2

Part of 2013 China National Olympiad

Problems(2)

Source: 2013 China Mathematical Olympaid P5

For any positive integer

n

0 \leqslant i \leqslant n

C_n^i \equiv c(n,i)\pmod{2}

c(n,i) \in \left\{ {0,1} \right\}

f(n,q) = \sum\limits_{i = 0}^n {c(n,i){q^i}}

m,n,q

are positive integers and

q + 1 \ne {2^\alpha }

\alpha \in \mathbb N

. Prove that if

f(m,q)\left| {f(n,q)} \right.

f(m,r)\left| {f(n,r)} \right.

for any positive integer

r

modular arithmeticalgebrapolynomialfunctionnumber theory proposednumber theory

Source: 2013 China Mathematical Olympaid P2

Find all nonempty sets

S

of integers such that

3m-2n \in S

for all (not necessarily distinct)

m,n \in S

inequalitiesalgebra proposedalgebra