Gulf Mathematical Olympiad 2012-19 (GMO) 7p

geometry problems from (Arab) Gulf Mathematical Olympiads (GMO)       
with aops links in the names

collected inside aop here

it did not take place in 2018

2012 - 2019

2012 Gulf MO p1
Let $X,\ Y$ and $Z$ be the midpoints of sides $BC,\ CA$, and $AB$ of the triangle $ABC$, respectively. Let $P$ be a point inside the triangle. Prove that the quadrilaterals $AZPY,\ BXPZ$, and $CYPX$ have equal areas if, and only if, $P$ is the centroid of $ABC$.

2013 Gulf MO p2
In triangle $ABC$, the bisector of angle $B$ meets the opposite side $AC$ at $B'$. Similarly, the bisector of angle $C$ meets the opposite side $AB$ at $C'$ . Prove that $A=60^{\circ}$ if, and only if, $BC'+CB'=BC$.

2014 Gulf MO p3
(i) $ABC$ is a triangle with a right angle at $A$, and $P$ is a point on the hypotenuse $BC$. The line $AP$ produced beyond $P$ meets the line through $B$ which is perpendicular to $BC$ at $U$. Prove that $BU = BA$ if, and only if, $CP = CA$.

(ii) $A$ is a point on the semicircle $CB$, and points $X$ and $Y$ are on the line segment $BC$. The line $AX$, produced beyond $X$, meets the line through $B$ which is perpendicular to $BC$ at $U$. Also the line $AY$, produced beyond $Y$, meets the line through $C$ which is perpendicular to $BC$ at $V$. Given that $BY = BA$ and $CX = CA$, determine the angle $\angle VAU$.

2015 Gulf MO p2
a) Let $UVW$ , $U'V'W'$ be two triangles such that $ VW = V'W' , UV = U'V' , \angle WUV = \angle W'U'V'.$ Prove that the angles $\angle VWU , \angle V'W'U'$ are equal or supplementary.

b) $ABC$ is a triangle where $\angle A$ is obtuse. take a point $P$ inside the triangle , and extend $AP,BP,CP$ to meet the sides $BC,CA,AB$ in $K,L,M$ respectively. Suppose that $PL = PM .$
i) If $AP$ bisects $\angle A$ , then prove that $AB = AC$ .
ii) Find the angles of the triangle $ABC$ if you know that $AK,BL,CM$ are angle bisectors of the triangle $ABC$ and that $2AK = BL$.

2016 Gulf MO p3
Consider the acute-angled triangle $ABC$. Let $X$ be a point on the side $BC$, and $Y$ be a point on the side $CA$. The circle $k_1$ with diameter $AX$ cuts $AC$ again at $E'$ .The circle $k_2$ with diameter $BY$ cuts $BC$ again at $B'$.
(i) Let $M$ be the midpoint of $XY$ . Prove that $A'M = B'M$.
(ii) Suppose that $k_1$ and $k_2$ meet at $P$ and $Q$. Prove that the orthocentre of $ABC$ lies on the line $PQ$.

2017 Gulf MO p3
Let $C_1$ and $C_2$ be two different circles , and let their radii be $r_1$ and $r_2$ , the two circles are passing through the two points $A$ and $B$
(i) Let $P_1$ on $C_1$ and $P_2$ on $C_2$ such that the line $P_1P_2$ passes through $A$. Prove that $P_1B \cdot  r_2 = P_2B \cdot r_1$
(ii) Let $DEF$ be a triangle that it's inscribed in $C_1$ , and let $D'E'F'$ be a triangle that is inscribed in $C_2$ . The lines $EE'$,$DD'$ and $FF'$ all pass through $A$ . Prove that the triangles $DEF$ and $D'E'F'$ are similar
(iii)The circle $C_3$ also passes through $A$ and $B$ . Let $l$ be a line that passes through $A$ and cuts circles $C_i$ in $M_i$ with $i = 1,2,3$ . Prove that the value of$$\frac{M_1M_2}{M_1M_3}$$is constant regardless of the position of $l$ Provided that $l$ is different from $AB$

it did not take place in 2018

2019 Gulf MO p1

Let $ABCD$ be a trapezium (trapezoid) with $AD$ parallel to $BC$  and $J$ be the intersection of the diagonals $AC$ and $BD$. Point $P$ a chosen on the side $BC$ such that the distance from $C$ to the line $AP$ is equal to the distance from $B$ to the line $DP$.

\The following three questions 1,  2 and 3 are independent, so that a condition in one question does not apply in another question.

1.Suppose that  $Area( \vartriangle AJB) =6$ and that $Area(\vartriangle BJC) = 9$. Determine $Area(\vartriangle APD)$.
2. Find all points $Q$ on the plane of the trapezium such that  $Area(\vartriangle AQB) = Area(\vartriangle DQC)$.
3. Prove that $PJ$ is the angle bisector of $\angle APD$.