Find all real-valued functions $f$ satisfying $f(2x + f(y)) + f(f(y)) = 4x + 8y$ for all real numbers $x$ and $y$.

Solve the system of equations: $$ 2(3-2\cos y)^2+2(4-2\sin y)^2=2(3-x)^2+32=(x-2\cos y)^2+4\sin^2y$$

Let $I$ be the incenter of fixed triangle $ABC$, and $D$ be an arbitrary point on $BC$. The perpendicular bisector of $AD$ meets $BI,CI$ at $F$ and $E$ respectively. Find the locus of orthocenters of $\triangle IEF$ as $D$ varies.

Let $\{a_n\}$ be a sequence such that: $a_1 = \frac{1}{2}$, $a_{k+1}=-a_k+\frac{1}{2-a_k}$ for all $k = 1, 2,\ldots$. Prove that \[ \left(\frac{n}{2(a_1+a_2+\cdots+a_n)}-1\right)^n \leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n\left(\frac{1}{a_1}-1\right)\left(\frac{1}{a_2}-1\right)\cdots \left(\frac{1}{a_n}-1\right). \]

In a rectangle $ABCD$, let $M$ and $N$ be the midpoints of sides $BC$ and $CD$, respectively, such that $AM$ is perpendicular to $MN$. Given that the length of $AN$ is $60$, the area of rectangle $ABCD$ is $m \sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m+n$. [i]Proposed by Yannick Yao[/i]

Determine the smallest positive integer $M$ with the following property: For every choice of integers $a,b,c$, there exists a polynomial $P(x)$ with integer coefficients so that $P(1)=aM$ and $P(2)=bM$ and $P(4)=cM$. [i]Proposed by Gerhard Woeginger, Austria[/i]

Calculate the following indefinite integrals. (1) $ \int \frac {3x \plus{} 4}{x^2 \plus{} 3x \plus{} 2}dx$ (2) $ \int \sin 2x\cos 2x\cos 4x\ dx$ (3) $ \int xe^{x}dx$ (4) $ \int 5^{x}dx$

Find all triplets $(a, b, c)$ of positive integers, such that $a+bc, b+ac, c+ab$ are primes and all divide $(a^2+1)(b^2+1)(c^2+1)$.

Fill in the circles in the picture at right with the digits $1-8$, one digit in each circle with no digit repeated, so that no two circles that are connected by a line segment contain consecutive digits. In how many ways can this be done? [asy] defaultpen(linewidth(1)); real r = 0.2; pair[] dots = {(0,1),(1,0),(0,0),(-1,0),(1,-1),(0,-1),(-1,-1),(0,-2)}; draw((0,1)--(-1,0)--(-1,-1)--(0,-2)--(1,-1)--(1,0)--(0,1)--(0,-2)); draw((-1,0)--(1,0)--(0,-1)--cycle); draw((-1,-1)--(1,-1)); for(int i = 0; i < dots.length; ++i) filldraw(circle(dots[i], r), white);[/asy]

Let $a, b, c$ be positive reals. Prove that $\sqrt{2a^2+bc}+\sqrt{2b^2+ac}+\sqrt{2c^2+ab}\ge 3 \sqrt{ab+bc+ca}$

How many distinct values of $x$ satisfy $\lfloor x \rfloor ^2 – 3x + 2 = 0$ where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$? $\textbf{(A) } \text{an infinite number} \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 0$

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

Let $a,b,c$ be positive real numbers. Prove the inequality \[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.\] When does equality occur?

In a plane, there is an infinite triangular grid consists of equilateral triangles whose lengths of the sides are equal to $ 1$, call the vertices of the triangles the lattice points, call two lattice points are adjacent if the distance between the two points is equal to $ 1;$ A jump game is played by two frogs $ A,B,$ "A jump" is called if the frogs jump from the point which it is lying on to its adjacent point, " A round jump of $ A,B$" is called if first $ A$ jumps and then $ B$ by the following rules: Rule (1): $ A$ jumps once arbitrarily, then $ B$ jumps once in the same direction, or twice in the opposite direction; Rule (2): when $ A,B$ sits on adjacent lattice points, they carry out Rule (1) finishing a round jump, or $ A$ jumps twice continually, keep adjacent with $ B$ every time, and $ B$ rests on previous position; If the original positions of $ A,B$ are adjacent lattice points, determine whether for $ A$ and $ B$,such that the one can exactly land on the original position of the other after a finite round jumps.

Find the sum : $C^{n}_{1}$ - $\frac{1}{3} \cdot C^{n}_{3}$ + $\frac{1}{9} \cdot C^{n}_{5}$ - $\frac{1}{27} \cdot C^{n}_{9}$ + ...

One eastern country was ruled by an old Shah. The population of the country consisted of inhabitants and satraps. Each resident had his own place of residence (place of registration). Satraps moved around the country and carried out the decrees of the Shah. One day the Shah issued a decree containing the following points: 1) Some residents are bandits. 2) Every bandit must be destroyed. 3) Together with the bandit, all those residents who are located closer to the bandit than the Shah (in other words, than the location of the Shah’s palace) must be destroyed. Finding out which of the residents was a bandit was entrusted to the Shah's adviser, known for his connections with one hostile state. Prove that: a) if the country in question is on a plane, then the adviser has the opportunity to declare no more than six inhabitants bandits in such a way that all inhabitants of the country must be destroyed in accordance with the decrees; b) if the country is located on a sphere, then you can get by with five bandits.

Find infinitely many positive integers $m$ such that for each $m$, the number $\dfrac{2^{m-1}-1}{8191m}$ is an integer.

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

Let $k_1$ and $k_2$ be two circles with centers $O_1$ and $O_2$ and equal radius $r$ such that $O_1O_2 = r$. Let $A$ and $B$ be two points lying on the circle $k_1$ and being symmetric to each other with respect to the line $O_1O_2$. Let $P$ be an arbitrary point on $k_2$. Prove that \[PA^2 + PB^2 \geq 2r^2.\]

$37$ points are arbitrarily marked on the plane. Prove that among them there must be either two points at a distance greater than $6$, or two points at a distance less than $1.5$.

Let ${a_n}$ and ${b_n}$ be positive real sequence that satisfy the following condition: (i) $a_1=b_1=1$ (ii) $b_n=a_n b_{n-1}-2$ (iii) $n$ is an integer larger than $1$. Let ${b_n}$ be a bounded sequence. Prove that $\sum_{n=1}^{\infty} \frac{1}{a_1a_2\cdots a_n}$ converges. Find the value of the sum.

How many natural numbers smaller than $ 2017 $ can be uniquely (order of summands are not relevant) written as a sum of three powers of $ 2? $ [i]Andrei Eckstein[/i]

Consider the real number axis (i. e. the $x$-axis of a Cartesian coordinate system). We mark the points $1$, $2$, ..., $2n$ on this axis. A flea starts at the point $1$. Now it jumps along the real number axis; it can jump only from a marked point to another marked point, and it doesn't visit any point twice. After the ($2n-1$)-th jump, it arrives at a point from where it cannot jump any more after this rule, since all other points are already visited. Hence, with its $2n$-th jump, the flea breaks this rule and gets back to the point $1$. Assume that the sum of the (non-directed) lengths of the first $2n-1$ jumps of the flea was $n\left(2n-1\right)$. Show that the length of the last ($2n$-th) jump is $n$.

a) Let $ ABC$ be a triangle, and $ O$ be its circumcenter. $ BO$ and $ CO$ intersect with $ AC,AB$ at $ B',C'$. $ B'C'$ intersects the circumcircle at two points $ P,Q$. Prove that $ AP\equal{}AQ$ if and only if $ ABC$ is isosceles. b) Prove the same statement if $ O$ is replaced by $ I$, the incenter.

Let $ABC$ be a triangle with $AB = 15$, $AC = 20$, and right angle at $A$. Let $D$ be the point on $\overline{BC}$ such that $\overline{AD}$ is perpendicular to $\overline{BC}$, and let $E$ be the midpoint of $\overline{AC}$. If $F$ is the point on $\overline{BC}$ such that $\overline{AD} \parallel \overline{EF}$, what is the area of quadrilateral $ADFE$?