MathDB
Delete a,b; Write 2a+b, 2b+a

Source: P28 [Combinatorics] - Turkish NMO 1st Round - 2014

May 23, 2014
invariantinequalitiesabsolute valuecombinatorics proposedcombinatorics

Problem Statement

The integers 1-1, 22, 3-3, 44, 5-5, 66 are written on a blackboard. At each move, we erase two numbers aa and bb, then we re-write 2a+b2a+b and 2b+a2b+a. How many of the sextuples (0,0,0,3,9,9)(0,0,0,3,-9,9), (0,1,1,3,6,6)(0,1,1,3,6,-6), (0,0,0,3,6,9)(0,0,0,3,-6,9), (0,1,1,3,6,9)(0,1,1,-3,6,-9), (0,0,2,5,5,6)(0,0,2,5,5,6) can be gotten?
<spanclass=latexbold>(A)</span> 1<spanclass=latexbold>(B)</span> 2<spanclass=latexbold>(C)</span> 3<spanclass=latexbold>(D)</span> 4<spanclass=latexbold>(E)</span> 5 <span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ 5