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$$\large 1•$$ 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 $${p_1}^2+ {p_2}^2+ {p_3}^2+4(q_1+q_2+q_3)=2(p_1p_2+p_2p_3+p_3p_1)$$

Note by Sanjeet Raria
3 years, 1 month ago

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2.) 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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