my JBMO Shortlist solutions

here I shall collect in one place links to my solutions from Junior Balkan Math Olympiad Shortlist

2010 JBMO Shortlist G1 (here)

Consider a triangle ${ABC}$ with${\angle ACB=90^{\circ}.}$ Let ${F}$ be the foot of the altitude from ${C}$. Circle ${\omega}$ touches the line segment ${FB}$at point ${P,}$  the altitude ${CF}$at point ${Q}$ and the circumcircle of ${ABC}$at point ${R.}$ Prove that points ${A,Q,R}$ are collinear and ${AP=AC}$.

2011 JBMO Shortlist G1 (here)
Let $ABC$ be an isosceles triangle with $AB=AC$. On the extension of the side ${CA}$ we consider the point ${D}$ such that ${AD<AC}$. The perpendicular bisector of the segment ${BD}$ meets the internal and the external bisectors of the angle $\angle BAC$ at the points ${E}$ and ${Z}$, respectively. Prove that the points ${A, E, D, Z}$ are concyclic. 

2011 JBMO Shortlist G2 (here)

Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$.

2011 JBMO Shortlist G3 (here)

Let $ABC$ be a triangle in which (${BL}$is the angle bisector of ${\angle{ABC}}$ $\left( L\in AC \right)$, ${AH}$ is an altitude of$\vartriangle ABC$ $\left( H\in BC \right)$ and ${M}$is the midpoint of the side ${AB}$. It is known  that the midpoints of the segments ${BL}$ and ${MH}$ coincides. Determine the internal angles of triangle $\vartriangle ABC$. 

2012 JBMO Shortlist G2 (here)

Let $ABC$ be an isosceles triangle with $AB=AC$. Let also ${c\left(K, KC\right)}$ be a circle tangent to the line ${AC}$ at point${C}$ which it intersects the segment ${BC}$ again at an interior point ${H}$. Prove that ${HK\perp AB}$.

2013 JBMO Shortlist G1 (here)
Let ${AB}$ be a diameter of a circle  ${\omega}$ and center ${O}$ ,  ${OC}$ a radius of ${\omega}$ perpendicular to $AB$,${M}$ be a point of the segment $\left( OC \right)$ . Let ${N}$ be the second intersection point of line ${AM}$ with ${\omega}$ and ${P}$ the intersection point of the tangents of ${\omega}$ at points ${N}$ and ${B.}$ Prove that points ${M,O,P,N}$ are cocyclic.

2014 JBMO Shortlist G1 (here)

Let $ABC$ be a triangle with $m\left( \angle B \right)=m\left( \angle C \right)={{40}^{{}^\circ }}$ Line bisector of ${\angle{B}}$ intersects ${AC}$ at point ${D}$. Prove that $BD+DA=BC$.

2015 JBMO Shortlist G1 (here)

Around the triangle $ABC$ the circle is circumscribed, and at the vertex ${C}$ tangent ${t}$ to this circle is drawn. The line ${p}$, which is parallel to this tangent intersects the lines ${BC}$ and ${AC}$ at the points ${D}$ and ${E}$, respectively. Prove that the points $A,B,D,E$ belong to the same circle.