3

Part of 2008 Poland - Second Round

Problems(2)

Not very attractive functional equation.

Source: Polish MO 2008 Second Round (1st day)

Find all functions

f: \mathbb{R} \rightarrow \mathbb{R}

for which the equality f(f(x)\minus{}y)\equal{}f(x)\plus{}f(f(y)\minus{}f(\minus{}x))\plus{}x holds for all real

x,y

functionalgebra proposedalgebra

Multiplications and digits

Source: Polish Mathematical Olympiad Second Round (day 2)

We have a positive integer

n

n \neq 3k

. Prove that there exists a positive integer

m

\forall_{k\in N \ k\geq m} \ k

can be represented as a sum of digits of some multiplication of

n

number theoryrelatively primenumber theory proposed