+98 Hi everybody! This is the first set of 5 problem sets of problems I will be giving out over a week.
P1 - Easy Combinatorics
Given $20$ apples and $5$ people, how many ways are there to distribute the apples among the people so that each person gets at least one?
P2 - Medium Number Theory
Show that $n^7-n$ is divisible by $42$ for every positive integer $n$.
Source: Putnam Number Theory
P3 - Very Hard Functional Equation
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that for any positive real numbers $x$ and $y$,
$$f(x+f(x)+f(y))=x+f(x+y)$$
Proposed by Athanasios Kontogeorgis, Grecce, and Dorlir Ahmeti, Kosovo
Source: FEOO Shortlist A5
Have fun solving them! If you are stuck, just message me and I'll send a hint.
