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

#Algebra #NumberTheory #Polynomials #QuadraticEquations #Incircle

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`2 \times 3`	$2 \times 3$
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Comments

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 $= 2s = \boxed{42}$ ~~~

Samuraiwarm Tsunayoshi - 6 years, 7 months ago

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Devendra Rathore - 6 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$