If $f,g: \mathbb{R} \to \mathbb{R}$ are functions that satisfy $f(x+g(y)) = 2x+y $ $\forall x,y \in \mathbb{R}$, then determine $g(x+f(y))$.

In square $ABCD$, points $E$ and $H$ lie on $\overline{AB}$ and $\overline{DA}$, respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\overline{BC}$ and $\overline{CD}$, respectively, and points $I$ and $J$ lie on $\overline{EH}$ so that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. See the figure below. Triangle $AEH$, quadrilateral $BFIE$, quadrilateral $DHJG$, and pentagon $FCGJI$ each has area $1.$ What is $FI^2$? [asy] real x=2sqrt(2); real y=2sqrt(16-8sqrt(2))-4+2sqrt(2); real z=2sqrt(8-4sqrt(2)); pair A, B, C, D, E, F, G, H, I, J; A = (0,0); B = (4,0); C = (4,4); D = (0,4); E = (x,0); F = (4,y); G = (y,4); H = (0,x); I = F + z * dir(225); J = G + z * dir(225); draw(A--B--C--D--A); draw(H--E); draw(J--G^^F--I); draw(rightanglemark(G, J, I), linewidth(.5)); draw(rightanglemark(F, I, E), linewidth(.5)); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, dir(90)); dot("$D$", D, dir(90)); dot("$E$", E, S); dot("$F$", F, dir(0)); dot("$G$", G, N); dot("$H$", H, W); dot("$I$", I, SW); dot("$J$", J, SW); [/asy] $\textbf{(A) } \frac{7}{3} \qquad \textbf{(B) } 8-4\sqrt2 \qquad \textbf{(C) } 1+\sqrt2 \qquad \textbf{(D) } \frac{7}{4}\sqrt2 \qquad \textbf{(E) } 2\sqrt2$

Let $M$ be the midpoint of the side $BC$ of a acute triangle $ABC$. Incircle of the triangle $ABM$ is tangent to the side $AB$ at the point $D$. Incircle of the triangle $ACM$ is tangent to the side $AC$ at the point $E$. Let $F$ be the such point, that the quadrilateral $DMEF$ is a parallelogram. Prove that $F$ lies on the bisector of $\angle BAC$.

Define $a_1=1$ and $a_{n+1}=\frac{a_n}{n}+\frac{n}{a_n}$ for $n\in\mathbb{N}$. Find the greatest integer not exceeding $a_{2000}$ and prove your claim.

Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all $x,y\in(0,1].$

Let $p,q, r, s$ be real numbers with $q \ne -1$ and $s \ne -1$. Prove that the quadratic equations $x^2 + px+q = 0$ and $x^2 +rx+s = 0$ have a common root, while their other roots are inverse of each other, if and only if $pr = (q+1)(s+1)$ and $p(q+1)s = r(s+1)q$. (A double root is counted twice.)

28. [b][15][/b] Find the shortest distance between the lines $\frac{x+2}{2}=\frac{y-1}{3}=\frac{z}{1}$ and $\frac{x-3}{-1}=\frac{y}{1}=\frac{z+1}{2}$ 29. [b][15][/b] Find the largest real number $k$ such that there exists a sequence of positive reals ${a_i}$ for which $\sum_{n=1}^{\infty}a_n$ converges but $\sum_{n=1}^{\infty}\frac{\sqrt{a_n}}{n^k}$ does not. 30. [b][15][/b] Find the largest integer $n$ such that the following holds: there exists a set of $n$ points in the plane such that, for any choice of three of them, some two are unit distance apart. 31. [b][17][/b] Two random points are chosen on a segment and the segment is divided at each of these two points. Of the three segments obtained, find the probability that the largest segment is more than three times longer than the smallest segment. 32. [b][17][/b] Find the sum of all positive integers $n\le 2015$ that can be expressed in the form $\left\lceil{\frac{x}{2}}\right \rceil +y+xy$, where $x$ and $y$ are positive integers. 33. [b][17][/b] How many ways are there to place four points in the plane such that the set of pairwise distances between the points consists of exactly $2$ elements? (Two configurations are the same if one can be obtained from the other via rotation and scaling.) 34. [b][20][/b] Let $n$ be the second smallest integer that can be written as the sum of two positive cubes in two different ways. Compute $n$. If your guess is $a$, you will receive $\max(25-5\cdot \max(\frac{a}{n},\frac{n}{a}),0)$, rounded up. 35. [b][20][/b] Let $n$ be the smallest positive integer such that any positive integer can be expressed as the sum of $n$ integer 2015th powers. Find $n$. If your answer is $a$, your score will be $\max(20-\frac{1}{5}|\log _{10} \frac{a}{n}|,0)$, rounded up. 36. [b][20][/b] Consider the following seven false conjectures with absurdly high counterexamples. Pick any subset of them, and list their labels in order of their smallest counterexample (the smallest $n$ for which the conjecture is false) from smallest to largest. For example, if you believe that the below list is already ordered by counterexample size, you should write ”PECRSGA”. - [b]P.[/b] (Polya’s conjecture) For any integer $n$, at least half of the natural numbers below $n$ have an odd number of prime factors. - [b]E.[/b] (Euler’s conjecture) There is no perfect cube $n$ that can be written as the sum of three positive cubes. - [b]C.[/b] (Cyclotomic) The polynomial with minimal degree whose roots are the primitive $n$th roots of unity has all coefficients equal to $-1$, $0$, or $1$. - [b]R.[/b] (Prime race) For any integer $n$, there are more primes below $n$ equal to $2(\mod 3)$ than there are equal to $1 (\mod 3)$. - [b]S.[/b] (Seventeen conjecture) For any integer $n$, $n^{17} + 9$ and $(n + 1)^{17} + 9$ are relatively prime. - [b]G.[/b] (Goldbach’s (other) conjecture) Any odd composite integer $n$ can be written as the sum of a prime and twice a square. - [b]A.[/b] (Average square) Let $a_1 = 1$ and $a_{k+1}=\frac{1+a_1^2+a_2^2+...+a_k^2}{k}$. Then $a_n$ is an integer for any n. If your answer is a list of $4\le n\le 7$ labels in the correct order, your score will be $(n-2)(n-3)$. Otherwise, your score will be $0$.

A plane is tiled with regular hexagons of side $1$. $A$ is a fixed hexagon vertex. Find the number of paths $P$ such that: (1) one endpoint of $P$ is $A$, (2) the other endpoint of $P$ is a hexagon vertex, (3) $P$ lies along hexagon edges, (4) $P$ has length $60$, and (5) there is no shorter path along hexagon edges from $A$ to the other endpoint of $P$.

Find all triples of nonnegative real numbers \((x, y, z)\) that satisfy the equality: \[ \frac{\left(x^2 - y\right)(1 - y)}{(x - y)^2} + \frac{\left(y^2 - z\right)(1 - z)}{(y - z)^2} + \frac{\left(z^2 - x\right)(1 - x)}{(z - x)^2} = 3 \] [i]Proposed by Vadym Solomka[/i]

In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\overline{AC}$ and $\overline{AB},$ respectively, so that $AP=PQ=QB=BC.$ Angle $ACB$ is $r$ times as large as angle $APQ,$ where $r$ is a positive real number. Find the greatest integer that does not exceed $1000r.$

Let $n > 3$ be an odd positive integer not divisible by $3$. Determine if it is possible to form an $n \times n$ array of numbers such that [list] [*] [b](a)[/b] the set of the numbers in each row is a permutation of $0, 1, \dots , n - 1$; the set of the numbers in each column is a permutation of $0, 1, \dots , n-1$; [*] [b](b)[/b] the board is [i]totally non-symmetric[/i]: for $1 \le i < j \le n$ and $1 \le i' < j' \le n$, if $(i, j) \neq (i', j')$ then $(a_{i,j} , a_{j,i}) \neq (a_{i',j'} , a_{j',i'})$ where $a_{i,j}$ denotes the entry in the $i^\text{th}$ row and $j^\text{th}$ column.[/list]

If $ S \equal{} i^n \plus{} i^{\minus{}n}$, where $ i \equal{} \sqrt{\minus{}1}$ and $ n$ is an integer, then the total number of possible distinct values for $ S$ is: $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ \text{more than 4}$

On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take? $\textbf{(A) }50\qquad\textbf{(B) }70\qquad\textbf{(C) }80\qquad\textbf{(D) }90\qquad \textbf{(E) }100$

Let an acute triangle $ ABC$ with curcumcircle $ (O)$. Call $ A_1,B_1,C_1$ are foots of perpendicular line from $ A,B,C$ to opposite side. $ A_2,B_2,C_2$ are reflect points of $ A_1,B_1,C_1$ over midpoints of $ BC,CA,AB$ respectively. Circle $ (AB_2C_2),(BC_2A_2),(CA_2B_2)$ cut $ (O)$ at $ A_3,B_3,C_3$ respectively. Prove that: $ A_1A_3,B_1B_3,C_1C_3$ are concurent.

Determine all positive integers $n$ for which the number \[ N = \frac{1}{n \cdot (n + 1)} \] can be represented as a finite decimal fraction.

Let $M$ be any point on the circumcircle of triangle $ABC$. Suppose the tangents from $M$ to the incircle meet $BC$ at two points $X_1$ and $X_2$. Prove that the circumcircle of triangle $MX_1X_2$ intersects the circumcircle of $ABC$ again at the tangency point of the $A$-mixtilinear incircle.

We all know the Fibonacci sequence. However, a slightly less known sequence is the $k$-bonacci sequence. In it, we have $F_1^{(k)} = F_2^{(k)} = \cdots = F_{k-1}^{(k)} = 0, F_k^{(k)} = 1$ and $$F^{(k)}_{n+k} = F^{(k)}_{n+k-1} + F^{(k)}_{n+k-2} + \cdots + F^{(k)}_n,$$for all $n \geq 1$. Find all positive integers $k$ for which there exists a constant $N$ such that $$F^{(k)}_{n-1}F^{(k)}_{n+1} - (F ^{(k)}_n)^2 = (-1)^n$$ for every positive integer $n \geq N$.

Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$. What is the minimum value of $2^a + 3^b$ ?

Let $ABC$ be an isosceles triangle in which $AB = AC$. On its sides $BC$ and $AC$ respectively are marked points $P$ and $Q$ so that $PQ\parallel AB$. Let $F$ be the center of the circle circumscribed about the triangle $PQC$, and $E$ the midpoint of the segment $BQ$. Prove that $\angle AEF = 90^o $.

For an integer $n>2$, the tuple $(1, 2, \ldots, n)$ is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers $n > 2$ for which all permutations of $\{1, 2,\ldots, n\}$ can appear on the blackboard in this way.

On a piece of paper, we write down all positive integers $n$ such that all proper divisors of $n$ are less than $30$. We know that the sum of all numbers on the paper having exactly one proper divisor is $2397$. What is the sum of all numbers on the paper having exactly two proper divisors? We say that $k$ is a proper divisor of the positive integer $n$ if $k | n$ and $1 < k < n$.

Let $n$ be a positive integer and $A$ a set of integers such that the set $\{x = a + b\ |\ a, b \in A\}$ contains $\{1^2, 2^2, \dots, n^2\}$. Prove that there is a positive integer $N$ such that if $n \ge N$, then $|A| > n^{0.666}$.

The diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect at $O$. If $S_1$ and $S_2$ are the areas of triangles $AOB$ and $COD$ and S that of $ABCD$, show that $\sqrt{S_1} + \sqrt{S_2} \le \sqrt{S}$. Prove that equality holds if and only if $AB$ and $CD$ are parallel.

Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m,n\in \mathbb{Z}$: \[f(m+f(n))=f(m)-n.\]

Find all $x$ for which the equation $ x^2 + y^2 + z^2 + 2xyz = 1$ (relative to $z$) has a valid solution for any $y$.