2022 USAMO

Part of USAMO

Subcontests

(3)

1

Invert Your Expectations

\mathbb{R}_{>0}

be the set of all positive real numbers. Find all functions

f:\mathbb{R}_{>0} \to \mathbb{R}_{>0}

such that for all

x,y\in \mathbb{R}_{>0}

f(x) = f(f(f(x)) + y) + f(xf(y)) f(x+y).

1

Essentially Increasing Functions

f: \mathbb{R}\to \mathbb{R}

is essentially increasing if

f(s)\leq f(t)

s\leq t

are real numbers such that

f(s)\neq 0

f(t)\neq 0

.
Find the smallest integer

k

such that for any 2022 real numbers

x_1,x_2,\ldots , x_{2022},

k

essentially increasing functions

f_1,\ldots, f_k

f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.

1

Imagine Having >=2 Friends

2022

users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.)
Starting now, Mathbook will only allow a new friendship to be formed between two users if they have at least two friends in common. What is the minimum number of friendships that must already exist so that every user could eventually become friends with every other user?