Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 2"

Revision as of 10:52, 27 April 2008

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

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

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

@@ Line 1: / Line 1: @@
 ==Problem==
-The operation <math>\alpha * \beta</math> is defined by <math>\alpha * \beta = \alpha^2 - \beta^2</math>  <math>\forall \alpha , \beta \in R</math>.
+The operation <math>\alpha*\beta</math> is defined by <math>\alpha*\beta=\alpha^2-\beta^2\forall\alpha,\beta\in\mathbb{R}</math>.
-The value of the expression <math>K = \left[\left(1+\sqrt{3}\right) * 2\right]*\sqrt{2} </math> is
+The value of the expression <math>K=\left[\left(1+\sqrt{3}\right)*2\right]*\sqrt{2}</math> is
-A. <math>3</math>
+<math>\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ 0\qquad\mathrm{(C)}\ \sqrt{3}\qquad\mathrm{(D)}\ 9\qquad\mathrm{(E)}\ 1</math>
-B. <math>0</math>
-C. <math>\sqrt{3}</math>
-D. <math>9</math>
-E. <math>1</math>
 ==Solution==
-<math>(1+\sqrt{3}) * 2 =(1+\sqrt{3})^2-2^2=1^2+2\sqrt{3}+3-4=2\sqrt{3}</math>
+<math>(1+\sqrt{3})*2=(1+\sqrt{3})^2-2^2=1^2+2\sqrt{3}+3-4=2\sqrt{3}</math>
-<math>2\sqrt{3} * \sqrt{2} = 12-2=10</math>
-I think that B was mistyped, and that it is supposed to be 10.
-{{wikify}}
+<math>2\sqrt{3}*\sqrt{2} = 12-2=10</math>
 ==See also==
 {{CYMO box|year=2006|l=Lyceum|num-b=1|num-a=3}}

2006 Cyprus MO, Lyceum (Problems)
Preceded by Problem 1	Followed by Problem 3
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Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 2"

Revision as of 10:52, 27 April 2008

Problem

Solution

See also