Problem 3

Part of 2000 Mongolian Mathematical Olympiad

Problems(2)

cutting up cube into unit cubes

Source: Mongolia MO 2000 Grade 10 P3

n

n^3

unit cubes, and m of these cubes are marked so that the centers of any three marked cubes do not form a right-angled triangle with legs parallel to sides of the cube. Find the maximum possible value of

m

geometry3D geometrycombinatorics

points moving around circles

Source: Mongolia MO 2000 Teachers P3

A

B

move around two different circles in the plane with the same angular velocity. Suppose that there is a point

C

which is equidistant from

A

B

at every moment. Prove that, at some moment,

A

B