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2006 Cyprus MO/Lyceum/Problem 2

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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

See also

2006 Cyprus MO, Lyceum (Problems)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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