A pool table has the shape of a triangle whose angles are in a rational ratio. A ball positioned at an interior point of the table is hit by a stick. The ball reflects from the sides of the triangle according to the law of reflection. Prove that the ball will move only along a finite number of segments. (It is assumed that the ball does not reach the vertices of the triangle.)

For the sequence of real numbers $a_1,a_2,\dots ,a_k$ we say it is [i]invested[/i] on the interval $[b,c]$ if there exists numbers $x_0,x_1,\dots ,x_k$ in the interval $[b,c]$ such that $|x_i-x_{i-1}|=a_i$ for $i=1,2,3,\dots k$ . A sequence is [i]normed[/i] if all its members are not greater than $1$ . For a given natural $n$ , prove : a)Every [i]normed[/i] sequence of length $2n+1$ is [i]invested[/i] in the interval $\left[ 0, 2-\frac{1}{2^n} \right ]$. b) there exists [i]normed[/i] sequence of length $4n+3$ wich is not [i]invested[/i] on $\left[ 0, 2-\frac{1}{2^n} \right ]$.

Let $S$ be the set of sides and diagonals of a regular pentagon. A pair of elements of $S$ are selected at random without replacement. What is the probability that the two chosen segments have the same length? $ \textbf{(A) }\frac25\qquad\textbf{(B) }\frac49\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac59\qquad\textbf{(E) }\frac45 $

Let $s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n$. (a) Show that $s(n)$ is an integer whenever $n$ is an integer. (b) How many integers $n$ with $0 < n \le 2008$ are such that $s(n)$ is divisible by $4$?

Let $ABC$ be a triangle such that $AB$ is not equal to $AC$. Let $M$ be the midpoint of $BC$ and $H$ be the orthocenter of triangle $ABC$. Let $D$ be the midpoint of $AH$ and $O$ the circumcentre of triangle $BCH$. Prove that $DAMO$ is a parallelogram.

A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are 57, 60, and 91. What is the fourth term of this sequence? $\textbf{(A) }190\qquad\textbf{(B) }194\qquad\textbf{(C) }198\qquad\textbf{(D) }202\qquad\textbf{(E) }206$

Let $AA_1$ and $BB_1$ be the altitudes of an acute-angled, non-isosceles triangle $ABC$. Also, let $A_0$ and $B_0$ be the midpoints of its sides $BC$ and $CA$, respectively. The line $A_1B_1$ intersects the line $A_0B_0$ at a point $C'$. Prove that the line $CC'$ is perpendicular to the Euler line of the triangle $ABC$ (this is the line that joins the orthocenter and the circumcenter of the triangle $ABC$).

A family $L$ of 2006 lines on the plane is given in such a way that it doesn't contain parallel lines and it doesn't contain three lines with a common point.We say that the line $l_1\in L$ is [i]bounding[/i] the line $l_2\in L$,if all intersection points of the line $l_2$ with other lines from $L$ lie on the one side of the line $l_1$. Prove that in the family $L$ there are two lines $l$ and $l'$ such that the following 2 conditions are satisfied simultaneously: [b]1)[/b] The line $l$ is bounding the line $l'$; [b]2)[/b] the line $l'$ is not bounding the line $l$.

Let $\triangle ABC$ be an acute triangle. Let us denote the foot of the altitudes from the vertices $A, B$ and $C$ to the opposite sides by $D, E$ and $F,$ respectively, and the intersection point of the altitudes of the triangle $ABC$ by $H.$ Let $P$ be the intersection of the line $BE$ and the segment $DF.$ A straight line passing through $P$ and perpendicular to $BC$ intersects $AB$ at $Q.$ Let $N$ be the intersection of the segment $EQ$ with the perpendicular drawn from $A.$ Prove that $N$ is the midpoint of segment $AH.$

In a tree with $n$ vertices, for each vertex $x_i$, denote the longest paths passing through it by $l_i^1,l_i^2,...,l_i^{k_i}$. $x_i$ cuts those longest paths into two parts with $(a_i^1,b_i^1),(a_i^2,b_i^2),...,(a_i^{k_i},b_i^{k_i})$ vertices respectively. If $\max_{j=1,...,k_i} \{a_i^j\times b_i^j\}=p_i$, find the maximum and minimum values of $\sum_{i=1}^{n} p_i$. [i]Proposed by Sina Rezaei[/i]

Let $s (n)$ denote the sum of digits of a positive integer $n$. Using six different digits, we formed three 2-digits $p, q, r$ such that $$p \cdot q \cdot s(r) = p\cdot s(q) \cdot r = s (p) \cdot q \cdot r.$$ Find all such numbers $p, q, r$.

Inside of the square $ABCD$ the point $P$ is given such that $|PA|:|PB|:|PC|=1:2:3$. Find $\angle APB$.

A triangle with sides $a,b,c$ and perimeter $2p$ is given. Is possible, a new triangle with sides $p-a$, $p-b$, $p-c$ is formed. The process is then repeated with the new triangle. For which original triangles can this process be repeated indefinitely?

Let $I$ be the incenter of $ABC$ and $I_A$ the excenter of the side $BC$, let $M$ be the midpoint of $CB$ and $N$ the midpoint of arc $BC$(with the point $A$). If $T$ is the symmetric of the point $N$ by the point $A$, prove that the quadrilateral $I_AMIT$ is cyclic.

We call a permutation $ \left(a_1, a_2, ..., a_n\right)$ of $ \left(1, 2, ..., n\right)$ [i]quadratic[/i] if there exists at least a perfect square among the numbers $ a_1$, $ a_1 \plus{} a_2$, $ ...$, $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n$. Find all natural numbers $ n$ such that all permutations in $ S_n$ are quadratic. [i]Remark.[/i] $ S_{n}$ denotes the $ n$-th symmetric group, the group of permutations on $ n$ elements.

There is a set of $ n$ coins with distinct integer weights $ w_1, w_2, \ldots , w_n$. It is known that if any coin with weight $ w_k$, where $ 1 \leq k \leq n$, is removed from the set, the remaining coins can be split into two groups of the same weight. (The number of coins in the two groups can be different.) Find all $ n$ for which such a set of coins exists.

Compute \[\sum_{k=0}^{2017}\dfrac{5+\cos\left(\frac{\pi k}{1009}\right)}{26+10\cos\left(\frac{\pi k}{1009}\right)}.\]

Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the $1000^{th}$ number in $S$. Find the remainder when $N$ is divided by $1000$.

China Mathematical Olympiad 2018 Q6 Given the positive integer $n ,k$ $(n>k)$ and $ a_1,a_2,\cdots ,a_n\in (k-1,k)$ ,if positive number $x_1,x_2,\cdots ,x_n$ satisfying:For any set $\mathbb{I} \subseteq \{1,2,\cdots,n\}$ ,$|\mathbb{I} |=k$,have $\sum_{i\in \mathbb{I} }x_i\le \sum_{i\in \mathbb{I} }a_i$ , find the maximum value of $x_1x_2\cdots x_n.$

Princeton has an endowment of $5$ million dollars and wants to invest it into improving campus life. The university has three options: it can either invest in improving the dorms, campus parties or dining hall food quality. If they invest $a$ million dollars in the dorms, the students will spend an additional $5a$ hours per week studying. If the university invests $b$ million dollars in better food, the students will spend an additional $3b$ hours per week studying. Finally, if the $c$ million dollars are invested in parties, students will be more relaxed and spend $11c - c^2$ more hours per week studying. The university wants to invest its $5$ million dollars so that the students get as many additional hours of studying as possible. What is the maximal amount that students get to study?

Numbers $1,\frac12,\frac13,\ldots,\frac1{2001}$ are written on a blackboard. A student erases two numbers $x,y$ and writes down the number $x+y+xy$ instead. Determine the number that will be written on the board after $2000$ such operations.

For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold: $x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$ $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$ $v_{1},v_{2},v_{3}$ satisfy triangle inequality $v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality. Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.

Can there be drawn on a circle of radius $1$ a number of $1975$ distinct points, so that the distance (measured on the chord) between any two points (from the considered points) is a rational number?

In triangle $ABC$, $AB=AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP=PC=CB$, then $\angle A =$ [asy] draw((0,0)--(8,0)--(4,12)--cycle); draw((8,0)--(1.6,4.8)); label("A", (4,12), N); label("B", (0,0), W); label("C", (8,0), E); label("P", (1.6,4.8), NW); dot((0,0)); dot((4,12)); dot((8,0)); dot((1.6,4.8)); [/asy] $ \textbf{(A)}\ 30^{\circ} \qquad\textbf{(B)}\ 36^{\circ} \qquad\textbf{(C)}\ 48^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 72^{\circ} $

suppose that $A$ is the set of all Closed intervals $[a,b] \subset \mathbb{R}$. Find all functions $f:\mathbb{R} \rightarrow A$ such that $\bullet$ $x \in f(y) \Leftrightarrow y \in f(x)$ $\bullet$ $|x-y|>2 \Leftrightarrow f(x) \cap f(y)=\varnothing$ $\bullet$ For all real numbers $0\leq r\leq 1$, $f(r)=[r^2-1,r^2+1]$ Proposed by Matin Yousefi