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3 questions with geometric sequences, arithmetic and polynomial related

Source: Gulf Mathematical Olympiad 2015 p4

August 23, 2019
geometric sequencealgebrapolynomialarithmetic sequence

Problem Statement

a) We have a geometric sequence of 33 terms. If the sum of these terms is 2626 , and their sum of squares is 364364 , find the terms of the sequence.
b) Suppose that a,b,c,u,v,wa,b,c,u,v,w are positive real numbers , and each of a,b,ca,b,c and u,v,wu,v,w are geometric sequences. Suppose also that a+u,b+v,c+wa+u,b+v,c+w are an arithmetic sequence. Prove that a=b=ca=b=c and u=v=wu=v=w
c) Let a,b,c,da,b,c,d be real numbers (not all zero), and let f(x,y,z)f(x,y,z) be the polynomial in three variables defined byf(x,y,z)=axyz+b(xy+yz+zx)+c(x+y+z)+df(x,y,z) = axyz + b(xy + yz + zx) + c(x+y+z) + d.Prove that f(x,y,z)f(x,y,z) is reducible if and only if a,b,c,da,b,c,d is a geometric sequence.
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