Statement 1: For positive p,q,r, ( p + q ) ( p + r ) ( q + r ) = 8 p 3 forces p = q = r . Statement 2: For positive a 1 , … , a n , their arithmetic mean is at least their geometric mean.

Question

Statement 1: For positive p,q,r, ( p + q ) ( p + r ) ( q + r ) = 8 p 3 forces p = q = r . Statement 2: For positive a 1 ​ , … , a n ​ , their arithmetic mean is at least their geometric mean.

TestHub · Accepted Answer

Statement 2 is AM-GM. Statement 1 fails: take p = 1 , q = 1/2 , r = ( 777 ​ − 9 ) /12 , a positive number unequal to p,q. Then ( 1 + q ) ( 1 + r ) ( q + r ) = 8 . The source argument proves only that p = q = r is sufficient, not necessary.

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