MathDB

While there do not exist pairwise distinct real numbers

a,b,c

satisfying

a^2+b^2+c^2 = ab+bc+ca

, there do exist complex numbers with that property. Let

a,b,c

be complex numbers such that

a^2+b^2+c^2 = ab+bc+ca

and

|a+b+c| = 21

. Given that

|a-b| = 2\sqrt{3}

|a| = 3\sqrt{3}

, compute

|b|^2+|c|^2

. [hide="Clarifications"]
[*] The problem should read

|a+b+c| = 21

. An earlier version of the test read

|a+b+c| = 7

; that value is incorrect. [*]

|b|^2+|c|^2

should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers

m

and

n

. Find

m+n

.''
Ray Li

Problem Statement