Difference between revisions of "Specimen Cyprus Seniors Provincial/2nd grade/Problems"
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== Problem 1 == | == Problem 1 == | ||
− | Let <math>\ | + | 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>\ | + | a) the distance of <math>\Gamma</math> from <math>(\epsilon)</math> is equal to the sum of the distances <math>\text{B}</math> , <math>\Delta</math> from <math>(\epsilon)</math>. |
− | b) | + | b) <math>\text{Area}(\text{B}\Gamma\Delta)=\text{Area}(\text{B} '\Gamma '\Delta ')</math>. |
[[Specimen Cyprus Seniors Provincial/2nd grade/Problem 1|Solution]] | [[Specimen Cyprus Seniors Provincial/2nd grade/Problem 1|Solution]] | ||
== Problem 2 == | == Problem 2 == | ||
− | + | ||
− | If <math>\alpha= | + | 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]] | [[Specimen Cyprus Seniors Provincial/2nd grade/Problem 2|Solution]] | ||
== Problem 3 == | == 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 | + | 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]] | [[Specimen Cyprus Seniors Provincial/2nd grade/Problem 3|Solution]] |
Latest revision as of 19:19, 18 January 2021
Contents
[hide]Problem 1
Let be a parallelogram. Let be a straight line passing through without cutting . If are the projections of on respectively, show that
a) the distance of from is equal to the sum of the distances , from .
b) .
Problem 2
If ,, and prove that
Problem 3
Prove that if are positive integers, then the equation has irrational roots.
Problem 4
If are the roots of equation then:
a) Prove that and
b) Calculate the value of: .