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 .

Solution

Problem 2

Find all three digit numbers (=100x+10y+z) for which .

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 , that has no parallel sides, is inscribed in a circle, its sides , meet at and its sides , meet at . If the bisectors of and intersect the sides of the quadrilateral at th points prove that

i)the bisectors intersect normally

ii)the points 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 of 50 who is sitting between two girls.

Solution

See also

• 2006 Seniors Pancyprian
• Seniors Pancyprian Competition
• Seniors Pancyprian Problems and Solutions
• Mathematics competition resources