5

Part of 1995 IMO Shortlist

Problems(3)

f(x + 1/x^2) = f(x) + [f(1/x)]^2

Source: IMO Shortlist 1995, A5

\mathbb{R}

be the set of real numbers. Does there exist a function

f: \mathbb{R} \mapsto \mathbb{R}

which simultaneously satisfies the following three conditions?
(a) There is a positive number

M

\forall x:

\minus{} M \leq f(x) \leq M. (b) The value of

f(1)

1

x \neq 0,

then f \left(x \plus{} \frac {1}{x^2} \right) \equal{} f(x) \plus{} \left[ f \left(\frac {1}{x} \right) \right]^2

functionalgebrafunctional equationIMO Shortlist

How many people are at the meeting?

Source: IMO Shortlist 1995, N5

At a meeting of

12k

people, each person exchanges greetings with exactly 3k\plus{}6 others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?

combinatoricsgraph theoryExtremal combinatoricsIMO Shortlist

Arithmetic progression of positive integers

Source: IMO Shortlist 1995, S5

For positive integers

n,

f(n)

are defined inductively as follows: f(1) \equal{} 1, and for every positive integer

n,

f(n\plus{}1) is the greatest integer

m

such that there is an arithmetic progression of positive integers a_1 < a_2 < \ldots < a_m \equal{} n for which f(a_1) \equal{} f(a_2) \equal{} \ldots \equal{} f(a_m). Prove that there are positive integers

a

b

such that f(an\plus{}b) \equal{} n\plus{}2 for every positive integer

n.

arithmetic sequencealgebrarecurrence relationfunctionIMO Shortlist