2

Part of 1990 China Team Selection Test

Problems(2)

polygon is cut by at least one of these m lines

Source: China TST 1990, problem 2

Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin

O

that cuts them both, then these polygons are called "properly placed". Find the least

m \in \mathbb{N}

, such that for any group of properly placed polygons,

m

lines can drawn through

O

and every polygon is cut by at least one of these

m

geometry unsolvedgeometry

Find three functions

Source: China TST 1990, problem 6

Find all functions

f,g,h: \mathbb{R} \mapsto \mathbb{R}

f(x) - g(y) = (x-y) \cdot h(x+y)

x,y \in \mathbb{R}.

functionalgebra unsolvedalgebra