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

m (Spacing of LaTeX)
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
 
==Problem==
 
==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==
 
==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==
 
==See also==
 
{{CYMO box|year=2006|l=Lyceum|num-b=1|num-a=3}}
 
{{CYMO box|year=2006|l=Lyceum|num-b=1|num-a=3}}

Latest revision as of 09:56, 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

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

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

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