1

Part of 2002 France Team Selection Test

Problems(2)

MA_1 is tangent to (A_1B_1C)

Source: French TST 2002

In an acute-angled triangle

ABC

A_1

B_1

are the feet of the altitudes from

A

B

respectively, and

M

is the midpoint of

AB

. a) Prove that

MA_1

is tangent to the circumcircle of triangle

A_1B_1C

. b) Prove that the circumcircles of triangles

A_1B_1C,BMA_1

AMB_1

have a common point.

geometrycircumcirclegeometric transformationreflectiongeometry proposed

Source:

There are three colleges in a town. Each college has

n

students. Any student of any college knows

n+1

students of the other two colleges. Prove that it is possible to choose a student from each of the three colleges so that all three students would know each other.

combinatoricsmaximum and minimum