Georgia TST 2005 4p

geometry problems from Team Selection Tests (TST) from Georgia with aops links in the names

2005

2005 Georgia TST 1.2

In triangle $ABC$ we have $\angle{ACB} = 2\angle{ABC}$ and there exists the point $D$ inside the triangle such that $AD = AC$ and $DB = DC$. Prove that $\angle{BAC} = 3\angle{BAD}$.

2005 Georgia TST 2.2

Let $ABCD$ be a convex quadrilateral. Points $P,Q$ and $R$ are the feets of the perpendiculars from point $D$ to lines $BC, CA$ and $AB$, respectively. Prove that $PQ=QR$ if and only if the bisectors of the angles $ABC$ and $ADC$ meet on segment $AC$.

2005 Georgia TST 3.2

In a convex quadrilateral $ABCD$ the points $P$ and $Q$ are chosen on the sides $BC$ and $CD$ respectively so that $\angle{BAP}=\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ABP$ and $ADQ$, is perpendicular to $AC$ if and only if the triangles $ABP$ and $ADQ$ have the same areas.

2005 Georgia TST 4.2

On the sides $AB, BC, CD$ and $DA$ of the rhombus $ABCD$, respectively, are chosen points $E, F, G$ and $H$ so, that $EF$ and $GH$ touch the incircle of the rhombus. Prove that the lines $EH$ and $FG$ are parallel.