Difference between revisions of "1974 IMO Problems/Problem 4"
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Similarly, you can find the other possibilities as <math>\{a_1,a_2,\ldots ,a_7\}=\{1,2,3,4,5,8,9\}</math> or <math>\{1,2,3,4,6,7,9\}</math> or <math>\{1,2,3,5,6,7,8\}</math>. Tilings are not hard to find. | Similarly, you can find the other possibilities as <math>\{a_1,a_2,\ldots ,a_7\}=\{1,2,3,4,5,8,9\}</math> or <math>\{1,2,3,4,6,7,9\}</math> or <math>\{1,2,3,5,6,7,8\}</math>. Tilings are not hard to find. | ||
+ | The above solution was posted and copyrighted by WakeUp. The original thread for this problem can be found here: [https://aops.com/community/p2554359] | ||
{{alternate solutions}} | {{alternate solutions}} |
Latest revision as of 16:03, 29 January 2021
Problem
Consider decompositions of an chessboard into
non-overlapping rectangles subject to the following conditions:
(i) Each rectangle has as many white squares as black squares.
(ii) If is the number of white squares in the
-th rectangle, then
Find the maximum value of for which such a decomposition is possible. For this value of
determine all possible sequences
Solution
Since each rectangle has the same number of black squares as white squares, . Clearly
for
to
so
so this forces
. It is possible to decompose the board into
rectangles, as we will show later. But first let us find all such sequences
.
Now
. For a rectangle to have
white squares, it will have an area of
so it's dimensions are either
or
- neither of which would fit on a
board. So
.
If (which could fit as a
rectangle) then
. Then
so
. So
are 6 numbers among 1-7. If
is the number that is not equal to any
, then
so
. Then
. Such a decomposition is possible. Take a
rectangle on the top left corner, where there are
squares horizontally and
vertically. Then directly below use a
and a
rectangle to cover the 3 rows below it. It's simple from there.
Similarly, you can find the other possibilities as or
or
. Tilings are not hard to find.
The above solution was posted and copyrighted by WakeUp. The original thread for this problem can be found here: [1]
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1974 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |