[b]Q.[/b] Consider in the plane $n>3$ different points. These have the properties, that all $3$ points can be included in a triangle with maximum area $1$. Prove that all the $n>3$ points can be included in a triangle with maximum area $4$. [i]Proposed by TuZo[/i]

Find the smallest positive integer $n$ with the following property: for every sequence of positive integers $a_1,a_2,\ldots , a_n$ with $a_1+a_2+\ldots +a_n=2013$, there exist some (possibly one) consecutive term(s) in the sequence that add up to $70$.

Denote the three real roots of the cubic $ x^3 \minus{} 3x \minus{} 1 \equal{} 0$ by $ x_1$, $ x_2$, $ x_3$ in order of increasing magnitude. (You may assume that the equation in fact has three distinct real roots.) Prove that $ x_3^2 \minus{} x_2^2 \equal{} x_3 \minus{} x_1$.

In a country with $ n $ ($ n \geq 3 $) airports, the government only licenses air travel to those airlines whose airline system meets the following conditions: a) Each airline must connect any two airports with one and only one one-way airline; b) For each airline there is an airport from which the passenger could fly off and fly back, using the services of only this airline. What is the maximum number of airlines with different airline systems?

Find all functions $f: \mathbb{Z} \to \mathbb{Z}$ such that \[ f(x) + f(y - 1) + f(f(y - f(x))) = 1 \] for all integers $x$ and $y$.

There are two piles of cards, one with $n$ cards and the other with $m$ cards. $A$ and $B$ play alternately, performing one of the following actions in each turn. following operations: a) Remove a card from a pile. b) Remove one card from each pile. c) Move a card from one pile to the other. Player $A$ always starts the game and whoever takes the last one letter wins . Determine if there is a winning strategy based on $m$ and $n$, so that one of the players following her can win always.

Let $ABC$ be a triangle. Let $B'$ be the symmetric of $B$ with respect to $C, C'$ the symmetry of $C$ with respect to $A$ and $A'$ the symmetry of $A$ with respect to $B$. a) Prove that the area of triangle $AC'A'$ is twice the area of triangle $ABC$. b) If we delete points $A, B, C$, how can they be reconstituted? Justify your reasoning.

Point $M$ is an arbitrary point in the plane and let points $G$ and $H$ be the intersection points of the tangents from point M and the circle $k$. Let $O$ be the center of the circle $k$ and let $K$ be the orthocenter of the triangle $MGH$. Prove that ${\angle}GMH={\angle}OGK$.

Let $x$ be a real number such that $$4^{2x}+2^{-x}+1=(129+8\sqrt2)(4^{x}+2^{-x}-2^{x}).$$ Find $10x$.

For $a>0$, find the minimum value of $I(a)=\int_1^e |\ln ax|\ dx.$

A drunken person walks randomly on a tree. Each time, he chooses uniformly at random a neighbouring node and walks there. Show that wherever his starting point and goal are, the expected number of steps the person takes to reach the goal is always an integer. [i]houkai[/i]

Find the sum of all integers from 1 to 1000 which contain at least one “7” in their digits.

Consider an acute-angled triangle $ABC$ with $AB<AC$ and let $\omega$ be its circumscribed circle. Let $t_B$ and $t_C$ be the tangents to the circle $\omega$ at points $B$ and $C$, respectively, and let $L$ be their intersection. The straight line passing through the point $B$ and parallel to $AC$ intersects $t_C$ in point $D$. The straight line passing through the point $C$ and parallel to $AB$ intersects $t_B$ in point $E$. The circumcircle of the triangle $BDC$ intersects $AC$ in $T$, where $T$ is located between $A$ and $C$. The circumcircle of the triangle $BEC$ intersects the line $AB$ (or its extension) in $S$, where $B$ is located between $S$ and $A$. Prove that $ST$, $AL$, and $BC$ are concurrent. $\text{Vangelis Psychas and Silouanos Brazitikos}$

How many ways are there for Nick to travel from $(0,0)$ to $(16,16)$ in the coordinate plane by moving one unit in the positive $x$ or $y$ direction at a time, such that Nick changes direction an odd number of times?

Given a non-isosceles triangle $ABC$, let $D,E$, and $F$ denote the midpoints of the sides $BC,CA$, and $AB$ respectively. The circle $BCF$ and the line $BE$ meet again at $P$, and the circle $ABE$ and the line $AD$ meet again at $Q$. Finally, the lines $DP$ and $FQ$ meet at $R$. Prove that the centroid $G$ of the triangle $ABC$ lies on the circle $PQR$. [i](United Kingdom) David Monk[/i]

At a movie theater tickets for adults cost $4$ dollars more than tickets for children. One afternoon the theater sold $100$ more child tickets than adult tickets for a total sales amount of $1475$ dollars. How many dollars would the theater have taken in if the same tickets were sold, but the costs of the child tickets and adult tickets were reversed?

Let $f(n) = n + \lfloor \sqrt{n} \rfloor$. Prove that for every positive integer $m$, the integer sequence $m, f(m), f(f(m)), \dots$ contains at least one square of an integer.

Find all positive integers $a$ and $b$ such that $ ab+1 \mid a^2-1$

Prove that $\forall x_{1},x_{2},\ldots,x_{2011},y_{1},y_{2},\ldots,y_{2011}\in\mathbb{Z_{+}}$ product: \[(2x_{1}^{2}+3y_{1}^{2})(2x_{2}^{2}+3y_{2}^{2})\ldots(2x_{2011}^{2}+3y_{2011}^{2})\] is not a perfect square.

A right hexagon with side length $n$ is divided into tiles of three types, which are shown in the image, which are rhombuses with side length $1$ each and the acute angle $60$. In one move you can choose three tiles, arranged as shown in the image on the left, and rearrange them, as shown in the image on the right [img]https://iili.io/dxEvyqN.jpg[/img] Moves are made until it is impossible to make a move. a) Prove that for the fixed initial arrangement of tiles the same amount of moves would be made independent of the moves. b) For each positive integer $n$ find the maximum number of moves among all possible initial arrangements [i]M. Zorka[/i]

Let $I$ be the incenter of the non-isosceles triangle $ABC$ and let $A',B',C'$ be the tangency points of the incircle with the sides $BC,CA,AB$ respectively. The lines $AA'$ and $BB'$ intersect in $P$, the lines $AC$ and $A'C'$ in $M$ and the lines $B'C'$ and $BC$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular. [i]Alternative formulation.[/i] The incircle of a non-isosceles triangle $ABC$ has center $I$ and touches the sides $BC$, $CA$ and $AB$ in $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$, respectively. The lines $AA^{\prime}$ and $BB^{\prime}$ intersect in $P$, the lines $AC$ and $A^{\prime}C^{\prime}$ intersect in $M$, and the lines $BC$ and $B^{\prime}C^{\prime}$ intersect in $N$. Prove that the lines $IP$ and $MN$ are perpendicular.

Decide whether or not there exists a nonconstant polynomial $Q(x)$ with integer coefficients with the following property: for every positive integer $n > 2$, the numbers \[ Q(0), \; Q(1), Q(2), \; \dots, \; Q(n-1) \] produce at most $0.499n$ distinct residues when taken modulo $n$. [i]Proposed by Yang Liu[/i]

Let $ ABC$ be an isosceles triangle with $ AB\equal{}AC$, and $ D$ be midpoint of $ BC$, and $ E$ be foot of altitude from $ C$. Let $ H$ be orthocenter of $ ABC$ and $ N$ be midpoint of $ CE$. $ AN$ intersects with circumcircle of triangle $ ABC$ at $ K$. The tangent from $ C$ to circumcircle of $ ABC$ intersects with $ AD$ at $ F$. Suppose that radical axis of circumcircles of $ CHA$ and $ CKF$ is $ BC$. Find $ \angle BAC$.

Prove for every irrational real number a, there are irrational numbers b and b' such that a+b and ab' are rational while a+b' and ab are irrational.

$a,b,c$ are nonnegative and $a+b+c=2\sqrt{abc}$. Prove $bc \geq b+c$