In a round robin chess tournament each player plays every other player exactly once. The winner of each game gets $ 1$ point and the loser gets $ 0$ points. If the game is tied, each player gets $ 0.5$ points. Given a positive integer $ m$, a tournament is said to have property $ P(m)$ if the following holds: among every set $ S$ of $ m$ players, there is one player who won all her games against the other $ m\minus{}1$ players in $ S$ and one player who lost all her games against the other $ m \minus{} 1$ players in $ S$. For a given integer $ m \ge 4$, determine the minimum value of $ n$ (as a function of $ m$) such that the following holds: in every $ n$-player round robin chess tournament with property $ P(m)$, the final scores of the $ n$ players are all distinct.

Let $ABC$ be a triangle not being isosceles at $A$. Let $(O)$ and $(I)$ denote the circumcircle and incircle of the triangle. $(I)$ touches $AC$ and $AB$ at $E, F$ respectively. Points $M$ and $N$ are on the circle $(I)$ such that $EM \parallel FN \parallel BC$. Let $P,Q$ be the intersections of $BM,CN$ and $(I)$. Prove that i) $BC,EP, FQ$ are concurrent, and denote by $K$ the point of concurrency. ii) the circumcircles of triangle $BPK, CQK$ are all tangent to $(I)$ and all pass through a common point on the circle $(O)$. Nguyễn Minh Hà

Prove that there exist $C>0$, which satisfies the following conclusion: For any infinite positive arithmetic integer sequence $a_1, a_2, a_3,\cdots$, if the greatest common divisor of $a_1$ and $a_2$ is squarefree, then there exists a positive integer $m\le C\cdot {a_2}^2$, such that $a_m$ is squarefree. Note: A positive integer $N$ is squarefree if it is not divisible by any square number greater than $1$. [i]Proposed by Qu Zhenhua[/i]

Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$. [i](Theresia Eisenkoelbl)[/i]

What is the smallest whole number larger than the perimeter of any triangle with a side of length $ 5$ and a side of length $19$? $\textbf{(A) }24\qquad\textbf{(B) }29\qquad\textbf{(C) }43\qquad\textbf{(D) }48\qquad \textbf{(E) }57$

$ (a)$ Prove that $ 91$ divides $ n^{37}\minus{}n$ for all integers $ n$. $ (b)$ Find the largest $ k$ that divides $ n^{37}\minus{}n$ for all integers $ n$.

155 birds $ P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\cdot)$ of their positions $ m(P_iP_j) \leq 10^{\circ}.$ Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $ \{x,y\}$ of mutually visible pairs of birds with $ x,y \in \{P_1, \ldots, P_{155}\}.$ One assumes that a position (point) on $ C$ can be occupied simultaneously by several birds, e.g. all possible birds.

What is the largest integer that must divide $n^5-5n^3+4n$ for all integers $n$? [i]2016 CCA Math Bonanza Lightning #2.4[/i]

Let $a<b<c<d$ be positive integers which satisfy $ad=bc.$ Prove that $2a+\sqrt{a}+\sqrt{d}<b+c+1.$ [i]Marius Mînea[/i]

Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which is divisible by $3$, such that $x^2+2y^2 = 3^k$.

For the triple $(a,b,c)$ of positive integers we say it is interesting if $c^2+1\mid (a^2+1)(b^2+1)$ but none of the $a^2+1, b^2+1$ are divisible by $c^2+1$. Let $(a,b,c)$ be an interesting triple, prove that there are positive integers $u,v$ such that $(u,v,c)$ is interesting and $uv<c^3$.

Let $\{a_n\}$ be a sequence of natural numbers such that each prime number greater than $1402$ divides a member of that. Prove that the set of prime divisors of members of sequence $\{b_n\}$ which $b_n=a_1a_2...a_n-1$ , is infinite. [i]Proposed by Navid Safaei[/i]

The $25$ member states of the European Union set up a committee with the following rules: 1) the committee should meet daily; 2) at each meeting, at least one member should be represented; 3) at any two different meetings, a different set of member states should be represented; 4) at $n^{th}$ meeting, for every $k<n$, the set of states represented should include at least one state that was represented at the $k^{th}$ meeting. For how many days can the committee have its meetings?

Can we find positive reals $a_1, a_2, \dots, a_{2002}$ such that for any positive integer $k$, with $1 \leq k \leq 2002$, every complex root $z$ of the following polynomial $f(x)$ satisfies the condition $|\text{Im } z| \leq |\text{Re } z|$, \[f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k,\] where $a_{2002+i}=a_i$, for $i=1,2, \dots, 2001$.

Prove that there exists a polynomial with integer coefficients satisfying the following conditions: (a)$f(x)=0$ has no rational root. (b) For any positive integer $n$, there always exists an integer $m$ such that $n\mid f(m)$.

Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.

Let $L$ be the line of intersection of the planes $~x+y=0~$ and $~y+z=0$. [b](a)[/b] Write the vector equation of $L$, i.e. find $(a,b,c)$ and $(p,q,r)$ such that $$L=\{(a,b,c)+\lambda(p,q,r)~~\vert~\lambda\in\mathbb{R}\}$$ [b](b)[/b] Find the equation of a plane obtained by $x+y=0$ about $L$ by $45^{\circ}$.

For what value of $n$ is $\frac{1}{2\cdot5}+\frac{1}{5\cdot8}+\frac{1}{8\cdot 11}+\frac{1}{n(n+3)}=\frac{25}{154}$?

Determine the integers $x$ such that $2^x + x^2 + 25$ is the cube of a prime number

In the diagram below, the larger square has side length $6$. Find the area of the smaller square.

Let $ABCD$ be a square with side length $6$. Let $E$ be a point on the perimeter of $ABCD$ such that the area of $\triangle{AEB}$ is $\frac{1}{6}$ the area of $ABCD$. Find the maximum possible value of $CE^2$. [i]Proposed by Anthony Yang[/i]

Prove that there exist infinitely many natural numbers $ n$ such that the numerator of $ 1 \plus{} \frac {1}{2} \plus{} \frac {1}{3} \plus{} \frac {1}{4} \plus{} ... \plus{} \frac {1}{n}$ in the lowest terms is not a power of a prime number.

Three lights are placed horizontally on a line on the ceiling. All the lights are initially off. Every second, Neil picks one of the three lights uniformly at random to switch: if it is off, he switches it on; if it is on, he switches it off. When a light is switched, any lights directly to the left or right of that light also get turned on (if they were off) or off (if they were on). The expected number of lights that are on after Neil has flipped switches three times can be expressed in the form $m/ n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

In the diagram below, a triangular array of three congruent squares is configured such that the top row has one square and the bottom row has two squares. The top square lies on the two squares immediately below it. Suppose that the area of the triangle whose vertices are the centers of the three squares is $100.$ Find the area of one of the squares. [asy] unitsize(1.25cm); draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0)); draw((1,0)--(2,0)--(2,1)--(1,1)); draw((1.5,1)--(1.5,2)--(0.5,2)--(0.5,1)); draw((0.5,0.5)--(1.5,0.5)--(1,1.5)--(0.5,0.5),dashed); [/asy]

Given a subset of $\{1,2,...,n\}$, we define its [i]alternating sum [/i] in the following way: we order the elements of the subset in descending order and, starting with the largest, we alternately add and subtract the successive numbers. For example, the [i]alternating sum[/i] of the set $\{1,3,4,6,8\}$ is $8-6+4-3+1 = 4$. Determines the sum of the alternating sums of all subsets of $\{1,2,...,10\}$ with an odd number of elements.