\(\large 1•\) If \(x^n + py^n + qz^n\) is exactly divisibly by \(x^2-(ay +bz)x + abyz\) then find the value of \(\frac{p}{a^n}+\frac{q}{b^n}+1\)

\(\large 2•\) In \(\triangle ABC\), the incircle touches the sides \(BC, CA\) and \(AB\) respectively at \(D\), \(E\) and \(F\). If the radius of incircle is \(4\) units and \(BD, CE\) and \(AF\) are consecutive integers, then find the perimeter of \(\triangle ABC\).

\(\large 3•\) If each pair of the three equations \(x^2 +p_1x + q_1 = 0\), \(x^2 + p_2x + q_2 = 0\) and \(x^2 + p_3 + q_3 = 0\) have a common root, then prove that \({p_1}^2+ {p_2}^2+ {p_3}^2+4(q_1+q_2+q_3)=2(p_1p_2+p_2p_3+p_3p_1)\)

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TopNewest2.) Let \(BD = x-1, CE = x, AF = x+1\)

Since the incircle is tangent to \(\overline{BC},\overline{CA},\overline{AB}\) at \(D,E,F\), we get

\(BF=BD=x-1, CD=CE=x, AE=AF=x+1\)

and \([\triangle ABC] = rs\) where \(s = \displaystyle \frac{a+b+c}{2} = 3x\) and \(r = 4\).

From Heron's formula, we get

\([\triangle ABC]=\sqrt{s(s-a)(s-b)(s-c)}\)

\(\sqrt{(3x)(x+1)(x-1)(x)} = 4\times(3x)\)

Solve the equation we get \(x = 7\).

Therefore, perimeter \(= 2s = \boxed{42}\) ~~~

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