Olympiad problems

2011 Laurențiu Duican, 2

Let be four real numbers $ x,y,z,t $ satisfying the following system: $$ \left\{ \begin{matrix} \sin x+\sin y+\sin z +\sin t =0 \\ \cos x+\cos y+\cos z+\cos t=0 \end{matrix} \right. $$ Prove that $$ \sin ((1+2k)x) +\sin ((1+2k)y) +\sin ((1+2k)z) +\sin ((1+2k)t) =0, $$ for any integer $ k. $ [i]Aurel Bârsan[/i]

2019 Middle European Mathematical Olympiad, 5

Tags: circumcircle , sinus , perpendicular bisector , geometry

Let $ABC$ be an acute-angled triangle such that $AB<AC$. Let $D$ be the point of intersection of the perpendicular bisector of the side $BC$ with the side $AC$. Let $P$ be a point on the shorter arc $AC$ of the circumcircle of the triangle $ABC$ such that $DP \parallel BC$. Finally, let $M$ be the midpoint of the side $AB$. Prove that $\angle APD=\angle MPB$. [i]Proposed by Dominik Burek, Poland[/i]

2007 Mathematics for Its Sake, 1

Tags: sinus , geometric relation , trigonometry , geometry

Find the angles of a triangle $ ABC $ in which $ \frac{\sin A}{\sin B} +\frac{\sin B}{\sin C} +\frac{\sin C}{\sin A} =3. $