MathDB

1994 IberoAmerican

Part of IberoAmerican

Subcontests

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9th ibmo - brazil 1994/q3.

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.

9th ibmo - brazil 1994/q6.

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.
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9th ibmo - brazil 1994/q2.

Let ABCD ABCD a cuadrilateral inscribed in a circumference. Suppose that there is a semicircle with its center on AB AB, that is tangent to the other three sides of the cuadrilateral. (i) Show that AB \equal{} AD \plus{} BC. (ii) Calculate, in term of x \equal{} AB and y \equal{} CD, the maximal area that can be reached for such quadrilateral.

9th ibmo - brazil 1994/q5.

Let nn and rr two positive integers. It is wanted to make rr subsets A1, A2,…,ArA_1,\ A_2,\dots,A_r from the set {0,1,⋯ ,n−1}\{0,1,\cdots,n-1\} such that all those subsets contain exactly kk elements and such that, for all integer xx with 0≤x≤n−10\leq{x}\leq{n-1} there exist x1∈A1, x2∈A2…,xr∈Arx_1\in{}A_1,\ x_2\in{}A_2 \dots,x_r\in{}A_r (an element of each set) with x=x1+x2+⋯+xrx=x_1+x_2+\cdots+x_r.
Find the minimum value of kk in terms of nn and rr.
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