4

Part of 1991 Romania Team Selection Test

Problems(2)

(a_m,a_n) = a_{(m,n)}, a_n = \prod_{d|n} b_d

Source: Romania BMO TST 1991 p4

(a_n)

of positive integers satisfies

(a_m,a_n) = a_{(m,n)}

m,n

. Prove that there is a unique sequence

(b_n)

of positive integers such that

a_n = \prod_{d|n} b_d

number theoryProductSequence

f : S \to (0,1) with $f(S_1 \cup S_2) = f(S_1)+ f(S_2), from set of polygonals

Source: Romania IMO TST 1991 p4

S

be the set of all polygonal areas in a plane. Prove that there is a function

f : S \to (0,1)

which satisfies

f(S_1 \cup S_2) = f(S_1)+ f(S_2)

S_1,S_2 \in S

which have common points only on their borders

functioncombinatorial geometrygeometry