MathDB

3

Part of 1994 IberoAmerican

Problems(2)

9th ibmo - brazil 1994/q3.

Source: Spanish Communities

5/7/2006
In each square of an n×nn\times{n} grid there is a lamp. If the lamp is touched it changes its state every lamp in the same row and every lamp in the same column (the one that are on are turned off and viceversa). At the begin, all the lamps are off. Show that lways is possible, with an appropriated sequence of touches, that all the the lamps on the board end on and find, in function of nn the minimal number of touches that are necessary to turn on every lamp.
functioncombinatorics unsolvedcombinatorics
9th ibmo - brazil 1994/q6.

Source: Spanish Communities

5/7/2006
Show that every natural number n≤21  000  000n\leq2^{1\;000\;000} can be obtained first with 1 doing less than 1  100  0001\;100\;000 sums; more precisely, there is a finite sequence of natural numbers x_0,\ x_1,\dots,\ x_k\mbox{ with }k\leq1\;100\;000,\ x_0=1,\ x_k=n such that for all i=1, 2,…, ki=1,\ 2,\dots,\ k there exist r, sr,\ s with 0≤r≤s<i0\leq{r}\leq{s}<i such that xi=xr+xsx_i=x_r+x_s.
inductionalgebra unsolvedalgebra