MathDB

1996 IberoAmerican

Part of IberoAmerican

Subcontests

(3)
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11st ibmo - costa rica 1996/q3.

We have a grid of k2−k+1k^2-k+1 rows and k2−k+1k^2-k+1 columns, where k=p+1k=p+1 and pp is prime. For each prime pp, 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 kk 0's, on each column there are exactly kk 0's, and there is no rectangle with sides parallel to the sides of the grid with 0s on each four vertices.

11st ibmo - costa rica 1996/q6.

There are nn different points A1,…,AnA_1, \ldots , A_n in the plain and each point AiA_i it is assigned a real number λi\lambda_i distinct from zero in such way that (AiAj‾)2=λi+λj(\overline{A_i A_j})^2 = \lambda_i + \lambda_j for all the ii,jj with i≠ji\neq{}j} Show that:
(1) n≤4n \leq 4 (2) If n=4n=4, then 1λ1+1λ2+1λ3+1λ4=0\frac{1}{\lambda_1} + \frac{1}{\lambda_2} + \frac{1}{\lambda_3}+ \frac{1}{\lambda_4} = 0
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11st ibmo - costa rica 1996/q2.

Let △ABC\triangle{ABC} be a triangle, DD the midpoint of BCBC, and MM be the midpoint of ADAD. The line BMBM intersects the side ACAC on the point NN. Show that ABAB is tangent to the circuncircle to the triangle △NBC\triangle{NBC} if and only if the following equality is true: BMMN=(BC)2(BN)2.\frac{{BM}}{{MN}} =\frac{({BC})^2}{({BN})^2}.

11st ibmo - costa rica 1996/q5.

Three tokens AA, BB, CC are, each one in a vertex of an equilateral triangle of side nn. Its divided on equilateral triangles of side 1, such as it is shown in the figure for the case n=3n=3
Initially, all the lines of the figure are painted blue. The tokens are moving along the lines painting them of red, following the next two rules:
(1) First AA moves, after that BB moves, and then CC, by turns. On each turn, the token moves over exactly one line of one of the little triangles, form one side to the other. (2) Non token moves over a line that is already painted red, but it can rest on one endpoint of a side that is already red, even if there is another token there waiting its turn.
Show that for every positive integer nn it is possible to paint red all the sides of the little triangles.
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