2006 Seniors Pancyprian/2nd grade/Problems
Problem 1
Let 
be a given triangle and 
the midpoint of the side 
. The circle with diameter 
cuts 
at 
and form we bring 
( is out of the triangle). Prove that the area of the quadrilateral 
is equal to the area of the triangle .
Problem 2
Find all three digit numbers 
(=100x+10y+z) for which 
.
Problem 3
i)Convert $\Alpha=sin(x-y)+sin(y-z)+sin(z-x)$ (Error compiling LaTeX. Unknown error_msg) into product.
ii)Prove that: If in a triangle $\Alpha\Beta\Gamma$ (Error compiling LaTeX. Unknown error_msg) is true that $\alpha sin \Beta + \beta sin \Gamma + \gamma sin \Alpha= \frac {\alpha+\beta+\gamma}{2}$ (Error compiling LaTeX. Unknown error_msg), then the triangle is isosceles.
Problem 4
A quadrilateral $\Alpha\Beta\Gamma\Delta$ (Error compiling LaTeX. Unknown error_msg), that has no parallel sides, is inscribed in a circle, its sides $\Delta\Alpha$ (Error compiling LaTeX. Unknown error_msg), $\Gamma\Beta$ (Error compiling LaTeX. Unknown error_msg) meet at $\Epsilon$ (Error compiling LaTeX. Unknown error_msg) and its sides $\Beta\Alpha$ (Error compiling LaTeX. Unknown error_msg), 
meet at $\Zeta$ (Error compiling LaTeX. Unknown error_msg).
If the bisectors of of $\angle\Delta\Epsilon\Gamma$ (Error compiling LaTeX. Unknown error_msg) and $\angle\Gamma\Zeta\Beta$ (Error compiling LaTeX. Unknown error_msg) intersect the sides of the quadrilateral at th points $\Kappa, \Lambda, \Mu, \Nu$ (Error compiling LaTeX. Unknown error_msg) prove that
i)the bisectors intersect normally
ii)the points $\Kappa, \Lambda, \Mu, \Nu$ (Error compiling LaTeX. Unknown error_msg) are vertices of a rhombus.
Problem 5
Fifty persons, twenty five boys and twenty five girls are sitting around a table. Prove that there is a person out out of 50, who is sitting between two girls.
