Specimen Cyprus Seniors Provincial/2nd grade/Problems

Problem 1

Let $\text{A}\text{B}\Gamma\Delta$ be a parallelogram. Let $(\epsilon)$ be a straight line passing through $\text{A}$ without cutting $\text{A}\text{B}\Gamma\Delta$ . If $\text{B} ', \Gamma ', \Delta '$ are the projections of $\text{B}, \Gamma, \Delta$ on $(\epsilon)$ respectively, show that

a) the distance of $\Gamma$ from $(\epsilon)$ is equal to the sum of the distances $\text{B}$ , $\Delta$ from $(\epsilon)$ .

b) $\text{Area}(\text{B}\Gamma\Delta)=\text{Area}(\text{B} '\Gamma '\Delta ')$ .

Solution

Problem 2

If $\alpha=\sin x_{1}$ , $\beta=\cos x_{1}\sin x_{2}$ , $\gamma=\cos x_{1}\cos x_{2} \sin x_{3}$ and $\delta=\cos x_{1}\cos x_{2}\cos x_{3}$ 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 irrational roots.