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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 [/*]
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[*]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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* 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;
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* 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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[b]Note.[/b] 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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'''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.

Revision as of 20:07, 26 December 2019

Find all integers $n$ for which each cell of $n \times n$ table can be filled with one of the letters $I,M$ and $O$ in such a way that:

  • in each row and each column, one third of the entries are $I$, one third are $M$ and one third are $O$; and;
  • in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are $I$, one third are $M$ and one third are $O$.

Note. The rows and columns of an $n \times n$ table are each labelled $1$ to $n$ in a natural order. Thus each cell corresponds to a pair of positive integer $(i,j)$ with $1 \le i,j \le n$. For $n>1$, the table has $4n-2$ diagonals of two types. A diagonal of first type consists all cells $(i,j)$ for which $i+j$ is a constant, and the diagonal of this second type consists all cells $(i,j)$ for which $i-j$ is constant.

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