Specimen Cyprus Seniors Provincial/2nd grade/Problems
Problem 1
Let $\Alpha\Beta\Gamma\Delta$ (Error compiling LaTeX. Unknown error_msg) be a parallelogram. Let 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
respectively, show that
a) the distance of from
is equal to the sum of the distances $\Beta, \Delta$ (Error compiling LaTeX. Unknown error_msg) from
.
b)Area$(\Beta\Gamma\Delta)$ (Error compiling LaTeX. Unknown error_msg)=Area$(\Beta '\Gamma '\Delta ')$ (Error compiling LaTeX. Unknown error_msg).
Problem 2
Problem
If ,
,
and
prove that
Problem 3
Prove that if are positive integers, then the equation
has irratioanl roots.
Problem 4
If are the roots of equation
then:
a) Prove that and
b) Calculate the value of: .