MathDB

1

Part of BIMO 2021

Problems(2)

non-trivial divisors add up to n-1

Source: Own. IMO 2021 Malaysian Training Camp 1

12/31/2020
Given a natural number nn, call a divisor dd of nn to be <spanclass=′latex−italic′>nontrivial</span><span class='latex-italic'>nontrivial</span> if d>1d>1. A natural number nn is <spanclass=′latex−italic′>good</span><span class='latex-italic'>good</span> if one or more distinct nontrivial divisors of nn sum up to n−1n-1.
Prove that every natural number nn has a multiple that is good.
number theoryDivisorsPerfect Numbers
f(x^2+f(y))=f(f(y)-x^2)+f(xy)

Source: Own. IMO 2021 Malaysian Training Camp 1

12/31/2020
Find all continuous functions f:R→R f : \mathbb{R} \rightarrow \mathbb{R} such that for all real numbers x,y x, y f(x2+f(y))=f(f(y)−x2)+f(xy) f(x^2+f(y))=f(f(y)-x^2)+f(xy)
[Extra: Can you solve this without continuity?]
algebra