# [Solutions] International Olympiad of Metropolises 2017

1. 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 $BF||AC$.
2. 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 $d$?
3. 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 $0.001$.
4. 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 $100$.
5. 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 $b$).
6. Let $ABCDEF$ 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.
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