A mind boggling question

Algebra Level pending

For positive real numbers a , b , c a,b,c which of the following statements necessarily implies a = b = c a = b = c

( I ) . a ( b 3 + c 3 ) = b ( a 3 + c 3 ) = c ( a 3 + b 3 ) (I). a(b^{3}+c^{3})=b(a^{3}+c^{3})=c(a^{3}+b^{3}) ( I I ) . a ( a 3 + b 3 ) = b ( b 3 + c 3 ) = c ( c 3 + a 3 ) (II). a(a^{3}+b^{3})=b(b^{3}+c^{3})=c(c^{3}+a^{3})

Source:INMO(Indian National Mathematics Olympiad) 2016

None of the statements Statement 1 Statement 2 Both The Statements

Ashutosh Tripathi by Ashutosh Tripathi , 31, India

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1 solution

The question is easier than what many think. For statement (I),if we take a = b = 1 a=b=1 then we learn it is not necessary for c c to be equal to 1. Another possible value for c is c = 5 − 1 2 c =\frac{\sqrt{5}-1}{2} ,hence proving the conclusion for this statement is false. On the other hand for (II) Let a = m a x ( a , b , c ) a=max(a,b,c) (for cyclic reasons)and see that a ( a 3 + b 3 ) = b ( b 3 + c 3 ) a(a^{3}+b^{3})=b(b^{3}+c^{3}) and since a ≥ b , a 3 + b 3 ≥ b 3 + c 3 a\geq b,a^{3}+b^{3} \geq b^{3}+c^{3} we see that this could happen only when a = b = c a = b = c Hence the only possible statement that is true for the conclusion is S t a t e m e n t 2 Statement 2 and hence the answer is A \large A

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