MathDB

1

Part of 2012 Indonesia MO

Problems(2)

Even non-negative integer

Source: Indonesia National Science Olympiad 2012 D1 P1

9/4/2012
Show that for any positive integers aa and bb, the number n=LCM(a,b)+GCD(a,b)−a−bn=\mathrm{LCM}(a,b)+\mathrm{GCD}(a,b)-a-b is an even non-negative integer.
Proposer: Nanang Susyanto
number theoryleast common multiplegreatest common divisornumber theory proposed
Same matrices

Source: Indonesia National Science Olympiad D2 P1

9/5/2012
Given positive integers mm and nn. Let PP and QQ be two collections of m×nm \times n numbers of 00 and 11, arranged in mm rows and nn columns. An example of such collections for m=3m=3 and n=4n=4 is [111011000000].\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 PP, from left to right, the numbers are non-increasing, (ii) On each column of QQ, from top to bottom, the numbers are non-increasing, (iii) The sum of numbers on the row in PP equals to the same row in QQ, (iv) The sum of numbers on the column in PP equals to the same column in QQ. Show that the number on row ii and column jj of PP equals to the number on row ii and column jj of QQ for i=1,2,…,mi=1,2,\dots,m and j=1,2,…,nj=1,2,\dots,n.
Proposer: Stefanus Lie
combinatorics proposedcombinatorics