Find the greatest possible value of the expression

Art of Problem Solving

AoPS Online

College Math Topics in undergraduate and graduate studies

Topics in undergraduate and graduate studies

3 M G

BBookmark VNew Topic kLocked

College Math Topics in undergraduate and graduate studies

Topics in undergraduate and graduate studies

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

Find the greatest possible value of the expression

G H J

Reveal topic tags

complex numbers

L

G H BBookmark kLocked kLocked NReply

Source: Yandex Uzbekistan Coding and Math Contest 2025

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

409 posts

#1 h Apr 22, 2025, 11:56 AM

Y by

Let $a, b, c, d$ be complex numbers with $|a| \le 1, |b| \le 1, |c| \le 1, |d| \le 1$ . Find the greatest possible value of the expression $|ac+ad+bc-bd|.$

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

10587 posts

#2 h Apr 23, 2025, 6:34 AM • 1 Y

Y by kiyoras_2001

We have $|c(a+b)+d(a-b)|\le |a+b|+|a-b|$ and for fixed $a,b$ we can obtain equality by a suitable rotation. Hence we maximise this over $|a|,|b|\le 1$ . We have $|a+b|^2+|a-b|^2=2(|a|^2+|b|^2)\le 4$ and with $x:=|a+b|,y:=|a-b|$ we have $x^2+y^2\le 4\implies x+y\le 2\sqrt{2}$ .
Equality when $a=1,b=i$ .

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a