Consider $n$ persons, each of them speaking at most $3$ languages. From any $3$ persons there are at least two which speak a common language. i) For $n \le 8$, exhibit an example in which no language is spoken by more than two persons. ii) For $n \ge 9$, prove that there exists a language which is spoken by at least three persons

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?

Let $n$ be a positive integer. Each point $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers with $x+y<n$, is coloured red or blue, subject to the following condition: if a point $(x,y)$ is red, then so are all points $(x',y')$ with $x'\leq x$ and $y'\leq y$. Let $A$ be the number of ways to choose $n$ blue points with distinct $x$-coordinates, and let $B$ be the number of ways to choose $n$ blue points with distinct $y$-coordinates. Prove that $A=B$.

$20$ players enter a chess tournament in which each player will play every other player exactly once. Some competitors are cheaters and will cheat in every game they play, but the rest of the competitors are not cheaters. A game is cheating if both players cheat, and a game is half-cheating if one player cheats and one player does not. If there were $68$ more half-cheating games than cheating games, how many of the players are cheaters?

Let $ABCD$ be a square, and let $E$ be an internal point on side $AD$. Let $F$ be the foot of the perpendicular from $B$ to $CE$. Suppose $G$ is a point such that $BG = FG$, and the line through $G$ parallel to $BC$ passes through the midpoint of $EF$. Prove that $AC < 2 \cdot FG$.

Let $\omega_1$ and $\omega_2$ be two non-intersecting circles. Let one of its internal tangents touches $\omega_1$ and $\omega_2$ at $A_1$ and $A_2$, respectively, and let one of its external tangents touches $\omega_1$ and $\omega_2$ at $B_1$ and $B_2$, respectively. Prove that if $A_1B_2 = A_2B_1$, then $A_1B_2 \perp A_2B_1$.

In obtuse triangle $ABC$ where $\angle B > 90^\circ$ let $H$ and $O$ be its orthocenter and circumcenter respectively. Let $D$ be the foot of the altitude from $A$ to $HC$ and $E$ be the foot of the altitude from $B$ to $AC$ such that $O,E,D$ lie on a line. If $OC=8$ and $OE=4$, find the area of triangle $HAB$. [i]Proposed by Harry Kim[/i]

Prove that there are infinitely many natural numbers $n$ such that we can divide $1,2,\ldots ,3n$ into three sequences $(a_n),(b_n)$ and $(c_n)$, with $n$ terms in each, satisfying the following conditions: i) $a_1+b_1+c_1= a_2+b_2+c_2=\ldots =a_n+b_n+c_n$ and $a_1+b_1+c_1$ is divisible by $6$; ii) $a_1+a_2+\ldots +a_n= b_1+b_2+\ldots +b_n=c_1+c_2+\ldots +c_n,$ and $a_1+a_2+\ldots +a_n$ is divisible by $6$.

$a,b$ are real numbers such that: \[ a^3+b^3=8-6ab. \] Find the maximal and minimal value of $a+b$.

Prove that for all integers $ m$ and $ n$, the inequality \[ \dfrac{\phi(\gcd(2^m \plus{} 1,2^n \plus{} 1))}{\gcd(\phi(2^m \plus{} 1),\phi(2^n \plus{} 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}\] holds. [i]Nanang Susyanto, Jogjakarta [/i]

A $\text{22.0 kg}$ suitcase is dragged in a straight line at a constant speed of $\text{1.10 m/s}$ across a level airport floor by a student on the way to the 40th IPhO in Merida, Mexico. The individual pulls with a $\text{1.00} \times \text{10}^2 \text{N}$ force along a handle with makes an upward angle of $\text{30.0}$ degrees with respect to the horizontal. What is the coefficient of kinetic friction between the suitcase and the floor? (A) $\mu_\text{k} = \text{0.013}$ (B) $\mu_\text{k} = \text{0.394}$ (C) $\mu_\text{k} = \text{0.509}$ (D) $\mu_\text{k} = \text{0.866}$ (E) $\mu_\text{k} = \text{1.055}$

Alexa wrote the first $16$ numbers of a sequence: \[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\] Then she continued following the same pattern, until she had $2015$ numbers in total. What was the last number she wrote?

In a plane, the point $(x,y)$ has temperature $x^2+y^2-6x+4y$. Determine the coldest point of the plane and its temperature.

Determine whether or not there exist 15 integers $m_1,\ldots,m_{15}$ such that~ $$\displaystyle \sum_{k=1}^{15}\,m_k\cdot\arctan(k) = \arctan(16). \eqno(1)$$ (Proposed by Gerhard Woeginger, Eindhoven University of Technology)

Let $ P(x)$ be a polynomial such that the following inequalities are satisfied: \[ P(0) > 0;\]\[ P(1) > P(0);\]\[ P(2) > 2P(1) \minus{} P(0);\]\[ P(3) > 3P(2) \minus{} 3P(1) \plus{} P(0);\] and also for every natural number $ n,$ \[ P(n\plus{}4) > 4P(n\plus{}3) \minus{} 6P(n\plus{}2)\plus{}4P(n \plus{} 1) \minus{} P(n).\] Prove that for every positive natural number $ n,$ $ P(n)$ is positive.

The incircle of triangle $ABC$ centered at $I$ touches $CA,AB$ at points $E,F$ respectively. Let points $M,N$ of line $EF$ be such that $CM=CE$ and $BN=BF$. Lines $BM$ and $CN$ meet at point $P$. Prove that $PI$ bisects segment $MN$.

A grasshopper jumps on the plane from an integer point (point with integer coordinates) to another integer point according to the following rules: His first jump is of length $\sqrt{98}$, his second jump is of length $\sqrt{149}$, his next jump is of length $\sqrt{98}$, and so on, alternatively. What is the least possible odd number of moves in which the grasshopper could return to his starting point?

The hour and minute hands on a certain 12-hour analog clock are indistinguishable. If the hands of the clock move continuously, compute the number of times strictly between noon and midnight for which the information on the clock is not sufficient to determine the time. [i]Proposed by Lewis Chen[/i]

Given a set of integers $S$ satisfies that: for any $a,b,c\in S$ ($a,b,c$ can be the same), $ab+c\in S$\\ Find all pairs of integers $(x,y)$ such that if $x,y\in S$, then $S=\mathbb{Z}$.

On each side of triangle $ABC$, two distinct points are marked. It is known that these points are the feet of the altitudes and of the bisectors. a) Using only a ruler determine which points are the feet of the altitudes and which points are the feet of the bisectors. b) Solve p.a) drawing only three lines.

In a triangle $ABC$, $I$ is the incenter. $D$ is the reflection of $A$ to $I$. the incircle is tangent to $BC$ at point $E$. $DE$ cuts $IG$ at $P$ ($G$ is centroid). $M$ is the midpoint of $BC$. prove that a) $AP||DM$.(15 points) b) $AP=2DM$. (10 points)

The table of dimensions $n\times n$, $n\in\mathbb{N}$, is filled with numbers from $1$ to $n^2$, but the difference any two numbers on adjacent fields is at most $n$, and that for every $k = 1, 2,\dots , n^2$ set of fields whose numbers are $1, 2,\dots , k$ is connected, as well as the set of fields whose numbers are $k, k + 1,\dots , n^2$. Neighboring fields are fields with a common side, while a set of fields is considered connected if from each field to every other field of that set can be reached going only to the neighboring fields within that set. We call a pair of adjacent numbers, ie. numbers on adjacent fields, good, if their absolute difference is exactly $n$ (one number can be found in several good pairs). Prove that the table has at least $2 (n - 1)$ good pairs.

Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

An unfair $6$-sided die has faces labeled $1$, $2$, $3$, $4$, $5$, and $6$. The probability that a die lands with a certain face up is proportional to the number on the face. What is the probability that at least one of the first three rolls is a $1$ or a $2$?

Given a parabola $C: y^2=4px\ (p>0)$ with focus $F(p,\ 0)$. Let two lines $l_1,\ l_2$ passing through $F$ intersect orthogonaly each other, $C$ intersects with $l_1$ at two points $P_1,\ P_2$ and $C$ intersects with $l_2$ at two points $Q_1,\ Q_2$. Answer the following questions. (1) Set the equation of $l_1$ as $x=ay+p$ and let the coordinates of $P_1,\ P_2$ as $(x_1,\ y_1),\ (x_2,\ y_2)$, respectively. Express $y_1+y_2,\ y_1y_2$ in terms of $a,\ p$. (2) Show that $\frac{1}{P_1P_2}+\frac{1}{Q_1Q_2}$ is constant regardless of way of taking $l_1,\ l_2$.