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.

CitrusCornflakes Feb 13, 2021