3

Part of 2021 239 Open Mathematical Olympiad

Problems(2)

Number Theory about sequence

Source: 239 MO (8-9).2/(10-11).3

Given are two distinct sequences of positive integers

(a_n)

(b_n)

, such that their first two members are coprime and smaller than

1000

, and each of the next members is the sum of the previous two. 8-9 grade Prove that if

a_n

is divisible by

b_n

n<50

10-11 grade Prove that if

a_n^{100}

is divisible by

b_n

n<5000

Graph Theory about odd cycles

Source: 239 MO 2021 (8-9).3

Given is a simple graph with

239

vertices, such that it is not bipartite and each vertex has degree at least

3

. Find the smallest

k

, such that each odd cycle has length at most

k