2006 Seniors Pancyprian/2nd grade/Problems

Problem 1

Let $\alpha\beta\gamma$ be a given triangle and $\mu$ the midpoint of the side $\beta\gamma$ . The circle with diameter $\alpha\beta$ cuts $\alpha\gamma$ at $\delta$ and form $\delta$ we bring $\delta\zeta=\mu\gamma$ ( $\delta$ is out of the triangle). Prove that the area of the quadrilateral $\alpha\mu\gamma\zeta$ is equal to the area of the triangle $\alpha\beta\gamma$ .

Solution

Problem 2

Find all three digit numbers $\overline{xyz}$ (=100x+10y+z) for which $\frac {7}{4}(\overline{xyz})=\overline{zyx}$ .

Solution

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.

Solution

Problem 4

A quadrilateral $\alpha\beta\gamma\delta$ , that has no parallel sides, is inscribed in a circle, its sides $\delta\alpha$ , $\gamma\beta$ meet at $\epsilon$ and its sides $\beta\alpha$ , $\gamma\delta$ meet at $\zeta$ . If the bisectors of $\angle\delta\epsilon\gamma$ and $\angle\gamma\zeta\beta$ intersect the sides of the quadrilateral at th points $\kappa, \lambda, \mu, \nu$ prove that

i)the bisectors intersect normally

ii)the points $\kappa, \lambda, \mu, \nu$ are vertices of a rhombus.

Solution

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.

Solution

Page

Toolbox

Search

2006 Seniors Pancyprian/2nd grade/Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See also