Suppose that in a certain society, each pair of persons can be classified as either [i]amicable [/i]or [i]hostile[/i]. We shall say that each member of an amicable pair is a [i]friend[/i] of the other, and each member of a hostile pair is a [i]foe[/i] of the other. Suppose that the society has $\, n \,$ persons and $\, q \,$ amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include $\, q(1 - 4q/n^2) \,$ or fewer amicable pairs.

Calculate the lower bound of the areas of convex hexagons whose vertices all have integer coordinates.

Let $ d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $ n$ vertices (where $ n>3$). Let $ p$ be its perimeter. Prove that: \[ n\minus{}3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n\plus{}1\over 2}\Bigr]\minus{}2,\] where $ [x]$ denotes the greatest integer not exceeding $ x$.

A circle is divided into $7$ arcs. The sum of the angles subtending any two neighbouring arcs is no more than $103^o$. Find the maximal number $A$ such that any of the $7$ arcs is subtended by no less than $A^o$. Prove that this value $A$ is really maximal. (A. Tolpygo, Kiev)

Given a scalene triangle $ABC$ inscribed in the circle $(O)$. Let $(I)$ be its incircle and $BI,CI$ cut $AC,AB$ at $E,F$ respectively. A circle passes through $E$ and touches $OB$ at $B$ cuts $(O)$ again at $M$. Similarly, a circle passes through $F$ and touches $OC$ at $C$ cuts $(O)$ again at $N$. $ME,NF$ cut $(O)$ again at $P,Q$. Let $K$ be the intersection of $EF$ and $BC$ and let $PQ$ cuts $BC$ and $EF$ at $G,H$, respectively. Show that the median correspond to $G$ of the triangle $GHK$ is perpendicular to $IO$.

At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different?

A set $M$ is formed of $\binom{2n}{n}$ men, $n=1,2,\ldots$. Prove that we can choose a subset $P$ of the set $M$ consisting of $n+1$ men such that one of the following conditions is satisfied: $(1)$ every member of the set $P$ knows every other member of the set $P$; $(2)$ no member of the set $P$ knows any other member of the set $P$.

On a board the following six vectors are written: $$(1, 0, 0), (-1, 0, 0), (0, 1, 0), (0, -1, 0), (0, 0, 1), (0, 0, -1).$$ Given two vectors $v$ and $w$ on the board, a move consists of erasing $v$ and $w$ and replacing them with $\frac{1}{\sqrt2} (v + w)$ and $\frac{1}{\sqrt2} (v - w)$. After some number of moves, the sum of the six vectors on the board is $u$. Find, with proof, the maximum possible length of $u$.

In prism $JHOPKINS$, quadrilaterals $JHOP$ and $KINS$ are parallel and congruent bases that are kites, where $JH = JP = KI = KS$ and $OH = OP = NI = NS$; the longer two sides of each kite have length $\tfrac{4 + \sqrt{5}}{2}$, and the shorter two sides of each kite have length $\tfrac{5 + \sqrt{5}}{4}$. Assume that $\overline{JK}$, $\overline{HI}$, $\overline{ON}$, and $\overline{PS}$ are congruent edges of $JHOPKINS$ perpendicular to the planes containing $JHOP$ and $KINS$. Vertex $J$ is part of a regular pentagon $JAZZ'Y$ that can be inscribed in prism $JHOPKINS$ such that $A \in \overline{HI}$, $Z \in \overline{NI}$, $Z' \in \overline{NS}$, $Y \in \overline{PS}$, $AI = YS$, and $ZI = Z'S$. Compute the height of $JHOPKINS$ (that is, the distance between the bases).

A triangle $ ABC$ is inscribed in a circle $ C(O,R)$ and has incenter $ I$. Lines $ AI,BI,CI$ meet the circumcircle $ (O)$ of triangle $ ABC$ at points $ D,E,F$ respectively. The circles with diameter $ ID,IE,IF$ meet the sides $ BC,CA, AB$ at pairs of points $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$ respectively. Prove that the six points $ A_1,A_2, B_1, B_2, C_1, C_2$ are concyclic. Babis

On side $AB$ of triangle $ABC$ is marked point $K$ such that $AB = CK$. Points $N$ and $M$ are the midpoints of $AK$ and $BC$, respectively. The segments $NM$ and $CK$ intersect in point $P$. Prove that $KN = KP$.

Does there exist a quadratic trinomial $f(x)$ such that $f(1/2017)=1/2018$, $f(1/2018)=1/2017$, and two of its coefficients are integers? (A. Khrabrov)

The function $f(x) = ax^2 + bx + c$ satisfies the following conditions: $f(\sqrt2)=3$ and $ |f(x)| \le 1$ for all $x \in [-1, 1]$. Evaluate the value of $f(\sqrt{2013})$

Each coefficient of a polynomial is an integer with absolute value not exceeding $2015$. Prove that every positive root of this polynomial exceeds $\frac{1}{2016}$. [i]($6$ points)[/i]

We are going to colour the cells of a $2015 \times 2015$ board such that there are none of the following: $1)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the right of the upper cell, $2)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the left of the lower cell. What is the minimum number of colours $k$ required to make such a colouring possible?

The two sides of the triangle are equal to $1$ and $x$, and $ x \ge 1$. The values $a$ and $b$ are the largest and smallest angles of this triangle, respectively. Find the greatest value of $\cos a$ and the smallest value of $\cos b$.

$ABCD$ - convex quadrilateral. Bisectors of angles $A$ and $D$ intersect in $K$, Bisectors of angles $B$ and $C$ intersect in $L$. Prove $$2KL \geq |AB-BC+CD-DA|$$

How many handles has the Riemann surface of the function \[w=\sqrt{1+z^n}\]

The sequence $ (x_n)_{n \in \mathbb{N}}$ is defined by $ x_1\equal{}2, x_2\equal{}3,$ and $ x_{2m\plus{}1}\equal{}x_{2m}\plus{}x_{2m\minus{}1}$ for $ m \ge 1;$ $ x_{2m}\equal{}x_{2m\minus{}1}\plus{}2x_{2m\minus{}2}$ for $ m \ge 2.$ Determine $ x_n$ as a function of $ n$.

For how many integers $n$ with $1 \le n \le 2012$ is the product \[ \prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right) \] equal to zero?

A line $l$ parallel to the side $BC$ of triangle $ABC$ touches its incircle and meets its circumcircle at points $D$ and $E$. Let $I$ be the incenter of $ABC$. Prove that $AI^2 = AD \cdot AE$.

Let triangle $ABC$ have altitudes $BE$ and $CF$ which meet at $H$. The reflection of $A$ over $BC$ is $A'$. Let $(ABC)$ meet $(AA'E)$ at $P$ and $(AA'F)$ at $Q$. Let $BC$ meet $PQ$ at $R$. Prove that $EF \parallel HR$. [i]Proposed by Daniel Hu[/i]

Define a sequence ${(n_k)_{k\ge 0}}$ by ${n_{0 }= n_{1} = 1}$, and ${n_{2k} = n_k + n_{k-1} }$ and ${n_{2k+1} = n_k}$ for ${k \ge 1}$. Let further ${q_k = n_k }$ / ${ n_{k-1} }$ for each ${k \ge 1}$. Show that every positive rational number is present exactly once in the sequence ${(q_k)_{k\ge 1}}$

In the square $ABCD$, points $E$ and $F$ were chosen on the sides $AB$ and $BC$ respectively, such that $BE=BF$. Let $L$ be midpoint of $EF$, $N$ be midpoint of $DF$, $O$ be the center of the square and $K=AL \cap DF$ (look at picture). Prove that points $C, K, L, O, N$ are lies on one circle. [i]A. Paleev[/i]

Determine all rational numbers $a$ for which the matrix $$\begin{pmatrix} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a\\ 0 & 1 & a & -a \end{pmatrix}$$ is the square of a matrix with all rational entries. [i]Proposed by Daniël Kroes, University of California, San Diego[/i]