Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions: (i) The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$ (ii) The interiors of any two different rectangles $ R_i$ are disjoint. (iii) Each rectangle $ R_i$ has at least one side of integral length. Prove that $ R$ has at least one side of integral length. Variant: Same problem but with rectangular parallelepipeds having at least one integral side.
Problem
Source: IMO Shortlist 1989, Problem 8, ILL 20
Tags: geometry, rectangle, calculus, integration, number theory, IMO Shortlist
ZetaX
18.09.2008 16:54
One way is to color the big rectangle in a chess-board like pattern with distance $ \frac 12$, starting at the lower left corner. Then show that every subrectangle $ R_i$ having an integer side contains the same amount of each color, so in total the big rectangle $ R$ has the same amount of each color. This contradicts if $ R$ doesn't have a side of integer length. PS: moved to combinatorics (wouldn't call this number theory ).
kevinatcausa
18.09.2008 21:58
Stan Wagon compiled a series of 14 proofs of this result in his article "14 proofs of a result about tiling a triangle" (American Mathematical Monthly, Aug-Sep 1987)
Blitzkrieg97
21.10.2014 21:18
i think the easiest solution is to consider this $ ABCD$ rectangle as a $ XOY $ coordinate system and the bottom $ A $ vertex was $ (0 ; 0) $ point($AB$ and $AD$ are on the $ OY $ and $ OY$,respectively.).Now let us consider two sets,$1st$ set contains the vertices which have at least one integral coordinate,and $2nd$ set contains the rectangles.Now let us connect each point from $1st$ set to the rectangles in $2nd$ set,if this rectangle consists of this point.Which means the amount of lines going out from $1st$ set to $2nd$ set is even,if we won't consider lines with $A,B,C,D$ points.This is true,since 1 point is mutual for $0$,$2$ or $4$ rectangle,if this point isn't $A,B,C,D$ (then it is mutual for 1).And the lines,coming out from $2nd$ set must be also even,but since (0 ; 0) is in $1st$ set,it means the lines coming out from $1st$ set is odd,if this set doesn't contain at least one more vertex from $ABCD$ rectangle.However,this set contains $B$ or $C$ point.Hence $ABCD$ has at least one integer side.$\blacksquare$
enescu
22.10.2014 00:59
Blitzkrieg97 wrote: i think the easiest solution is ... So, finally, you succeeded in reposting the problem ( http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=610014, http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=609286, and maybe more ). Here is one link to an article quoted in this thread: http://qcpages.qc.cuny.edu/~zakeri/mat208/wagon.pdf
Blitzkrieg97
22.10.2014 16:04
enescu wrote: Blitzkrieg97 wrote: i think the easiest solution is ... So, finally, you succeeded in reposting the problem ( http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=610014, http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=609286, and maybe more ). Here is one link to an article quoted in this thread: http://qcpages.qc.cuny.edu/~zakeri/mat208/wagon.pdf thanks for reminding