AM-GM-HM inequality

The ubiquitous AM-GM inequality has many applications. It almost crops up in unlikely situations and the solutions using AM-GM are truly elegant . Recall that for n positive reals a_(i) I = 1,2 …, n, the AM-GM inequality tells (overset(n) underset(1)suma_i)/n ge ( overset(n)underset(1)proda_i)^((1)/(n)) The special in which the inequality turns into equality help solves many problems where at first we seem to have not informantion to arrive at the answer . If the equation x^(4) - 4x^(3) + ax^(2) + bx + 1 = 0 has four positive roots , then the value of (|a|+|b|)/(a+b) is

The ubiquitous AM-GM inequality has many applications. It almost crops up in unlikely situations and the solutions using AM-GM are truly elegant . Recall that for n positive reals a_(i) I = 1,2 …, n, the AM-GM inequality tells (overset(n) underset(1)suma_i)/n ge ( overset(n)underset(1)proda_i)^((1)/(n)) The special in which the inequality turns into equality help solves many problems where at first we seem to have not informantion to arrive at the answer . The number of ordered pairs (x,y) pf real numbers satisfying the equation x^(8) + 6= 8 |xy|-y^(8) is

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If the AM and HM of two numbers are 27 and 12 respectively, then what is their GM equal to ?

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If A,G,H are AM,GM,HM between the two numbers a and b ,then A ,G,H are in

If A^(x)=G^(y)=H^(z) , where A,G,H are AM,GM and HM between two given quantities, then prove that x,y,z are in HP.