$f : \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $m+f(m+f(n+f(m))) = n + f(m)$ for every integers $m,n$. If $f(6) = 6$, then $f(2012) = ?$ $ \textbf{(A)}\ -2010 \qquad \textbf{(B)}\ -2000 \qquad \textbf{(C)}\ 2000 \qquad \textbf{(D)}\ 2010 \qquad \textbf{(E)}\ 2012$

In rectangle $ABCD$, $AB = 40$ and $AD = 30$. Let $C' $ be the reflection of $C$ over $BD$. Find the length of $AC'$.

Let $ ABC $ be a triangle, and let $M$ be midpoint of $BC$. Let $ I_b $ and $ I_c $ be incenters of $ AMB $ and $ AMC $. Prove that the second intersection of circumcircles of $ ABI_b $ and $ ACI_c $ distinct from $A$ lies on line $AM$.

$n$ is a natural number that $\frac{x^{n}+1}{x+1}$ is irreducible over $\mathbb Z_{2}[x]$. Consider a vector in $\mathbb Z_{2}^{n}$ that it has odd number of $1$'s (as entries) and at least one of its entries are $0$. Prove that these vector and its translations are a basis for $\mathbb Z_{2}^{n}$

Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.

Find all real number $x$ which could be represented as $x = \frac{a_0}{a_1a_2 . . . a_n} + \frac{a_1}{a_2a_3 . . . a_n} + \frac{a_2}{a_3a_4 . . . a_n} + . . . + \frac{a_{n-2}}{a_{n-1}a_n} + \frac{a_{n-1}}{a_n}$ , with $n, a_1, a_2, . . . . , a_n$ are positive integers and $1 = a_0 \leq a_1 < a_2 < . . . < a_n$

Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? $ \textbf{(A)}\ \frac{1}{5} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{2}{5} \qquad \textbf{(D)}\ \frac{2}{3} \qquad \textbf{(E)}\ \frac{4}{5}$

Let there be positive integers $a, c$. Positive integer $b$ is a divisor of $ac-1$. For a positive rational number $r$ which is less than $1$, define the set $A(r)$ as follows. $$A(r) = \{m(r-ac)+nab| m, n \in \mathbb{Z} \}$$ Find all rational numbers $r$ which makes the minimum positive rational number in $A(r)$ greater than or equal to $\frac{ab}{a+b}$.

Let $ABC$ be triangle with $A$-excenter $J$. The reflection of $J$ in $BC$ is $K$. The points $E$ and $F$ are on $BJ, CJ$ such that $\angle EAB=\angle CAF=90^{\circ}$. Prove that $\angle FKE+\angle FJE=180^{\circ}$.

Suppose that $u_0 , u_1 ,\ldots$ is a sequence of real numbers such that $$u_n = \sum_{k=1}^{\infty} u_{n+k}^{2}\;\;\; \text{for} \; n=0,1,2,\ldots$$ Prove that if $\sum u_n$ converges, then $u_k=0$ for all $k$.

Triangles $ABC$ and $BDC$ are such that $\angle{ABC}=\angle{BDC}=90^{\circ}$ and $\angle{DBC}=\angle{CAB}$. Let $Q$ be a point on $\overline{BD}$ such that $\overline{QC}\perp\overline{AD}$. Suppose that $BD=15$. Then $DQ$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Greg plays a game in which he is given three random $1$ digit numbers, each between $0$ and $9$, inclusive, with repeats allowed. He is to put these three numbers into any order. Exactly one ordering of the three numbers is correct, and if he guesses the correct ordering, he wins $\$150$. What are Greg's expected winnings for this game, given that he randomly guesses one valid ordering when he plays?

Given a triangle $ \triangle{ABC} $ whose incenter is $ I $ and $ A $-excenter is $ J $. $ A' $ is point so that $ AA' $ is a diameter of $ \odot\left(\triangle{ABC}\right) $. Define $ H_{1}, H_{2} $ to be the orthocenters of $ \triangle{BIA'} $ and $ \triangle{CJA'} $. Show that $ H_{1}H_{2} \parallel BC $

Let $\triangle ABC$ be an acute triangle. The bisector of $\angle ACB$ intersects side $AB$ in $D$. The circumcircle of triangle $ADC$ intersects side $BC$ in $C$ and $E$ with $C \neq E$. The line parallel to $AE$ which passes through $B$ intersects line $CD$ in $F$. Prove that the triangle $\triangle AFB$ is isosceles.

Given two positive integers $a$ and $b$, an [i]legal move[/i] consists in choosing a proper divisor of one of them and adding it to $a$ or adding it to $b$. Two players, Agustin and Ian, take turns making an legal move; Agustin plays first. Whoever gets a number greater than or equal to $2015$ wins the game. [list=a] [*]Determine which of the players has a winning strategy if $a=3, b=5$. [*]Determine which of the players has a winning strategy if $a=6, b=7$. [/list]

Let $ x_1,x_2,..,x_n$ be non-negative numbers whose sum is $ 1$ . Show that there exist numbers $ a_1,a_2,\ldots ,a_n$ chosen from amongst $ 0,1,2,3,4$ such that $ a_1,a_2,\ldots ,a_n$ are different from $ 2,2,\ldots ,2$ and $ 2\leq a_1x_1\plus{}a_2x_2\plus{}\ldots\plus{}a_nx_n\leq 2\plus{}\frac{2}{3^n\minus{}1}$.

$(HUN 3)$ In the plane $4000$ points are given such that each line passes through at most $2$ of these points. Prove that there exist $1000$ disjoint quadrilaterals in the plane with vertices at these points.

1990 mathematicians attend a meeting, every mathematician has at least 1327 friends (the relation of friend is reciprocal). Prove that there exist four mathematicians among them such that any two of them are friends.

Let $ABC$ be a triangle. The line through $A$ tangent to the circumcircle of $ABC$ intersects line $BC$ at point $W$. Points $X,Y \neq A$ lie on lines $AC$ and $AB$, respectively, such that $WA=WX=WY$. Point $X_1$ lies on line $AB$ such that $\angle AXX_1 = 90^{\circ}$, and point $X_2$ lies on line $AC$ such that $\angle AX_1X_2 = 90^{\circ}$. Point $Y_1$ lies on line $AC$ such that $\angle AYY_1 = 90^{\circ}$, and point $Y_2$ lies on line $AB$ such that $\angle AY_1Y_2 = 90^{\circ}$. Let lines $AW$ and $XY$ intersect at point $Z$, and let point $P$ be the foot of the perpendicular from $A$ to line $X_2Y_2$. Let line $ZP$ intersect line $BC$ at $U$ and the perpendicular bisector of segment $BC$ at $V$. Suppose that $C$ lies between $B$ and $U$. Let $x$ be a positive real number. Suppose that $AB=x+1$, $AC=3$, $AV=x$, and $\frac{BC}{CU}=x$. Then $x=\frac{\sqrt{k}-m}{n}$ for positive integers $k$,$m$, and $n$ such that $k$ is not divisible by the square of any integer greater than $1$. Compute $100k+10m+n$. [i]Proposed by Ankit Bisain, Luke Robitaille, and Brandon Wang[/i]

Let $M_n$ be the $n\times n$ matrix with entries as follows: for $1\leq i \leq n$, $m_{i,i}=10$; for $1\leq i \leq n-1, m_{i+1,i}=m_{i,i+1}=3$; all other entries in $M_n$ are zero. Let $D_n$ be the determinant of matrix $M_n$. Then $\displaystyle \sum_{n=1}^{\infty} \dfrac{1}{8D_n+1}$ can be represented as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. Note: The determinant of the $1\times 1$ matrix $[a]$ is $a$, and the determinant of the $2\times 2$ matrix $\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]=ad-bc$; for $n\geq 2$, the determinant of an $n\times n$ matrix with first row or first column $a_1\ a_2\ a_3 \dots\ a_n$ is equal to $a_1C_1 - a_2C_2 + a_3C_3 - \dots + (-1)^{n+1} a_nC_n$, where $C_i$ is the determinant of the $(n-1)\times (n-1)$ matrix found by eliminating the row and column containing $a_i$.

There are numerous sets of $17$ distinct positive integers that sum to $2018$, such that each integer has the same sum of digits in base $10$. Let $M$ be the maximum possible integer that could exist in any such set. Find the sum of $M$ and the number of such sets that contain $M$.

Let $ABC$ be an acute triangle with $AB<AC$. The angle bisector of $BAC$ intersects the side $BC$ and the circumcircle of $ABC$ at $D$ and $M\neq A$, respectively. Points $X$ and $Y$ are chosen so that $MX \perp AB$, $BX \perp MB$, $MY \perp AC$, and $CY \perp MC$. Prove that the points $X,D,Y$ are collinear.

Let $n$ be a given positive integer. Prove that there exist infinitely many integer triples $(x,y,z)$ such that \[nx^2+y^3=z^4,\ \gcd (x,y)=\gcd (y,z)=\gcd (z,x)=1.\]

Three flower beds overlap as shown. Bed A has 500 plants, bed B has 450 plants, and bed C has 350 plants. Beds A and B share 50 plants, while beds A and C share 100. The total number of plants is [asy] draw((0,0)--(3,0)--(3,1)--(0,1)--cycle); draw(circle((.3,-.1),.7)); draw(circle((2.8,-.2),.8)); label("A",(1.3,.5),N); label("B",(3.1,-.2),S); label("C",(.6,-.2),S);[/asy] $ \text{(A)}\ 850\qquad\text{(B)}\ 1000\qquad\text{(C)}\ 1150\qquad\text{(D)}\ 1300\qquad\text{(E)}\ 1450 $

At a recent math contest, Evan was asked to find $2^{2016} \pmod{p}$ for a given prime number $p$ with $100 < p < 500$. Evan has forgotten what the prime $p$ was, but still remembers how he solved it: [list] [*] Evan first tried taking $2016$ modulo $p - 1$, but got a value $e$ larger than $100$. [*] However, Evan noted that $e - \frac{1}{2}(p - 1) = 21$, and then realized the answer was $-2^{21} \pmod{p}$. [/list] What was the prime $p$?