2002 Greece Junior Math Olympiad

Part of Greece Junior Math Olympiad

Subcontests

(4)

1

Find the ratio.

In the Mathematical Competition of HMS (Hellenic Mathematical Society) take part boys and girls who are divided into two groups : Juniors and seniors.The number of the boys taking part of this year competition is 55% of the number of all participants. The ratio of the number of juniors boys to the number of senior boys is equal to the ratio of the number of juniors to the number of seniors. Find the ratio of the number of junior boys to the number of junior girls.

1

Triangle

In the exterior of an equilateral triangle

ABC

\alpha

we construct an isosceles right-angled triangle

ACD

\angle CAD=90^0.

DA

CB

E

. (a) Find the angle

\angle DBC.

(b) Express the area of triangle

CDE

\alpha.

(c) Find the length of

BD.

1

Inequality

1\cdot2\cdot3\cdots 2002<\left(\frac{2003}{2}\right)^{2002}.

1

Triples

Determine all triples of positive integers

(x,y,z)

x \leq y \leq z

xy+yz+zx-xyz=2.