3

Part of 2001 Hungary-Israel Binational

Problems(2)

f (f (x)) = f (x) + x(maybe is old)

Source: 12-th Hungary-Israel Binational Mathematical Competition 2001

Find all continuous functions

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

such that for all

x \in\mathbb{ R}

f (f (x)) = f (x)+x.

functionalgebra proposedalgebra

prove that $G_n$ contains $C_4$

Source: 12-th Hungary-Israel Mathematical Competition 2001

G_{n}

denotes a simple undirected graph with

n

K_{n}

denotes the complete graph with

n

K_{n,m}

the complete bipartite graph whose components have

m

n

C_{n}

n

vertices. The number of edges in the graph

G_{n}

e(G_{n})

e(G_{n}) \geq\frac{n\sqrt{n}}{2}+\frac{n}{4}

G_{n}

C_{4}

graph theorycombinatorics proposedcombinatorics