Specimen Cyprus Seniors Provincial/2nd grade/Problems

Problem 1

Let $\Alpha\Beta\Gamma\Delta$ (Error compiling LaTeX. Unknown error_msg) be a parallelogram. Let $(\epsilon)$ be a straight line passing through $\Alpha$ (Error compiling LaTeX. Unknown error_msg) without cutting $\Alpha\Beta\Gamma\Delta$ (Error compiling LaTeX. Unknown error_msg). If $\Beta ', \Gamma ', \Delta '$ (Error compiling LaTeX. Unknown error_msg) are the projections of $\Beta, \Gamma, \Delta$ (Error compiling LaTeX. Unknown error_msg) on $(\epsilon)$ respectively, show that

a) the distance of $\Gamma$ from $(\epsilon)$ is equal to the sum of the distances $\Beta, \Delta$ (Error compiling LaTeX. Unknown error_msg) from $(\epsilon)$ .

b)Area$(\Beta\Gamma\Delta)$ (Error compiling LaTeX. Unknown error_msg)=Area$(\Beta '\Gamma '\Delta ')$ (Error compiling LaTeX. Unknown error_msg).

Solution

Problem 2

Problem

If $\alpha=sinx_{1}$ , $\beta=cosx_{1}$ $sinx_{2}$ , $\gamma=cosx_{1}cosx_{2} sinx_{3}$ and $\delta=cosx_{1}cosx_{2}cosx_{3}$ prove that $\alpha^2+\beta^2+\gamma^2+\delta^2=1$

Solution

Problem 3

Prove that if $\kappa, \lambda, \nu$ are positive integers, then the equation $x^2-(\nu +2)\kappa\lambda x+\kappa^2\lambda^2 = 0$ has irratioanl roots.