View source for Specimen Cyprus Seniors Provincial/2nd grade/Problems

== Problem 1 ==
Let <math>\text{A}\text{B}\Gamma\Delta</math> be a parallelogram. Let <math>(\epsilon)</math> be a straight line passing through <math>\text{A}</math> without cutting <math>\text{A}\text{B}\Gamma\Delta</math>. If <math>\text{B} ', \Gamma ', \Delta ' </math> are the projections of <math>\text{B}, \Gamma, \Delta</math> on <math>(\epsilon)</math> respectively, show that

a) the distance of <math>\Gamma</math> from <math>(\epsilon)</math> is equal to the sum of the distances <math>\Beta, \Delta</math> from <math>(\epsilon)</math>.

b) <math>\text{Area}(\text{B}\Gamma\Delta)=\text{Area}(\text{B} '\Gamma '\Delta ')</math>.

[[Specimen Cyprus Seniors Provincial/2nd grade/Problem 1|Solution]]

== Problem 2 ==

If <math>\alpha=\sin x_{1}</math>,<math>\beta=\cos x_{1}\sin x_{2}</math>, <math>\gamma=\cos x_{1}\cos x_{2} \sin x_{3}</math> and <math>\delta=\cos x_{1}\cos x_{2}\cos x_{3}</math> prove that <math>\alpha^2+\beta^2+\gamma^2+\delta^2=1</math>

[[Specimen Cyprus Seniors Provincial/2nd grade/Problem 2|Solution]]

== Problem 3 ==
Prove that if <math>\kappa, \lambda, \nu</math> are positive integers, then the equation <math>x^2-(\nu +2)\kappa\lambda x+\kappa^2\lambda^2 = 0</math> has irrational roots.

[[Specimen Cyprus Seniors Provincial/2nd grade/Problem 3|Solution]]

== Problem 4 ==
If <math>\rho_{1}, \rho_{2}</math> are the roots of equation <math>x^2-x+1=0</math> then:

a) Prove that <math>\rho_{1}^3=\rho_{2}^3 = -1</math> and

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

[[Specimen Cyprus Seniors Provincial/2nd grade/Problem 4|Solution]]

== See also ==
* [[Specimen Cyprus Seniors Provincial]]
* [[Cyprus Seniors Provincial competition]]
* [[Cyprus Seniors Provincial Problems and Solutions]]
* [[Mathematics competition resources]]