Olympiad problems

2011 Iran MO (3rd Round), 5

Given triangle $ABC$, $D$ is the foot of the external angle bisector of $A$, $I$ its incenter and $I_a$ its $A$-excenter. Perpendicular from $I$ to $DI_a$ intersects the circumcircle of triangle in $A'$. Define $B'$ and $C'$ similarly. Prove that $AA',BB'$ and $CC'$ are concurrent. [i]proposed by Amirhossein Zabeti[/i]

2009 China Team Selection Test, 1

Tags: geometry , circumcircle , trigonometry , reflection , geometric transformation , trapezoid , symmetry

Given that points $ D,E$ lie on the sidelines $ AB,BC$ of triangle $ ABC$, respectively, point $ P$ is in interior of triangle $ ABC$ such that $ PE \equal{} PC$ and $ \bigtriangleup DEP\sim \bigtriangleup PCA.$ Prove that $ BP$ is tangent of the circumcircle of triangle $ PAD.$

2004 Postal Coaching, 10

Tags: angle bisector , geometric transformation , geometry , perpendicular bisector , reflection

A convex quadrilateral $ABCD$ has an incircle. In each corner a circle is inscribed that also externally touches the two circles inscribed in the adjacent corners. Show that at least two circles have the same size.

2000 All-Russian Olympiad, 7

Tags: trigonometry , geometry , geometric transformation , reflection , angle bisector

A quadrilateral $ABCD$ is circumscribed about a circle $\omega$. The lines $AB$ and $CD$ meet at $O$. A circle $\omega_1$ is tangent to side $BC$ at $K$ and to the extensions of sides $AB$ and $CD$, and a circle $\omega_2$ is tangent to side $AD$ at $L$ and to the extensions of sides $AB$ and $CD$. Suppose that points $O$, $K$, $L$ lie on a line. Prove that the midpoints of $BC$ and $AD$ and the center of $\omega$ also lie on a line.

2010 Tournament Of Towns, 4

Tags: geometry , combinatorics , rectangle , analytic geometry , geometric transformation , reflection

A rectangle is divided into $2\times 1$ and $1\times 2$ dominoes. In each domino, a diagonal is drawn, and no two diagonals have common endpoints. Prove that exactly two corners of the rectangle are endpoints of these diagonals.

1992 AIME Problems, 11

Tags: geometric transformation , modular arithmetic , rotation , reflection , geometry

Lines $l_1$ and $l_2$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l$, the transformation $R(l)$ produces another line as follows: $l$ is reflected in $l_1$, and the resulting line is reflected in $l_2$. Let $R^{(1)}(l)=R(l)$ and $R^{(n)}(l)=R\left(R^{(n-1)}(l)\right)$. Given that $l$ is the line $y=\frac{19}{92}x$, find the smallest positive integer $m$ for which $R^{(m)}(l)=l$.