Problem 4

Part of 2000 Mongolian Mathematical Olympiad

Problems(2)

functional inequality over R

Source: Mongolia MO 2000 Grade 10 P4

Suppose that a function

f:\mathbb R\to\mathbb R

satisfies the following conditions:
(i)

\left|f(a)-f(b)\right|\le|a-b|

a,b\in\mathbb R

f(f(f(0)))=0

f(0)=0

functional equationfeFunctional inequalityalgebra

distance between towns in country

Source: Mongolia MO 2000 Teachers P4

In a country with

n

towns, the distance between the towns numbered

i

j

x_{ij}

. Suppose that the total length of every cyclic route which passes through every town exactly once is the same. Prove that there exist numbers

a_i,b_i

i=1,\ldots,n

x_{ij}=a_i+b_j

for all distinct

i,j