MathDB

2

Part of 2000 CentroAmerican

Problems(2)

Tiling a 15xn board

Source: Central American Olympiad 2000, problem 2

8/8/2007
Determine all positive integers n n such that it is possible to tile a 15×n 15 \times n board with pieces shaped like this: [asy]size(100); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(1,2)); draw((2,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); draw((5,0)--(6,0)); draw((4,1)--(7,1)); draw((4,2)--(7,2)); draw((5,3)--(6,3)); draw((4,1)--(4,2)); draw((5,0)--(5,3)); draw((6,0)--(6,3)); draw((7,1)--(7,2));[/asy]
modular arithmetic
Circles with AB and AC as diameters

Source: Central American Olympiad 2000, problem 5

8/9/2007
Let ABC ABC be an acute-angled triangle. C1 C_{1} and C2 C_{2} are two circles of diameters AB AB and AC AC, respectively. C2 C_{2} and AB AB intersect again at F F, and C1 C_{1} and AC AC intersect again at E E. Also, BE BE meets C2 C_{2} at P P and CF CF meets C1 C_{1} at Q Q. Prove that AP=AQ AP=AQ.
trigonometry