This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 6

2015 Geolympiad Spring, 3
Tags: Geolympiad Spring 2015
Let $ABC$ be an acute triangle with orthocenter $H$, incenter $I$, and excenters $I_A, I_B, I_C$. Show that $II_A * II_B * II_C \ge 8 AH * BH * CH$.

2015 Geolympiad Spring, 1
Tags: Geolympiad Spring 2015
Let $ABC$ be a triangle. Suppose $P,Q$ are on lines $AB, AC$ (on the same side of A) with $AP=AC$ and $AB=AQ$. Now suppose points $X,Y$ move along the sides $AB, AC$ of $ABC$ so that $XY || PQ$. Determine the locus of the circumcenters of the variable triangle $AXY$.

2015 Geolympiad Spring, 4
Tags: Geolympiad Spring 2015
Let $ABC$ be an acute triangle with $\angle A = 60$ and altitudes $BE, CF$. Suppose $BE, CF$ are reflected across the perpendicular bisector of $BC$ and the two new segments $B'E', C'F'$ intersect at a point $X$. If $A$ is reflected across $BC$ to form $A'$, show that $AX$ is bisected by the internal angle bisector of $A$.

2015 Geolympiad Spring, 6
Tags: Geolympiad Spring 2015
Let $ABC$ be a triangle, $X$ the midpoint of arc $BC$ on the circumcircle. The tangents from $X$ to the incircle meet the circumcircle again at $X_1,X_2$, and $X_1X_2$ intersects the incircle at $P,Q$. Let $M$ be the midpoint of $PQ$, and let $A_1$ be the tangency point of the $A$-mixtillinear incircle with the circumcircle. Show that $A,M,A_1$ are collinear.

2015 Geolympiad Spring, 5
Tags: Geolympiad Spring 2015
Let $ABC$ be a triangle with circumcircle $w_1$ and incenter $I$. Suppose $w_2$ is a circle tangent to $AB,AC$ at $X,Y$, and internally tangent to $w$ at $D$. Let the parallel to the exterior angle bisector of $A$ through $D$ meet $w_2$ at $P$. Show that $AP, DI$ intersect on $w_2$.

2015 Geolympiad Spring, 2
Tags: Geolympiad Spring 2015
Let $ABC$ be a triangle and $w$ its incircle. $w$ touches $BC,CA$ at $A_1,B_1$ respectively. The second intersection of $AA_1$ and $w$ is $A_2$, similarly define $B_2$. Then $AB,A_1B_1,A_2B_2$ concur at a point $C_3$.