Difference between revisions of "2016 IMO Problems/Problem 2"
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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: | ||
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| − | * 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 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>. | * 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>. | ||
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'''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. | '''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. | ||
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| + | ==Solution== | ||
| + | {{solution}} | ||
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| + | ==See Also== | ||
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| + | {{IMO box|year=2016|num-b=1|num-a=3}} | ||
Latest revision as of 00:35, 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:
|
, one third are 
and one third are 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
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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 | ||
