Geometry (Thailand Math POSN 2nd round)

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1. Let $H$ be orthocenter of $\triangle ABC$ and point $I,J,K$ be a midpoint between each vertex $A,B,C$ and $H$ respectively. If $P$ is a point on circumcircle of $\triangle ABC$ not on the vertex $A,B,C$, and $M$ be the midpoint between $P,H$. Prove that $I,J,K,M$ lie on the same circle.

2. Let the tangents of circumcircle of $\triangle ABC$ at point $B,C$ intersect at point $D$. Prove that $\overline{AD}$ is symmedian line of $ABC$.

3. Let $P,Q$ be 2 points that are isogonal conjugate to each other in $\triangle ABC$. If $\overline{PP_{1}}$ and $\overline{QQ_{1}}$ is perpendicular to $\overline{BC}$ at point $P_{1}$ and $Q_{1}$ respectively, $\overline{PP_{2}}$ and $\overline{QQ_{2}}$ is perpendicular to $\overline{CA}$ at point $P_{2}$ and $Q_{2}$ respectively, and $\overline{PP_{3}}$ and $\overline{QQ_{3}}$ is perpendicular to $\overline{AB}$ at point $P_{3}$ and $Q_{3}$ respectively. Prove that $P_{1},P_{2},P_{3},Q_{1},Q_{2},Q_{3}$ lie on the same circle, and the center of that circle lies in the midpoint of $P$ and $Q$.

4. Let $I,N,H$ be the center of incircle, nine-point circle, and orthocenter of $\triangle ABC$ respectively. Construct $\overline{ID}, \overline{NM}$ perpendicular to $\overline{BC}$ at point $D,M$ respectively. If $\overline{AH}$ intersect circumcircle of $\triangle ABC$ at point $K$ such that $Y$ is a midpoint of $\overline{AK}$. Prove that $|ID - NM| = \displaystyle \left |r - \displaystyle \frac{AY}{2}\right|$ where $r$ is an inradius of $\triangle ABC$.

5. Let $U$ be a foot of altitude of $\triangle ABC$ from point $A$. If $U',U''$ are reflection of $U$ by $\overline{CA},\overline{AB}$ respectively, such that $\overline{U'U''}$ intersect $\overline{CA},\overline{AB}$ at point $V,W$ respectively. Prove that $\overline{BV},\overline{CW}$ is perpendicular to $\overline{CA},\overline{AB}$ respectively.

This note is a part of Thailand Math POSN 2nd round 2015.