Let A,B,C be strictly positive real numbers satisfying

Charlotte Holden

Charlotte Holden

Answered question

2022-04-15

Let A,B,C be strictly positive real numbers satisfying A+B+C=1 and let a,b,c be real variables. Suppose a,b,c satisfy the following system of equations: A2a+B2b+C2c=1,anda+b+c=1.

Answer & Explanation

folytonr4qx

folytonr4qx

Beginner2022-04-16Added 14 answers

The solution is unique only if a, b, c>0. This is because by Titu's lemma:
1=A2a+B2b+C2c(A+B+C)2a+b+c=121=1, and you have equality which means the = must hold, and this occurs when
A=ak,B=bk,C=ckk=1a=A,b=B,c=C claimed. For if they are not required to be positive then there is another solution, namely:
a=A2B21C2,b=B2A21C2,c=1.

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