2

Part of 2008 Czech and Slovak Olympiad III A

Problems(2)

czech republic,czech and slovakia MO(third round),2008 p2

Source:

2/25/2012

Two disjoint circles

W_1(S_1,r_1)

and

W_2(S_2,r_2)

are given in the plane. Point

A

is on circle

W_1

and

AB,AC

touch the circle

W_2

B,C

respectively. Find the loci of the incenter and orthocenter of triangle

ABC

geometryincentergeometry proposed

czech republic,czech and slovakia MO(third round),2008 p5

Source:

2/25/2012

At one moment, a kid noticed that the end of the hour hand, the end of the minute hand and one of the twelve numbers (regarded as a point) of his watch formed an equilateral triangle. He also calculated that

t

hours would elapse for the next similar case. Suppose that the ratio of the lengths of the minute hand (whose length is equal to the distance from the center of the watch plate to any of the twelve numbers) and the hour hand is

k>1

. Find the maximal value of

t

rationumber theory proposednumber theory