3

Part of 1996 IberoAmerican

Problems(2)

11st ibmo - costa rica 1996/q3.

Source: Spanish Communities

We have a grid of

k^2-k+1

k^2-k+1

k=p+1

p

is prime. For each prime

p

, give a method to put the numbers 0 and 1, one number for each square in the grid, such that on each row there are exactly

k

0's, on each column there are exactly

k

0's, and there is no rectangle with sides parallel to the sides of the grid with 0s on each four vertices.

geometryrectanglecombinatorics unsolvedcombinatorics

11st ibmo - costa rica 1996/q6.

Source: Spanish Communities

n

different points

A_1, \ldots , A_n

in the plain and each point

A_i

it is assigned a real number

\lambda_i

distinct from zero in such way that

(\overline{A_i A_j})^2 = \lambda_i + \lambda_j

i

j

i\neq{}j

} Show that:
(1)

n \leq 4

n=4

\frac{1}{\lambda_1} + \frac{1}{\lambda_2} + \frac{1}{\lambda_3}+ \frac{1}{\lambda_4} = 0

geometrycircumcircletrigonometrygeometry unsolved