$5000$ movie fans gathered at a convention. Each participant had watched at least one movie. The participants should be split into discussion groups of two kinds. In each group of the first kind, the members would discuss a movie they all watched. In each group of the second kind, each member would tell about the movie that no one else in this group had watched. Prove that the chairman can always split the participants into exactly 100 groups. (A group consisting of one person is allowed; in this case this person submits a report).

Let $ABC$ be a scalene acute triangle with circumcenter $O$. Let $K$ be a point on the side $\overline{BC}$. Define $M$ as the second intersection of $\overleftrightarrow{OK}$ with the circumcircle of $BOC$. Let $L$ be the reflection of $K$ by $\overleftrightarrow{AC}$. Show that the circumcircles of the triangles $LCM$ and $ABC$ are tangent if, and only if, $\overline{AK} \perp \overline{BC}$.

Find all triples ( $\alpha, \beta,\theta$) of acute angles such that the following inequalities are fulfilled at the same time $$(\sin \alpha + \cos \beta + 1)^2 \ge 2(\sin \alpha + 1)(\cos \beta + 1)$$ $$(\sin \beta + \cos \theta + 1)^2 \ge 2(\sin \beta + 1)(\cos \theta + 1)$$ $$(\sin \theta + \cos \alpha + 1)^2 \ge 2(\sin \theta + 1)(\cos \alpha + 1).$$

Max flips $2020$ fair coins. Let the probability that there are at most $505$ heads be $p$. Estimate $-\log_2(p)$ to 5 decimal places, in the form $x.abcde$ where $x$ is a positive integer and $a, b, c, d, e$ are decimal digits.

[b]i.)[/b] Calculate $x$ if \[ x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})} \] [b]ii.)[/b] For each positive number $x,$ let \[ k = \frac{\left( x + \frac{1}{x} \right)^6 - \left( x^6 + \frac{1}{x^6} \right) - 2}{\left( x + \frac{1}{x} \right)^3 - \left( x^3 + \frac{1}{x^3} \right)} \] Calculate the minimum value of $k.$

For how many triples of positive integers $(a,b,c)$ with $1\leq a,b,c\leq 5$ is the quantity \[(a+b)(a+c)(b+c)\] not divisible by $4$?

Point $P$ is $\sqrt3$ units away from plane $A$. Let $Q$ be a region of $A$ such that every line through $P$ that intersects $A$ in $Q$ intersects $A$ at an angle between $30^o$ and $60^o$ . What is the largest possible area of $Q$?

The roots of the equation $x^2+5x-7=0$ are $x_1$ and $x_2$. What is the value of $x_1^3+5x_1^2-4x_1+x_1^2x_2-4x_2$ ? $\textbf{(A)}\ -15 \qquad\textbf{(B)}\ 175+25\sqrt{53} \qquad\textbf{(C)}\ -50 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ \text{None}$

Let $V = \left\{ 1, \dots, 8 \right\}$. How many permutations $\sigma : V \to V$ are automorphisms of some tree? (A $\emph{graph}$ consists of some set of vertices and some edges between pairs of distinct vertices. It is $\emph{connected}$ if every two vertices in it are connected by some path of one or more edges. A $\emph{tree}$ $G$ on $V$ is a connected graph with vertex set $V$ and exactly $|V|-1$ edges, and an $\emph{automorphism}$ of $G$ is a permutation $\sigma : V \to V$ such that vertices $i,j \in V$ are connected by an edge if and only if $\sigma(i)$ and $\sigma(j)$ are.)

Given $$a=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99}}}}, \,\,and\,\,\, b=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99+\dfrac{1}{100}}}}}$$ Prove that $$|a-b| <\frac{1}{99! 100!}$$ (G Galperin, Moscow)

Let $A,B,C,D$ be points on the line $d$ in that order and $AB = CD$. Denote $(P)$ as some circle that passes through $A, B$ with its tangent lines at $A, B$ are $a,b$. Denote $(Q)$ as some circle that passes through $C, D$ with its tangent lines at $C, D$ are $c,d$. Suppose that $a$ cuts $c, d$ at $K, L$ respectively and $b$ cuts $c, d$ at $M, N$ respectively. Prove that four points $K, L, M,N$ belong to a same circle $(\omega)$ and the common external tangent lines of circles $(P)$, $(Q)$ meet on $(\omega)$.

Ben has a big blackboard, initially empty, and Francisco has a fair coin. Francisco flips the coin $2013$ times. On the $n^{\text{th}}$ flip (where $n=1,2,\dots,2013$), Ben does the following if the coin flips heads: (i) If the blackboard is empty, Ben writes $n$ on the blackboard. (ii) If the blackboard is not empty, let $m$ denote the largest number on the blackboard. If $m^2+2n^2$ is divisible by $3$, Ben erases $m$ from the blackboard; otherwise, he writes the number $n$. No action is taken when the coin flips tails. If probability that the blackboard is empty after all $2013$ flips is $\frac{2u+1}{2^k(2v+1)}$, where $u$, $v$, and $k$ are nonnegative integers, compute $k$. [i]Proposed by Evan Chen[/i]

Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to factorials of some positive integers. Proposed by [i]Nikola Velov, Macedonia[/i]

Prove that there doesn't exist a function $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $(m+f(n))^2 \geq 3f(m)^2+n^2$ for all $m, n \in \mathbb{N}$.

Let $n\in \Bbb{N}, n \geq 4.$ Determine all sets $ A = \{a_1, a_2, . . . , a_n\} \subset \Bbb{N}$ containing $2015$ and having the property that $ |a_i - a_j|$ is prime, for all distinct $i, j\in \{1, 2, . . . , n\}.$

Higher Secondary P2 Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$. Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$.

Prove that $\frac1{2002}<\frac12\cdot\frac34\cdot\frac56\cdots\frac{2001}{2002}<\frac1{44}$.

Let $\theta$ be a positive real number. Show that if $k\in \mathbb{N}$ and both $\cosh k \theta$ and $\cosh(k+1) \theta$ are rational, then so is $\cosh \theta$.

Compute the remainder when $\binom{205}{101}$ is divded by $101 \times 103$. [i]Proposed by Brandon Xu[/i]

Sa se determine functia $f: [0,\infty)\rightarrow\mathbb{R}$, astfel incat \[f(x)+\sqrt{f^{2}([x])+f^{2}(\{x\})}=x,\] oricare ar fi $x\in [0,\infty).$

Let $w$ be a circle and $AB$ a line not intersecting $w$. Given a point $P_{0}$ on $w$, define the sequence $P_{0},P_{1},\ldots $ as follows: $P_{n\plus{}1}$ is the second intersection with $w$ of the line passing through $B$ and the second intersection of the line $AP_{n}$ with $w$. Prove that for a positive integer $k$, if $P_{0}\equal{}P_{k}$ for some choice of $P_{0}$, then $P_{0}\equal{}P_{k}$ for any choice of $P_{0}$. [i]Gheorge Eckstein[/i]

If $a, b, c \ge 4$ are integers, not all equal, and $4abc = (a+3)(b+3)(c+3)$ then what is the value of $a+b+c$ ?

Can $29$ boys and $31$ girls be lined up holding hands so no one is holding hands with two girls?

In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits? [i]Proposed by A. Shapovalov[/i]

Given two non-constant polynomials $P(x),Q(x)$ with real coefficients. For a real number $a$, we define $$P_a= \{z \in C : P(z) = a\}, Q_a =\{z \in C : Q(z) = a\}$$ Denote by $K$ the set of real numbers $a$ such that $P_a = Q_a$. Suppose that the set $K$ contains at least two elements, prove that $P(x) = Q(x)$.