1977 Dutch Mathematical Olympiad

Part of Dutch Mathematical Olympiad

Subcontests

(4)

1

even number of points in a plane, half red half blue

There are an even number of points in a plane. No three of them lie on one straight line. Half of the points are red, the other half are blue. Prove that there exists a connecting line of a red and a blue point such that in each of the half-planes bounded by that line the number of red points is equal to the number of blue points.

1

4 masts stand on a flat horizontal piece of land

Four masts stand on a flat horizontal piece of land at the vertices of a square

ABCD

. The height of the mast on

A

7

meters, of the mast on

B

13

meters, and of the mast on

C

15

meters. Within the square there is a point

P

on the ground equidistant from each of the tops of these three masts. (a) What length must the sides of the square be at least for this to be possible? (b) The distance from

P

to the top of the mast on

D

is equal to the distance from

P

to each of the tops of the three other masts. Calculate the height of the mast at

D

1

number of integers whose sum is a multiple of $7

\{a_1,a_2,...,a_7\} \subset Z

one can choose a number of elements whose sum is a multiple of

7

1

diophantine 2x^2y = (x+2)^2(y + 1)> 1000

Solve, for integers

x

y

2x^2y = (x+2)^2(y + 1),

(x+2)^2(y + 1)> 1000