Difference between revisions of "1994 OIM Problems/Problem 5"
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Let <math>n</math> and <math>r</math> be two positive integers. We wish to construct <math>r</math> subsets <math>A_1, A_2, \cdots , A_r</math> of <math>{0,1,... ,n-1}</math> each of them with exactly <math>k</math> elements and such that, for each integer <math>x</math>, <math>0 \le x \le n-1</math>, there exist <math>x_1</math> in <math>A_1</math>, <math>x_2</math> in <math>A_2</math>, <math>\cdots</math>, <math>x_r</math> in <math>A_r</math> (one element in each set) with | Let <math>n</math> and <math>r</math> be two positive integers. We wish to construct <math>r</math> subsets <math>A_1, A_2, \cdots , A_r</math> of <math>{0,1,... ,n-1}</math> each of them with exactly <math>k</math> elements and such that, for each integer <math>x</math>, <math>0 \le x \le n-1</math>, there exist <math>x_1</math> in <math>A_1</math>, <math>x_2</math> in <math>A_2</math>, <math>\cdots</math>, <math>x_r</math> in <math>A_r</math> (one element in each set) with | ||
| − | <cmath>x = | + | <cmath>x = x_1+x_2+ \cdots +x_r</cmath> |
Find the smallest possible value of <math>k</math> as a function of <math>n</math> and <math>r</math>. | Find the smallest possible value of <math>k</math> as a function of <math>n</math> and <math>r</math>. | ||
Latest revision as of 14:32, 13 December 2023
Problem
Let 
and 
be two positive integers. We wish to construct subsets 
of 
each of them with exactly 
elements and such that, for each integer 
, 
, there exist 
in 
, 
in 
, 
, 
in 
(one element in each set) with
![\[x = x_1+x_2+ \cdots +x_r\]](http://latex.artofproblemsolving.com/c/1/6/c1684f7101c5c4274f37f327593deaa16724981b.png)
Find the smallest possible value of as a function of and .
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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