Difference between revisions of "2006 Cyprus Seniors Provincial/2nd grade/Problems"
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== Problem 3 == | == Problem 3 == | ||
− | If <math>\ | + | If <math>\alpha=\frac{1-\cos \theta}{\sin \theta}</math> and <math>\beta=\frac{1-sin\theta}{cos\theta}</math>, prove that |
− | <math>\frac{\ | + | <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]] |
Latest revision as of 22:07, 10 April 2013
Problem 1
If with
, prove that
i)
ii) .
Problem 2
Let be consecutive points on a straight line
. We construct equilateral triangles
and
to the same side of
.
a) Prove that
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 and
, prove that
.
Problem 4
Find all integers pairs (x,y) that verify at the same time the inequalities and
.