Let $a_1, a_2, \ldots$ be a sequence of real numbers such that $a_1=4$ and $a_2=7$ such that for all integers $n$, $\frac{1}{a_{2n-1}}, \frac{1}{a_{2n}}, \frac{1}{a_{2n+1}}$ forms an arithmetic progression, and $a_{2n}, a_{2n+1}, a_{2n+2}$ forms an arithmetic progression. Find, with proof, the prime factorization of $a_{2019}$.

In convex hexagon $ABCDEF, AB$ is parallel to $CF, CD$ is parallel to $BE$ and $EF$ is parallel to $AD$. Prove that the areas of triangles $ACE$ and $BDF$ are equal .

Let $ n$ be a positive integer, and let $ M\equal{}\left\{ 1,2,...,n\right\}$. Two players take turns at the following game: Each player, at his turn, has to select an element of $ M$ and remove all divisors of this element (including this element itself) from the set $ M$. [b]a)[/b] Assume that the player who cannot move anymore (because the set $ M$ is empty when it's his move) wins. For which values of $ n$ does the first player have a winning strategy? [b]b)[/b] Assume that the player who cannot move anymore (because the set $ M$ is empty when it's his move) loses. For which values of $ n$ does the first player have a winning strategy?

Curves $A,B,C$ and $D$ are defined in the plane as follows: \begin{align*} A &= \left\{ (x,y): x^2-y^2 = \frac{x}{x^2+y^2} \right\}, \\ B &= \left\{ (x,y): 2xy + \frac{y}{x^2+y^2} = 3 \right\}, \\ C &= \left\{ (x,y): x^3-3xy^2+3y=1 \right\}, \\ D &= \left\{ (x,y): 3x^2 y - 3x - y^3 = 0\right\}. \end{align*} Prove that $A \cap B = C \cap D$.

Let $A$ be a finite set of functions $f: \Bbb{R}\to \Bbb{R.}$ It is known that: [list] [*] If $f, g\in A$ then $f (g (x)) \in A.$ [*] For all $f \in A$ there exists $g \in A$ such that $f (f (x) + y) = 2x + g (g (y) - x),$ for all $x, y\in \Bbb{R}.$ [/list] Let $i:\Bbb{R}\to \Bbb{R}$ be the identity function, ie, $i (x) = x$ for all $x\in \Bbb{R}.$ Prove that $i \in A.$

In a triangle $\vartriangle ABC$, $\angle B = 2\angle C$. Let $P$ and $Q$ be points on the perpendicular bisector of segment $BC$ such that rays $AP$ and $AQ$ trisect $\angle A$. Prove that $PQ$ is smaller than $AB$ if and only if $\angle B$ is obtuse.

Let $ f : [0, 1] \to\ R $ be a function such that :- $1.)$ $f(x) \ge 0$ for all $x$ in $[0,1]$ . $2.)$ $f(1) = 1$ . $3.)$ If $x_1 , x_2$ are in $[0,1]$ such that $x_1 + x_2 \le 1$ , then $f(x_1) + f(x_2) \le f(x_1 + x_2)$ . Show that $f(x) \le 2x $ for all $x$ in $ [0,1] $.

A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length $ \textbf{(A)}\ 3\sqrt3 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 6\sqrt3 \qquad \textbf{(D)}\ 12\sqrt3 \qquad \textbf{(E)}\ \text{ none of these}$

Let $h(t)$ and $f(t)$ be polynomials such that $h(t)=t^2$ and $f_n(t)=h(h(h(h(h...h(t))))))-1$ where $h(t)$ occurs $n$ times. Prove that $f_n(t)$ is a factor of $f_N(t)$ whenever $n$ is a factor of $N$

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 $\cos b$.