angle BMA$ such that $\angle TMX = 90^o$ and $TX = BX$. Show that $\angle MTB - \angle CTM$ does not depend on $X$.

A "[size=100][i]walking sequence[/i][/size]" is a sequence of integers with $a_{i+1} = a_i \pm 1$ for every $i$ .Show that there exists a sequence $b_1, b_2, . . . , b_{2016}$ such that for every walking sequence $a_1, a_2, . . . , a_{2016}$ where $1 \leq a_i \leq1010$, there is for some $j$ for which $a_j = b_j$ .

Given a positive prime number $p$. Prove that there exist a positive integer $\alpha$ such that $p|\alpha(\alpha-1)+3$, if and only if there exist a positive integer $\beta$ such that $p|\beta(\beta-1)+25$.

If $i^2=-1$, then $(i-i^{-1})^{-1}=$ $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ -2i \qquad\textbf{(C)}\ 2i \qquad\textbf{(D)}\ -\frac{i}{2} \qquad\textbf{(E)}\ \frac{i}{2}$

Evaluate $1^3-2^3+3^3-4^3+5^3-\cdots+101^3$.