Difference between revisions of "2006 Cyprus Seniors Provincial/2nd grade/Problems"
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ii) <math>\frac{1}{\beta^2 + \gamma^2 - \alpha^2} + \frac{1}{\gamma^2 + \alpha^2 - \beta^2} + \frac{1}{\alpha^2 + \beta^2 - \gamma^2} = 0</math>. | ii) <math>\frac{1}{\beta^2 + \gamma^2 - \alpha^2} + \frac{1}{\gamma^2 + \alpha^2 - \beta^2} + \frac{1}{\alpha^2 + \beta^2 - \gamma^2} = 0</math>. | ||
| − | [[ 2006 Cyprus Seniors Provincial/2nd grade/Problem 1|Solution]] | + | [[2006 Cyprus Seniors Provincial/2nd grade/Problem 1|Solution]] |
== Problem 2 == | == Problem 2 == | ||
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<math>\frac{x_{1}}{x_{2}} = \frac{Area(\Alpha\Gamma\Delta)}{Area(\Alpha\Gamma\Epsilon)} = \frac{\Alpha\Beta}{\Beta\Gamma}</math>. | <math>\frac{x_{1}}{x_{2}} = \frac{Area(\Alpha\Gamma\Delta)}{Area(\Alpha\Gamma\Epsilon)} = \frac{\Alpha\Beta}{\Beta\Gamma}</math>. | ||
| − | [[ 2006 Cyprus Seniors Provincial/2nd grade/Problem 2|Solution]] | + | [[2006 Cyprus Seniors Provincial/2nd grade/Problem 2|Solution]] |
== Problem 3 == | == Problem 3 == | ||
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<math>\frac{\Alpha^2}{(1+\Alpha^2)^2} + \frac{\Beta^2}{(1+\Beta^2)^2} = \frac{1}{4}</math>. | <math>\frac{\Alpha^2}{(1+\Alpha^2)^2} + \frac{\Beta^2}{(1+\Beta^2)^2} = \frac{1}{4}</math>. | ||
| − | [[ 2006 Cyprus Seniors Provincial/2nd grade/Problem 3|Solution]] | + | [[2006 Cyprus Seniors Provincial/2nd grade/Problem 3|Solution]] |
== Problem 4 == | == Problem 4 == | ||
Find all integers pairs (x,y) that verify at the same time the inequalities <math>x^2\leq\frac{y^2+2x-1}{2}</math> and <math>y^2\leq\frac{x^2-2y-1}{2}</math>. | Find all integers pairs (x,y) that verify at the same time the inequalities <math>x^2\leq\frac{y^2+2x-1}{2}</math> and <math>y^2\leq\frac{x^2-2y-1}{2}</math>. | ||
| − | [[ 2006 Cyprus Seniors Provincial/2nd grade/Problem 4|Solution]] | + | [[2006 Cyprus Seniors Provincial/2nd grade/Problem 4|Solution]] |
== See also == | == See also == | ||
Revision as of 06:50, 12 November 2006
Problem 1
If 
with 
, prove that
i) 
ii) 
.
Problem 2
Let $\Alpha, \Beta, \Gamma$ (Error compiling LaTeX. Unknown error_msg) be consecutive points on a straight line 
. We construct equilateral triangles $\Alpha\Beta\Delta$ (Error compiling LaTeX. Unknown error_msg) and $\Beta\Gamma\Epsilon$ (Error compiling LaTeX. Unknown error_msg) to the same side of .
a) Prove that $\angle\Alpha\Epsilon\Beta = \angle\Delta\Gamma\Beta$ (Error compiling LaTeX. Unknown error_msg)
b) If 
is the distance of 
form 
and 
is the distance of 
form $\Alpha\Gamma$ (Error compiling LaTeX. Unknown error_msg) prove that
$\frac{x_{1}}{x_{2}} = \frac{Area(\Alpha\Gamma\Delta)}{Area(\Alpha\Gamma\Epsilon)} = \frac{\Alpha\Beta}{\Beta\Gamma}$ (Error compiling LaTeX. Unknown error_msg).
Problem 3
If $\Alpha=\frac{1-cos\theta}{sin\theta}$ (Error compiling LaTeX. Unknown error_msg) and $\Beta=\frac{1-sin\theta}{cos\theta}$ (Error compiling LaTeX. Unknown error_msg), prove that $\frac{\Alpha^2}{(1+\Alpha^2)^2} + \frac{\Beta^2}{(1+\Beta^2)^2} = \frac{1}{4}$ (Error compiling LaTeX. Unknown error_msg).
Problem 4
Find all integers pairs (x,y) that verify at the same time the inequalities 
and 
.
