Determine the number of sequences of points $(x_1, y_1),(x_2, y_2), \dots ,(x_{4570}, y_{4570})$ on the plane satisfying the following two properties: $\text{(i)}$ $\{x_1,x_2,\dots,x_{4570}\}=\{1,2,\dots,2014\}$ and $\{y_1,y_2,\dots,y_{4570}\}=\{1,2,\dots,2557\}$ $\text{(ii)} $ For each $i = 1, 2,\dots , 4569$, exactly one of $x_i = x_{i+1}$ and $y_i = y_{i+1}$ holds.

The circle having $(0,0)$ and $(8,6)$ as the endpoints of a diameter intersects the $x$-axis at a second point. What is the $x$-coordinate of this point? $\textbf{(A)}\ 4\sqrt2 \qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 5\sqrt2 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 6\sqrt2$

$\triangle ABC$ has a right angle at $B$, $AB = 12$, and $BC = 16$. Let $M$ be the midpoint of $AC$. Let $\omega_1$ be the incircle of $\triangle ABM$ and $\omega_2$ be the incircle of $\triangle BCM$. The line externally tangent to $\omega_1$ and $\omega_2$ that is not $AC$ intersects $AB$ and $BC$ at $X$ and $Y$, respectively. If the area of $\triangle BXY$ can be expressed as $\frac{m}{n}$, compute is $m+n$. [i]Proposed by Alex Li[/i]

The game of pool includes $15$ balls that fit within a triangular rack as shown: [asy] // thanks Ritwin for this diagram :D unitsize(0.6cm); pair pos(real i, real j) { return i*dir(60) + (j,0); } for (int i = 0; i <= 4; ++i) { for (int j = 0; j <= 4-i; ++j) { draw(circle(pos(i,j), .5)); } } pair A = pos(0,0); pair B = pos(0,4); pair C = pos(4,0); pair dd = dir(270) * .5; pair ul = dir(150) * .5; pair ur = dir( 30) * .5; real S = 1.75; draw(A+dd -- B+dd ^^ B+ur -- C+ur ^^ C+ul -- A+ul ); draw(A+dd*S -- B+dd*S ^^ B+ur*S -- C+ur*S ^^ C+ul*S -- A+ul*S); draw(arc(A, A+ul*S, A+dd*S)); draw(arc(B, B+dd*S, B+ur*S)); draw(arc(C, C+ur*S, C+ul*S)); [/asy] Seven of the balls are "striped" (not colored with a single color) and eight are "solid" (colored with a single color). Prove that no matter how the $15$ balls are arranged in the rack, there must always be a pair of striped balls adjacent to each other.

Evaluate $\sum_{n=2017}^{2030}\sum_{k=1}^{n}\left\{\frac{\binom{n}{k}}{2017}\right\}$. [i]Note: $\{x\}=x-\lfloor x\rfloor$ for every real numbers $x$.[/i]

Let $a_1,a_2,\cdots, a_n$ and $b_1,b_2,\cdots, b_n$ be two permutations of the numbers $1,2,\cdots, n$. Show that \[\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2\]

Let $A$ and $B$ be two triangles on the plane such that the interior of both contains the origin and for each circle $C_r$ centered at the origin $|C_r \cap A|=|C_r\cap B|$ (where $|\cdot |$ is the arc-length measure). Prove that $A$ and $B$ are congruent. Does this statement remain true if the origin is on the border of $A$ or $B$? (translated by Miklós Maróti)

Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.

Let $\alpha$ be a real number, $1<\alpha<2$. (a) Show that $\alpha$ can uniquely be represented as the infinte product \[ \alpha = \left(1+\dfrac1{n_1}\right)\left(1+\dfrac1{n_2}\right)\cdots \] with $n_i$ positive integers satisfying $n_i^2\le n_{i+1}$. (b) Show that $\alpha\in\mathbb{Q}$ iff from some $k$ onwards we have $n_{k+1}=n_k^2$.

A triangle with sides a, b, c has area S. The distances of its centroid from the vertices are x, y, z. Show that: if (x + y + z)^2 ≤ (a^2 + b^2 + c^2)/2 + 2S√3, then the triangle is equilateral.

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$. (Nigeria)

Call a positive integer [i]challenging[/i] if it can be expressed as $2^a(2^b+1)$, where $a,b$ are positive integers. Prove that if $X$ is a set of challenging numbers smaller than $2^n (n$ is a given positive integer) and $|X|\ge \frac{4}{3}(n-1)$, there exist two disjoint subsets $A,B\subset X$ such that $|A|=|B|$ and $\sum_{a\in A}a=\sum_{b \in B}b$.

Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation $$x(x-y)+y(y-z)+z(z-x) = 1?$$$\textbf{(A)}\: x>y$ and $y=z$ $\textbf{(B)}\: x=y-1$ and $y=z-1$ $\textbf{(C)} \: x=z+1$ and $y=x+1$ $\textbf{(D)} \: x=z$ and $y-1=x$ $\textbf{(E)} \: x+y+z=1$

Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.

Let $ABCDE$ be a regular pentagon. Let point $F$ be intersection of segments $AC$ and $BD$. Let point $G$ be in segment $AD$ such that $2AD=3AG$. Let point $H$ be the midpoint of side $DE$. Show that the points $F,G,H$ lie on a line.

An equilateral triangle is drawn with a side of length $ a$. A new equilateral triangle is formed by joining the midpoints of the sides of the first one. Then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. The limit of the sum of the perimeters of all the triangles thus drawn is: $ \textbf{(A)}\ \text{Infinite} \qquad\textbf{(B)}\ 5\frac {1}{4}a \qquad\textbf{(C)}\ 2a \qquad\textbf{(D)}\ 6a \qquad\textbf{(E)}\ 4\frac {1}{2}a$

Let $\varphi$ denote the Euler phi-function. Prove that for every positive integer $n$ $$2^{n(n+1)} | 32 \cdot \varphi \left( 2^{2^n} - 1 \right).$$

Andrea flips a fair coin repeatedly, continuing until she either flips two heads in a row (the sequence $\texttt{HH}$) or flips tails followed by heads (the sequence $\texttt{TH}$). What is the probability that she will stop after flipping $\texttt{HH}$?

In the plane let two circles be given which intersect at two points $A, B$; Let $PT$ be one of the two common tangent line of these circles ($P, T$ are points of tangency). Tangents at $P$ and $T$ of the circumcircle of triangle $APT$ meet each other at $S$. Let $H$ be a point symmetric to $B$ under $PT$. Show that $A, S, H$ are collinear.

There are $ n$ points on the plane, no three of which are collinear. Each pair of points is joined by a red, yellow or green line. For any three points, the sides of the triangle they form consist of exactly two colours. Show that $ n<13$.

Let $a,b$ be arbitrary co-prime natural numbers. Alice writes the natural number $t < b$ on a blackboard. Every second she replaces the number on the blackboard, say $x$, with the smallest natural number in $\{x \pm a, x \pm b \}$ that she has not yet ever written. She keeps doing this as long as possible. Prove that this process goes on indefinitely and that Alice will write down every natural number. [i]~Pranjal Srivastava and Rohan Goyal[/i]

You are given $n\geq 4$ positive real numbers. Consider all $\frac{n(n-1)}{2}$ pairwise sums of these numbers. Show that some two of these sums differ in at most $\sqrt[n-2]{2}$ times. [i]Proposed by Anton Trygub[/i]

Given a triangle $ABC$, incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M$ be a point inside $ABC$. Prove that $M$ lie on $(I)$ if and only if one number among $\sqrt{AE\cdot S_{BMC}},\sqrt{BF\cdot S_{CMA}},\sqrt{CD\cdot S_{AMB}}$ is sum of two remaining numbers ($S_{ABC}$ denotes the area of triangle $ABC$)

Median $AM$ and the angle bisector $CD$ of a right triangle $ABC$ ($\angle B=90^o$) intersect at the point $O$. Find the area of the triangle $ABC$ if $CO=9, OD=5$.

[i]Biribol[/i] is a game played between two teams of 4 people each (teams are not fixed). Find all the possible values of $ n$ for which it is possible to arrange a tournament with $ n$ players in such a way that every couple of people plays a match in opposite teams exactly once.