3

Part of 2007 Tuymaada Olympiad

Problems(2)

altitudes and circles tangent to the circumcircle

Source: Tuymaada 2007, Problem 3

AA_{1}

BB_{1}

CC_{1}

are altitudes of an acute triangle

ABC

. A circle passing through

A_{1}

B_{1}

touches the arc

AB

of its circumcircle at

C_{2}

A_{2}

B_{2}

are defined similarly. Prove that the lines

AA_{2}

BB_{2}

CC_{2}

are concurrent.

geometrycircumcircleGauss

Sasha playing with knights...

Source: Tuymaada 2007, Problem 7

Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a

14\times 16

rectangle. What maximum number of knights can this rectangle contain?

geometryrectangleanalytic geometrycombinatorics proposedcombinatorics