IOM 2017

Let $ABCD$ be a parallelogram in which angle at $B$ is obtuse and $AD>AB$ . Points $K$ and $L$ on $AC$ such that $\angle ADL=\angle KBA$ (the points $A, K, C, L$ are all different, with $K$ between $A$ and $L$ ). The line $BK$ intersects the circumcircle $\omega$ of $ABC$ at points $B$ and $E$ , and the line $EL$ intersects $\omega$ at points $E$ and $F$ . Prove that $BF||AC$ .

In a country there are two-way non-stopflights between some pairs of cities. Any city can be reached from any other by a sequence of at most $100$ flights. Moreover, any city can be reached from any other by a sequence of an even number of flights. What is the smallest $d$ for which one can always claim that any city can be reached from any other by a sequence of an even number of flights not exceeding $d$ ?

Let $Q$ be a quadriatic polynomial having two different real zeros. Prove that there is a non-constant monic polynomial $P$ such that all coefficients of the polynomial $Q(P(x))$ except the leading one are (by absolute value) less than $0.001$ .

Find the largest positive integer $N$ for which one can choose $N$ distinct numbers from the set ${1,2,3,...,100}$ such that neither the sum nor the product of any two different chosen numbers is divisible by $100$ .

Proposed by Mikhail Evdokimov

Let $x$ and $y$ be positive integers such that $[x+2,y+2]-[x+1,y+1]=[x+1,y+1]-[x,y]$ .Prove that one of the two numbers $x$ and $y$ divide the other.

(Here $[a,b]$ denote the least common multiple of $a$ and $b$ ).

Proposed by Dusan Djukic.

et $ABCDEF$ be a convex hexagon which has an inscribed circle and a circumcribed. Denote by $\omega_{A}, \omega_{B},\omega_{C},\omega_{D},\omega_{E}$ and $\omega_{F}$ the inscribed circles of the triangles $FAB, ABC, BCD, CDE, DEF$ and $EFA$ , respecitively. Let $l_{AB}$ , be the external of $\omega_{A}$ and $\omega_{B}$ ; lines $l_{BC}$ , $l_{CD}$ , $l_{DE}$ , $l_{EF}$ , $l_{FA}$ are analoguosly defined. Let $A_1$ be the intersection point of the lines $l_{FA}$ and $l_{AB}$ , $B_1, C_1, D_1, E_1, F_1$ are analogously defined. Prove that $A_1D_1, B_1E_1, C_1F_1$ are concurrent.