1

Part of 2011 International Zhautykov Olympiad

Problems(2)

International Zhautykov Olympiad 2011 - Problem 1

Source:

Given is trapezoid

ABCD

M

N

being the midpoints of the bases of

AD

BC

, respectively. a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment

MN

. b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line

MN

geometrytrapezoidcircumcircleparallelogramrectangleperpendicular bisectorPythagorean Theorem

International Zhautykov Olympiad 2011- Problem 4

Source:

Find the maximum number of sets which simultaneously satisfy the following conditions:
i) any of the sets consists of

4

elements,
ii) any two different sets have exactly

2

common elements,
iii) no two elements are common to all the sets.

combinatorics unsolvedcombinatorics