RMO practice problems from past year papers (polynomials)

$\begin{cases} P_{1}(x) = ax^{2}-bx-c \\ P_{2}(x)=bx^{2}-cx-a \\ P_{3}(x)=cx^2-ax-b\ \end{cases}$ Let the above be three quadratic polynomials where a,b,c are non-zero real numbers. Suppose there exists a real number $\alpha$ such that $P_{1}(\alpha)=P_{2}(\alpha)=P_{3}(\alpha)$ , then

Prove that $a=b=c$ .

Please post your answers after at least one week (although this one is quite simple).

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Math	Appears as
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`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}`	$\boxed{123}$

RMO practice problems from past year papers (polynomials)

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