MathDB

2

Part of 1996 IberoAmerican

Problems(2)

11st ibmo - costa rica 1996/q2.

Source: Spanish Communities

4/23/2006
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}.
geometrycircumcircle
11st ibmo - costa rica 1996/q5.

Source: Spanish Communities

4/23/2006
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.
combinatorics unsolvedcombinatorics