Difference between revisions of "2006 Seniors Pancyprian/2nd grade/Problems"
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== Problem 1 == | == Problem 1 == | ||
Let <math>\Alpha\Beta\Gamma</math> be a given triangle and <math>\Mu</math> the midpoint of the side <math>\Beta\Gamma</math>. The circle with diameter <math>\Alpha\Beta</math> cuts <math>\Alpha\Gamma</math> at <math>\Delta</math> and form <math>\Delta</math> we bring <math>\Delta\Zeta=//\Mu\Gamma</math> (<math>\Delta</math> is out of the triangle). Prove that the area of the quadrilateral <math>\Alpha\Mu\Gamma\Zeta</math> is equal to the area of the triangle <math>\Alpha\Beta\Gamma</math>. | Let <math>\Alpha\Beta\Gamma</math> be a given triangle and <math>\Mu</math> the midpoint of the side <math>\Beta\Gamma</math>. The circle with diameter <math>\Alpha\Beta</math> cuts <math>\Alpha\Gamma</math> at <math>\Delta</math> and form <math>\Delta</math> we bring <math>\Delta\Zeta=//\Mu\Gamma</math> (<math>\Delta</math> is out of the triangle). Prove that the area of the quadrilateral <math>\Alpha\Mu\Gamma\Zeta</math> is equal to the area of the triangle <math>\Alpha\Beta\Gamma</math>. | ||
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[[2006 Seniors Pancyprian/2nd grade/Problem 1|Solution]] | [[2006 Seniors Pancyprian/2nd grade/Problem 1|Solution]] | ||
Revision as of 12:30, 26 December 2006
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 and form
we bring $\Delta\Zeta=//\Mu\Gamma$ (Error compiling LaTeX. Unknown error_msg) (
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).
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.