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IMO LongList 1967, Soviet Union 5

Source: IMO LongList 1967, Soviet Union 5

December 16, 2004

geometryfunctionalgebraInequalityLinear FunctionIMO ShortlistIMO Longlist

Problem Statement

A linear binomial

l(z) = Az + B

with complex coefficients

A

and

B

is given. It is known that the maximal value of

|l(z)|

on the segment

-1 \leq x \leq 1

(y = 0)

of the real line in the complex plane

z = x + iy

is equal to

M.

Prove that for every

z

|l(z)| \leq M \rho,

where

\rho

is the sum of distances from the point

P=z

to the points

Q_1: z = 1

and

Q_3: z = -1.

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