Three little complex numbers

Algebra Level 3

You have three complex numbers A , B A, B and C C , such that ∣ A − 1 ∣ ≤ 1 |A-1| \leq 1 , ∣ B − 2 ∣ ≤ 2 |B-2| \leq 2 and ∣ C − 3 ∣ ≤ 3 |C-3| \leq 3 .

What is the greatest possible value of ∣ A + B + C ∣ |A+B+C| ?


The answer is 12.

Abha Vishwakarma by Abha Vishwakarma , 20, India

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1 solution

Pierre Connault
Oct 30, 2018

If A , B , C A,B,C are any such complex numbers, then we have ∣ A + B + C ∣ = ∣ ( A − 1 ) + ( B − 2 ) + ( C − 3 ) + 1 + 2 + 3 ∣ ≤ triangular inequality ∣ ( A − 1 ) ∣ + ∣ ( B − 2 ) ∣ + ∣ ( C − 3 ) ∣ + 1 + 2 + 3 ≤ 12 |A+B+C|=|(A-1)+(B-2)+(C-3)+1+2+3| \underset{ \text{ triangular inequality } }{\leq} |(A-1)|+|(B-2)|+|(C-3)|+1+2+3 \leq 12 and this inequality is an equality as soon as, for instance A = 2 , B = 4 , C = 6 A=2, B=4, \ C=6 . So the answer is 12 \boxed{12}

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