2006 Cyprus MO/Lyceum/Problem 2

Problem

The operation $\alpha*\beta$ is defined by $\alpha*\beta=\alpha^2-\beta^2\ \forall\alpha,\beta\in\mathbb{R}$ . The value of the expression $K=\left[\left(1+\sqrt{3}\right)*2\right]*\sqrt{2}$ is

$\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ 0\qquad\mathrm{(C)}\ \sqrt{3}\qquad\mathrm{(D)}\ 9\qquad\mathrm{(E)}\ 1$

Solution

$(1+\sqrt{3})*2=(1+\sqrt{3})^2-2^2=1^2+2\sqrt{3}+3-4=2\sqrt{3}$

$2\sqrt{3}*\sqrt{2} = 12-2=10$

2006 Cyprus MO, Lyceum (Problems)
Preceded by Problem 1	Followed by Problem 3
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2006 Cyprus MO/Lyceum/Problem 2

Problem

Solution

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