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 Summer, 2.
Tags: Geolympiad Summer 2015
Let $ABC$ be a triangle. Let line $\ell$ be the line through the tangency points that are formed when the tangents from $A$ to the circle with diameter $BC$ are drawn. Let line $m$ be the line through the tangency points that are formed when the tangents from $B$ to the circle with diameter $AC$ are drawn. Show that the $\ell$, $m$, and the $C$-altitude concur.

2015 Geolympiad Summer, 4.
Tags: Geolympiad Summer 2015
Let $ABC$ be a triangle and $I$ be its incenter. Let $D$ be the intersection of the exterior bisectors of $\angle BAC$ and $\angle BIC$, $E$ be the intersection of the exterior bisectors of $\angle ABC$ and $\angle AIC$, and $F$ be the intersection of the exterior bisectors of $\angle ACB$ and $\angle AIB$. Prove that $D$, $E$, $F$ are collinear

2015 Geolympiad Summer, 3.
Tags: Geolympiad Summer 2015
Let $ABC$ be an acute scalene triangle with incenter $I$, circumcircle $w_1$, and denote the circumcircle of $BIC$ as $w_2$. Suppose point $P$ lies on $w_2$ and is inside $w_1$. Let $X,Y$ lie on $BC$ with $XP \perp BP, YP \perp PC$. Circles $O_1, O_2$ are drawn tangent to $w_1$ at points on the same side of $BC$ as $A$ and tangent to $BC$ at $X,Y$ respectively. Let the centers of those two circles be $Z_1, Z_2$. Let $D$ be the point on $w_2$ opposite to $P$ and let $E$ be the foot of the altitude from $P$ to $BC$. Show that $DE \perp Z_1Z_2$

2015 Geolympiad Summer, 6.
Tags: Geolympiad Summer 2015
Let $w_1, w_2$ be non-intersecting, congruent circles with centers $O_1, O_2$ and let $P$ be in the exterior of both of them. The tangents from $P$ to $w_1$ meet $w_1$ at $A_1, B_1$ and define $A_2, B_2$ similarly. If lines $A_1B_1, A_2B_2$ meet at $Q$ show that the midpoint of $PQ$ is equidistant from $O_1, O_2$.

2015 Geolympiad Summer, 1.
Tags: Geolympiad Summer 2015
Show in an acute triangle $ABC$ that $\cot A + \cot B + \cot C \ge \dfrac{12[ABC]}{a^2+b^2+c^2}$.

2015 Geolympiad Summer, 5.
Tags: Geolympiad Summer 2015
Let $ABC$ be a triangle and $P$ be in its interior. Let $Q$ be the isogonal conjugate of $P$. Show that $BCPQ$ is cyclic if and only if $AP=AQ$.