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Difference between revisions of "2012 Indonesia MO Problems/Problem 5"

(Created page with "==Problem== Let <math>n</math> be a positive integer. Show that the equation<cmath>\sqrt{x}+\sqrt{y}=\sqrt{n}</cmath>have solution of pairs of positive integers <math>(x,y)</m...")
 
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==Problem==
 
==Problem==
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Let <math>n</math> be a positive integer. Show that the equation<cmath>\sqrt{x}+\sqrt{y}=\sqrt{n}</cmath>have solution of pairs of positive integers <math>(x,y)</math> if and only if <math>n</math> is divisible by some perfect square greater than <math>1</math>.
+
Given positive integers <math>m</math> and <math>n</math>. Let <math>P</math> and <math>Q</math> be two collections of <math>m \times n</math> numbers of <math>0</math> and <math>1</math>, arranged in <math>m</math> rows and <math>n</math> columns. An example of such collections for <math>m=3</math> and <math>n=4</math> is
 +
<cmath>\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right].</cmath>
 +
Let those two collections satisfy the following properties:
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(i) On each row of <math>P</math>, from left to right, the numbers are non-increasing,
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(ii) On each column of <math>Q</math>, from top to bottom, the numbers are non-increasing,
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(iii) The sum of numbers on the row in <math>P</math> equals to the same row in <math>Q</math>,
 +
(iv) The sum of numbers on the column in <math>P</math> equals to the same column in <math>Q</math>.
 +
Show that the number on row <math>i</math> and column <math>j</math> of <math>P</math> equals to the number on row <math>i</math> and column <math>j</math> of <math>Q</math> for <math>i=1,2,\dots,m</math> and <math>j=1,2,\dots,n</math>.
  
 
==Solution==
 
==Solution==
  
−
 
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let the collection X be named
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<math>\begin{bmatrix} X_{1,1}&X_{1,2}&\dots&X_{1,n}\\X_{2,1}&X{2,2}&\dots&X_{2,n}\\\vdots&\vdots&\vdots&\vdots\\X_{m,1}&X_{m,2}&\dots&\X_{m,n}\end{bmatrix}</math>
  
 
==See Also==
 
==See Also==

Revision as of 04:47, 24 December 2024

Problem

Given positive integers $m$ and $n$. Let $P$ and $Q$ be two collections of $m \times n$ numbers of $0$ and $1$, arranged in $m$ rows and $n$ columns. An example of such collections for $m=3$ and $n=4$ is \[\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right].\] Let those two collections satisfy the following properties: (i) On each row of $P$, from left to right, the numbers are non-increasing, (ii) On each column of $Q$, from top to bottom, the numbers are non-increasing, (iii) The sum of numbers on the row in $P$ equals to the same row in $Q$, (iv) The sum of numbers on the column in $P$ equals to the same column in $Q$. Show that the number on row $i$ and column $j$ of $P$ equals to the number on row $i$ and column $j$ of $Q$ for $i=1,2,\dots,m$ and $j=1,2,\dots,n$.

Solution

let the collection X be named $\begin{bmatrix} X_{1,1}&X_{1,2}&\dots&X_{1,n}\\X_{2,1}&X{2,2}&\dots&X_{2,n}\\\vdots&\vdots&\vdots&\vdots\\X_{m,1}&X_{m,2}&\dots&\X_{m,n}\end{bmatrix}$ (Error compiling LaTeX. Unknown error_msg)

See Also

2012 Indonesia MO (Problems)
Preceded by
Problem 4 		1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 Followed by
Problem 6
All Indonesia MO Problems and Solutions
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