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Specimen Cyprus Seniors Provincial/2nd grade/Problems

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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 \Beta, \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.

Solution

Problem 4

If \rho_{1}, \rho_{2} are the roots of equation x^2-x+1=0 then:

a) Prove that \rho_{1}^3=\rho_{2}^3 = -1 and

b) Calculate the value of: \rho_{1}^{2006} + \rho_{2}^{2006}.

Solution

See also

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