5

Part of 2023 Rioplatense Mathematical Olympiad

Problems(2)

Find out alpha to solve the FE

Source: Rioplatense L3 2023 #5

\mathbb{R}^{+}

be the set of positive real numbers. Determine all non-negative real number

\alpha

such that there exist a function

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

f(x^{\alpha}+y)=(f(x+y))^{\alpha}+f(y)

x,y

positive real numbers.

functionalgebra

34 works, 30 not works

Source: Rioplatense L2 2023 #5

A positive integer

N

is rioplatense if it satifies the following conditions: 1 -There exist

34

consecutive integers such that its product is divisible by

N

, but none of them is divisible by

N

. 2 - There not exist

30

consecutive integers such that its product is divisible by

N

, but none of them is divisible by

N

. Determine all rioplatense numbers.