1990 Hungary-Israel Binational

Part of Hungary-Israel Binational

Subcontests

(4)

1

Hungary-Israel Binational 1990_4

A rectangular sheet of paper with integer length sides is given. The sheet is marked with unit squares. Arrows are drawn at each lattice point on the sheet in a way that each arrow is parallel to one of its sides, and the arrows at the boundary of the paper do not point outwards. Prove that there exists at least one pair of neighboring lattice points (horizontally, vertically or diagonally) such that the arrows drawn at these points are in opposite directions.

1

Hungary-Israel Binational 1990_3

Prove that: \frac{1989}{2}\minus{}\frac{1988}{3}\plus{}\frac{1987}{4}\minus{}\cdots\minus{}\frac{2}{1989}\plus{}\frac{1}{1990}\equal{}\frac{1}{996}\plus{}\frac{3}{997}\plus{}\frac{5}{998}\plus{}\cdots\plus{}\frac{1989}{1990}

1

Hungary-Israel Binational 1990_2

ABC

be a triangle where \angle ACB\equal{}90^{\circ}. Let

D

be the midpoint of

BC

E

F

AC

such that CF\equal{}FE\equal{}EA. The altitude from

C

to the hypotenuse

AB

CG

, and the circumcentre of triangle

AEG

H

. Prove that the triangles

ABC

HDF

1

Squares

Prove that there are no positive integers

x

y

x^2+y+2

y^2+4x

are perfect squares