Difference between revisions of "2016 IMO Problems/Problem 2"
m |
|||
(2 intermediate revisions by one other user not shown) | |||
Line 1: | Line 1: | ||
+ | ==Problem== | ||
+ | |||
Find all integers <math>n</math> for which each cell of <math>n \times n</math> table can be filled with one of the letters <math>I,M</math> and <math>O</math> in such a way that: | Find all integers <math>n</math> for which each cell of <math>n \times n</math> table can be filled with one of the letters <math>I,M</math> and <math>O</math> in such a way that: | ||
+ | {| border="0" cellpadding="5" | ||
+ | | valign="top"| | ||
+ | * in each row and each column, one third of the entries are <math>I</math>, one third are <math>M</math> and one third are <math>O</math>; and | ||
+ | * in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are <math>I</math>, one third are <math>M</math> and one third are <math>O</math>. | ||
+ | |} | ||
+ | '''Note.''' The rows and columns of an <math>n \times n</math> table are each labelled <math>1</math> to <math>n</math> in a natural order. Thus each cell corresponds to a pair of positive integer <math>(i,j)</math> with <math>1 \le i,j \le n</math>. For <math>n>1</math>, the table has <math>4n-2</math> diagonals of two types. A diagonal of first type consists all cells <math>(i,j)</math> for which <math>i+j</math> is a constant, and the diagonal of this second type consists all cells <math>(i,j)</math> for which <math>i-j</math> is constant. | ||
+ | |||
+ | ==Solution== | ||
+ | {{solution}} | ||
− | + | ==See Also== | |
− | |||
− | + | {{IMO box|year=2016|num-b=1|num-a=3}} |
Latest revision as of 01:35, 19 November 2023
Problem
Find all integers for which each cell of
table can be filled with one of the letters
and
in such a way that:
|
Note. The rows and columns of an table are each labelled
to
in a natural order. Thus each cell corresponds to a pair of positive integer
with
. For
, the table has
diagonals of two types. A diagonal of first type consists all cells
for which
is a constant, and the diagonal of this second type consists all cells
for which
is constant.
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
See Also
2016 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |