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IMO Shortlist 2006 problem G7

In a triangle $ABC$ , let $M_{a}$ , $M_{b}$ , $M_{c}$ be the midpoints of the sides $BC$ , $CA$ , $AB$ , respectively, and $T_{a}$ , $T_{b}$ , $T_{c}$ be the midpoints of the arcs $BC$ , $CA$ , $AB$ of the circumcircle of $ABC$ , not containing the vertices $A$ , $B$ , $C$ , respectively. For $i \in \left\{a, b, c\right\}$ , let $w_{i}$ be the circle with $M_{i}T_{i}$ as diameter. Let $p_{i}$ be the common external common tangent to the circles $w_{j}$ and $w_{k}$ (for all $\left\{i, j, k\right\}= \left\{a, b, c\right\}$ ) such that $w_{i}$ lies on the opposite side of $p_{i}$ than $w_{j}$ and $w_{k}$ do.
Prove that the lines $p_{a}$ , $p_{b}$ , $p_{c}$ form a triangle similar to $ABC$ and find the ratio of similitude.

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IMO Shortlist 2004 problem G7

For a given triangle $ABC$ , let $X$ be a variable point on the line $BC$ such that $C$ lies between $B$ and $X$ and the incircles of the triangles $ABX$ and $ACX$ intersect at two distinct points $P$ and $Q.$ Prove that the line $PQ$ passes through a point independent of $X$ .

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An extension by Darij Grinberg can be found here.

IMO Shortlist 2004 problem G8

Given a cyclic quadrilateral $ABCD$ , let $M$ be the midpoint of the side $CD$ , and let $N$ be a point on the circumcircle of triangle $ABM$ . Assume that the point $N$ is different from the point $M$ and satisfies $\frac{AN}{BN}=\frac{AM}{BM}$ . Prove that the points $E$ , $F$ , $N$ are collinear, where $E=AC\cap BD$ and $F=BC\cap DA$ .

IMO Shortlist 2004 problem N7

Let $p$ be an odd prime and $n$ a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length $p^{n}$ . Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by $p^{n+1}$ .

IMO Shortlist 2006 problem G5

In triangle $ABC$ , let $J$ be the center of the excircle tangent to side $BC$ at $A_{1}$ and to the extensions of the sides $AC$ and $AB$ at $B_{1}$ and $C_{1}$ respectively. Suppose that the lines $A_{1}B_{1}$ and $AB$ are perpendicular and intersect at $D$ . Let $E$ be the foot of the perpendicular from $C_{1}$ to line $DJ$ . Determine the angles $\angle{BEA_{1}}$ and $\angle{AEB_{1}}$ .

IMO Shortlist 2006 problem G9

Points $A_{1}$ , $B_{1}$ , $C_{1}$ are chosen on the sides $BC$ , $CA$ , $AB$ of a triangle $ABC$ respectively. The circumcircles of triangles $AB_{1}C_{1}$ , $BC_{1}A_{1}$ , $CA_{1}B_{1}$ intersect the circumcircle of triangle $ABC$ again at points $A_{2}$ , $B_{2}$ , $C_{2}$ respectively ( $A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$ ). Points $A_{3}$ , $B_{3}$ , $C_{3}$ are symmetric to $A_{1}$ , $B_{1}$ , $C_{1}$ with respect to the midpoints of the sides $BC$ , $CA$ , $AB$ respectively. Prove that the triangles $A_{2}B_{2}C_{2}$ and $A_{3}B_{3}C_{3}$ are similar.

IMO Shortlist 2007 problem G8

Point $P$ lies on side $AB$ of a convex quadrilateral $ABCD$ . Let $\omega$ be the incircle of triangle $CPD$ , and let $I$ be its incenter. Suppose that $\omega$ is tangent to the incircles of triangles $APD$ and $BPC$ at points $K$ and $L$ , respectively. Let lines $AC$ and $BD$ meet at $E$ , and let lines $AK$ and $BL$ meet at $F$ . Prove that points $E$ , $I$ , and $F$ are collinear.

Author: Waldemar Pompe, Poland