IMO Shortlist 1967 problem 5
Dodao/la:
arhiva2. travnja 2012. A linear binomial
![l(z) = Az + B](/media/m/b/4/0/b4097c9b5439a19b43d01a3aa175832b.png)
with complex coefficients
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
is given. It is known that the maximal value of
![|l(z)|](/media/m/a/6/a/a6af8264cf101019d296a3f8331052f2.png)
on the segment
![(y = 0)](/media/m/0/3/6/0364f98916f41ca6dce92a179774b8dd.png)
of the real line in the complex plane
![z = x + iy](/media/m/e/2/d/e2d9fbcd5006fcabb288e5634c9b5c41.png)
is equal to
![M.](/media/m/6/2/9/62959f9153a30be543f678d987317439.png)
Prove that for every
where
![\rho](/media/m/5/5/f/55fdf949d51a428d16e54946e2077a37.png)
is the sum of distances from the point
![P=z](/media/m/2/2/0/220df4655eb732adf4c331e0ccd42de2.png)
to the points
![Q_1: z = 1](/media/m/7/4/e/74e40ad3e65bba557c26da0715069ed6.png)
and
%V0
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 $Q_3: z = -1.$
Izvor: Međunarodna matematička olimpijada, shortlist 1967