2006 Seniors Pancyprian/2nd grade/Problems

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Problem 1

Let $\Alpha\Beta\Gamma$ (Error compiling LaTeX. Unknown error_msg) be a given triangle and $\Mu$ (Error compiling LaTeX. Unknown error_msg) the midpoint of the side $\Beta\Gamma$ (Error compiling LaTeX. Unknown error_msg). The circle with diameter $\Alpha\Beta$ (Error compiling LaTeX. Unknown error_msg) cuts $\Alpha\Gamma$ (Error compiling LaTeX. Unknown error_msg) at $\Delta$ and form $\Delta$ we bring $\Delta\Zeta=//\Mu\Gamma$ (Error compiling LaTeX. Unknown error_msg) ($\Delta$ is out of the triangle). Prove that the area of the quadrilateral $\Alpha\Mu\Gamma\Zeta$ (Error compiling LaTeX. Unknown error_msg) is equal to the area of the triangle $\Alpha\Beta\Gamma$ (Error compiling LaTeX. Unknown error_msg). 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 $\Alpha\Beta\Gamma\Delta$ (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), $\Gamma\Delta$ 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.

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

See also