1994 Nordic

Part of Nordic

Subcontests

(4)

1

n^2 + (n+ 1)^2 is square of integer, n <200

Determine all positive integers

n < 200

n^2 + (n+ 1)^2

is the square of an integer.

1

folding a square, perimeter question

A piece of paper is the square

ABCD

. We fold it by placing the vertex

D

D'

BC

. We assume that

AD

moves on the segment

A' D'

A' D'

AB

E

. Prove that the perimeter of the triangle

EBD'

is one half of the perimeter of the square.

1

two-neighbour set with lattice point

We call a finite plane set

S

consisting of points with integer coefficients a two-neighbour set, if for each point

(p, q)

S

exactly two of the points

(p +1, q), (p, q +1), (p-1, q), (p, q-1)

S

. For which integers

n

there exists a two-neighbour set which contains exactly

n

1

inequality with cevians in equilateral

O

be an interior point in the equilateral triangle

ABC

, of side length

a

AO, BO

CO

intersect the sides of the triangle in the points

A_1, B_1

C_1

OA_1 + OB_1 + OC_1 < a