Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle. [i]Evan O'Dorney.[/i]

Determine the sum of all possible surface area of a cube two of whose vertices are $(1,2,0)$ and $(3,3,2)$.

Let $ABCD$ be a trapezium with $AB\parallel CD$. Let $P$ be a point on $AC$ such that $C$ is between $A$ and $P$; and let $X, Y$ be the midpoints of $AB, CD$ respectively. Let $PX$ intersect $BC$ in $N$ and $PY$ intersect $AD$ in $M$. Prove that $MN\parallel AB$.

Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.$

In acute triangle $ABC$, let $D$ and $E$ be points on sides $AB$ and $AC$ respectively. Let segments $BE$ and $DC$ meet at point $H$. Let $M$ and $N$ be the midpoints of segments $BD$ and $CE$ respectively. Show that $H$ is the orthocenter of triangle $AMN$ if and only if $B,C,E,D$ are concyclic and $BE\perp CD$.

Two players play a game, starting with a pile of $N$ tokens. On each player’s turn, they must remove $2^n$ tokens from the pile for some nonnegative integer $n$. If a player cannot make a move, they lose. For how many $N$ between $ 1$ and $2019$ (inclusive) does the first player have a winning strategy?

Map the exterior of the disc conformally onto the exterior of a given ellipse.

Let $ H $ denote the orthocenter of an acute triangle $ ABC, $ and $ A_1,A_2,A_3 $ denote the intersections of the altitudes of this triangle with its circumcircle, and $ A',B',C' $ denote the projections of the vertices of this triangle on their opposite sides. [b]a)[/b] Prove that the sides of the triangle $ A'B'C' $ are parallel to the sides of $ A_1B_1C_1. $ [b]b)[/b] Show that $ B_1C_1\cdot\overrightarrow{HA_1} +C_1A_1\cdot\overrightarrow{HB_1} +A_1B_1\cdot\overrightarrow{HC_1} =0. $ [i]Geoghe Duță[/i]

How many integers are there in $\{0,1, 2,..., 2014\}$ such that $C^x_{2014} \ge C^{999}{2014}$ ? (A): $15$, (B): $16$, (C): $17$, (D): $18$, (E) None of the above. Note: $C^{m}_{n}$ stands for $\binom {m}{n}$

For integers $a, b$ and $c$, define $\boxed{a, b, c}$ to mean $a^b-b^c+c^a$. Then $\boxed{1, -1, 2}$ equals $ \textbf{(A)}\ -4 \qquad\textbf{(B)}\ -2 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4 $

Let $b_0, b_1, b_2, \ldots$ be a sequence of pairwise distinct nonnegative integers such that $b_0=0$ and $b_n<2n$ for all positive integers $n$. Prove that for each nonnegative integer $m$ there exist nonnegative integers $k, \ell$ such that \begin{align*} b_k+b_{\ell}=m. \end{align*}

In a convex pentagon $ABCDE$ the sides have lengths $1,2,3,4,$ and $5$, though not necessarily in that order. Let $F,G,H,$ and $I$ be the midpoints of the sides $AB$, $BC$, $CD$, and $DE$, respectively. Let $X$ be the midpoint of segment $FH$, and $Y$ be the midpoint of segment $GI$. The length of segment $XY$ is an integer. Find all possible values for the length of side $AE$.

Let $b$, $c$ be integer numbers, and define $f(x)=(x+b)^2-c$. i) If $p$ is a prime number such that $c$ is divisible by $p$ but not by $p^{2}$, show that for every integer $n$, $f(n)$ is not divisible by $p^{2}$. ii) Let $q \neq 2$ be a prime divisor of $c$. If $q$ divides $f(n)$ for some integer $n$, show that for every integer $r$ there exists an integer $n'$ such that $f(n')$ is divisible by $qr$.

Find the smallest possible area of an ellipse passing through $(2,0)$, $(0,3)$, $(0,7)$, and $(6,0)$.

Consider a finite binary string $b$ with at least $2017$ ones. Show that one can insert some plus signs in between pairs of digits such that the resulting sum, when performed in base $2$, is equal to a power of two. [i]Proposed by David Stoner

$O$ is the circumcenter of triangle $ABC$. The bisector from $A$ intersects the opposite side in point $P$. Prove that the following is satisfied: $$AP^2 + OA^2 - OP^2 = bc.$$

Given natural numbers $k$ and $n, k \le n, n \ge 3,$ find the set of all values in the interval $(0, \pi)$ that the $k^{th}-$largest among the interior angles of a convex $n$-gon can take.

We are given a chessboard 100 x 100, $k$ barriers (each with length 1), and one ball. We want to put the barriers between the cells of the board and put the ball in some cell, in such way that the ball can get to each possible cell on the board. The only way that the ball can move is by lifting the board so it can go only forward, backward, to the left or to the right. The ball passes all cells on its way until it reaches a barrier or the edge of the board where it stops. What’s the least number of barriers we need so we can achieve that?

In acute triangle $ABC$, $AB > AC$. Let $M$ be the midpoint of side $BC$. The exterior angle bisector of $\widehat{BAC}$ meet ray $BC$ at $P$. Point $K$ and $F$ lie on line $PA$ such that $MF \perp BC$ and $MK \perp PA$. Prove that $BC^2 = 4 PF \cdot AK$. [asy] defaultpen(fontsize(10)); size(7cm); pair A = (4.6,4), B = (0,0), C = (5,0), M = midpoint(B--C), I = incenter(A,B,C), P = extension(A, A+dir(I--A)*dir(-90), B,C), K = foot(M,A,P), F = extension(M, (M.x, M.x+1), A,P); draw(K--M--F--P--B--A--C); pair point = I; pair[] p={A,B,C,M,P,F,K}; string s = "A,B,C,M,P,F,K"; int size = p.length; real[] d; real[] mult; for(int i = 0; i<size; ++i) { d[i] = 0; mult[i] = 1;} string[] k= split(s,","); for(int i = 0;i<p.length;++i) { label("$"+k[i]+"$",p[i],mult[i]*dir(point--p[i])*dir(d[i])); }[/asy]

[b]8.[/b] Does there exist a Euclidean ring which is properly contained in the field $V$ of real numbers, and whose quotient field is $V$? [b](A.21)[/b]

Let be a natural number $ a\ge 2. $ Prove that for any choice of primes which has the property that none of them divides any of the numbers $ N_n=1+a+a^2+a^3+\cdots +a^{2n} , $ with natural $ n, $ there is another prime not among this choice which doesn't divide any of the numbers $ N_n. $

Let $M$ and $m$ be, respectively, the greatest and the least ten-digit numbers that are rearrangements of the digits $0$ through $9$ such that no two adjacent digits are consecutive. Find $M - m$.

Prove that we can put $ \Omega(\frac1{\epsilon})$ points on surface of a sphere with radius 1 such that distance of each of these points and the plane passing through center and two of other points is at least $ \epsilon$.

A [i]slab[/i] is the set of points strictly between two parallel planes. Prove that a countable sequence of slabs, the sum of whose thicknesses converges, cannot fill space.

In base $10$, the number $2013$ ends in the digit $3$. In base $9$, on the other hand, the same number is written as $(2676)_9$ and ends in the digit $6$. For how many positive integers $b$ does the base-$b$ representation of $2013$ end in the digit $3$? $\textbf{(A) }6\qquad \textbf{(B) }9\qquad \textbf{(C) }13\qquad \textbf{(D) }16\qquad \textbf{(E) }18\qquad$