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2006 Cyprus Seniors Provincial/2nd grade/Problems

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Problem 1

If \alpha, \beta, \gamma \in \Re- \{0\} with \alpha + \beta + \gamma = 0, prove that

i) \alpha^2 + \beta^2 - \gamma^2 = -2(\beta + \gamma)(\alpha + \gamma)

ii) \frac{1}{\beta^2 + \gamma^2 - \alpha^2} + \frac{1}{\gamma^2 + \alpha^2 - \beta^2} + \frac{1}{\alpha^2 + \beta^2 - \gamma^2} =....

Solution

Problem 2

Let \text{A}, \text{B}, \Gamma be consecutive points on a straight line (\epsilon). We construct equilateral triangles \text{AB}\Delta and \text{B}\Gamma\text{E} to the same side of (\epsilon).

a) Prove that \angle \text{AEB} = \angle\Delta\Gamma\text{B}

b) If x_{1} is the distance of A form \Gamma\Delta and x_{2} is the distance of \Gamma form \Alpha\Gamma prove that

\frac{x_{1}}{x_{2}} = \frac{Area(\Alpha\Gamma\Delta)}{Area(\Alpha\Gamma\Epsilon)} = \frac{\Alpha\Beta}{\Beta\Gamma}.

Solution

Problem 3

If \text{A}=\frac{1-\cos \theta}{\sin \theta} and \Beta=\frac{1-sin\theta}{cos\theta}, prove that \frac{\Alpha^2}{(1+\Alpha^2)^2} + \frac{\Beta^2}{(1+\Beta^2)^2} = \frac{1}{4}.

Solution

Problem 4

Find all integers pairs (x,y) that verify at the same time the inequalities x^2\leq\frac{y^2+2x-1}{2} and y^2\leq\frac{x^2-2y-1}{2}.

Solution

See also

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