Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2016 IMO Problems/Problem 2"

(Created page)
 
m
Line 1: Line 1:
 
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:
−
[LIST]
+
 
 
[*] 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>.[/*]
−
[/LIST]
+
 
 
[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.
 
[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.

Revision as of 19:02, 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 $O$.[/*]

[b]Note.[/b] 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.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2016_IMO_Problems/Problem_2&oldid=113497"