$H$ is the orthocentre of $\triangle{ABC}$. $D$, $E$, $F$ are on the circumcircle of $\triangle{ABC}$ such that $AD \parallel BE \parallel CF$. $S$, $T$, $U$ are the semetrical points of $D$, $E$, $F$ with respect to $BC$, $CA$, $AB$. Show that $S, T, U, H$ lie on the same circle.

Three real numbers $a$, $b$, and $c$ between $0$ and $1$ are chosen independently and at random. What is the probability that $a + 2b + 3c > 5$?

The sequence $a_1, a_2, a_3, ...$ of positive reals is such that $\sum a_i$ diverges. Show that there is a sequence $b_1, b_2, b_3, ...$ of positive reals such that $\lim b_n = 0$ and $\sum a_ib_i$ diverges.

There is a box containing $100$ balls, each of which is either orange or black. The box is equally likely to contain any number of black balls between $0$ and $100$, inclusive. A random black ball rolls out of the box. The probability that the next ball to roll out of the box is also black can be written in the form $\tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

$2017$ voters vote by submitting a ranking of the integers $\{1, 2, ..., 38\}$ from favorite (a vote for that value in $1$st place) to least favorite (a vote for that value in $38$th/last place). Let $a_k$ be the integer that received the most $k$th place votes (the smallest such integer if there is a tie). Find the maximum possible value of $\Sigma_{k=1}^{38} a_k$.

Determine the smallest integer $n\geq 4$ for which one can choose four different numbers $a,b,c$ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$.

Prove that for any positive integer $t,$ \[1+2^t+3^t+\cdots+9^t - 3(1 + 6^t +8^t )\] is divisible by $18.$

Consider the sequence $x_n>0$ defined with the following recurrence relation: \[x_1 = 0\] and for $n>1$ \[(n+1)^2x_{n+1}^2 + (2^n+4)(n+1)x_{n+1}+ 2^{n+1}+2^{2n-2} = 9n^2x_n^2+36nx_n+32.\] Show that if $n$ is a prime number larger or equal to $5$, then $x_n$ is an integer.

Let $k$ be a positive integer. Prove that there exists an infinite monotone increasing sequence of integers $\{a_{n}\}_{n \ge 1}$ such that \[a_{n}\; \text{divides}\; a_{n+1}^{2}+k \;\; \text{and}\;\; a_{n+1}\; \text{divides}\; a_{n}^{2}+k\] for all $n \in \mathbb{N}$.

In an acute triangle $ABC$, $AC>AB$, $D$ is the point on $BC$ such that $AD=AB$. Let $\omega_1$ be the circle through $C$ tangent to $AD$ at $D$, and $\omega_2$ the circle through $C$ tangent to $AB$ at $B$. Let $F(\ne C)$ be the second intersection of $\omega_1$ and $\omega_2$. Prove that $F$ lies on $AC$.

Find all natural numbers $k$ such that there exist natural numbers $a_1,a_2,...,a_{k+1}$ with $ a_1!+a_2!+... +a_{k+1}!=k!$ Note that we do not consider $0$ to be a natural number.

In the cyclic quadrilateral $ABCD$, the diagonal $BD$ is divided in half by the diagonal $AC$. The points $E, F, G$ and $H{}$ are the midpoints of the sides $AB, BC, CD{}$ and $DA$ respectively. Let $P = AD \cap BC$ and $Q = AB \cap CD{}$. The bisectors of the angles $APC$ and $AQC$ intersect the segments $EG$ and $FH$ at the points $X{}$ and $Y{}$ respectively. Prove that $XY \parallel BD$.

Let $a\ge b$ and $c\ge d$ be real numbers. Prove that the equation \[(x+a)(x+d)+(x+b)(x+c)=0\] has real roots.

Let $ABC$ be a triangle inscribed in circle $\Gamma$ with center $O$. Let $H$ be the orthocenter of triangle $ABC$ and let $K$ be the midpoint of $OH$. Tangent of $\Gamma$ at $B$ intersects the perpendicular bisector of $AC$ at $L$. Tangent of $\Gamma$ at $C$ intersects the perpendicular bisector of $AB$ at $M$. Prove that $AK$ and $LM$ are perpendicular. by Michael Sarantis, Greece

Express the value of $\sin 3^\circ$ in radicals.

Find all nonzero polynomials $P(x)$ with real coefficents, that satisfies $$P(x)^3+3P(x)^2=P(x^3)-3P(-x)$$ for all real numbers $x$

Johny's father tells him: "I am twice as old as you will be seven years from the time I was thrice as old as you were". What is Johny's age?

[b]5.[/b] On a circle consider $n$ points among which there acts a repulsive force inversely proportional to the square of their distance. Prove that the point system is in stable equilibrium if and only if the points form a regular $n$-gon; in other words, considering the sum of the reciprocal distances of the $\binom{n}{2}$ pairs of points which can be chosen from among the $n$ given points, this sum is minimal if and only if the points lie at the vertices of a regular $n$-gon. [b](G. 2)[/b]

Let $ABC$ be an acute triangle with $AB<AC<BC, c$ it's circumscribed circle and $D,E$ be the midpoints of $AB,AC$ respectively. With diameters the sides $AB,AC$, we draw semicircles, outer of the triangle, which are intersected by line $D$ at points $M$ and $N$ respectively. Lines $MB$ and $NC$ intersect the circumscribed circle at points $T,S$ respectively. Lines $MB$ and $NC$ intersect at point $H$. Prove that: a) point $H$ lies on the circumcircle of triangle $AMN$ b) lines $AH$ and $TS$ are perpedicular and their intersection, let it be $Z$, is the circimcenter of triangle $AMN$

Catherine has a plate containing $300$ circular crumbling mooncakes, arranged as follows: [asy] unitsize(10); for (int i = 0; i < 16; ++i){ for (int j = 0; j < 3; ++j){ draw(circle((sqrt(3)*i,j),0.5)); draw(circle((sqrt(3)*(i+0.5),j-0.5),0.5)); } } dot((16*sqrt(3)+.5,.75)); dot((16*sqrt(3)+1,.75)); dot((16*sqrt(3)+1.5,.75)); [/asy] (This continues for $100$ total columns). She wants to pick some of the mooncakes to eat, however whenever she takes a mooncake all adjacent mooncakes will be destroyed and cannot be eaten. Let $M$ be the maximal number of mooncakes she can eat, and let $n$ be the number of ways she can pick $M$ mooncakes to eat (Note: the order in which she picks mooncakes does not matter). Compute the ordered pair ($M$, $n$).

Given a regular $(6n+3)$-gon, $3n$ of its vertices are used to form $n$ acute triangles with distinct vertices. Prove that the other $3n+3$ vertices can be used to form $n+1$ acute triangles with distinct vertices. [i]Lim Jeck[/i]

Let $ABCD$ be a rectangle with $AB<BC$ and circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) and let $Q$ be a point on the arc $CD$ (not containing $A$) such that $BP=CQ$. The circle with diameter $AQ$ intersects $AP$ again in $S$. The perpendicular to $AQ$ through $B$ intersects $AP$ in $X$. (a) Show that $XS=PS$. (b) Show that $AX=DQ$.

In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear. [i]Proposed by David B. Rush, USA[/i]

Given positive integers $a,b,c,d$ such that $a\mid c^d$ and $b\mid d^c$. Prove that \[ ab\mid (cd)^{max(a,b)} \]