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Turkish NMO First Round - 2012 Problem - 31 {Algebra}

Source:

July 1, 2012

Problem Statement

f:ZZf : \mathbb{Z} \rightarrow \mathbb{Z} satisfies m+f(m+f(n+f(m)))=n+f(m)m+f(m+f(n+f(m))) = n + f(m) for every integers m,nm,n. If f(6)=6f(6) = 6, then f(2012)=?f(2012) = ?
<spanclass=latexbold>(A)</span> 2010<spanclass=latexbold>(B)</span> 2000<spanclass=latexbold>(C)</span> 2000<spanclass=latexbold>(D)</span> 2010<spanclass=latexbold>(E)</span> 2012 <span class='latex-bold'>(A)</span>\ -2010 \qquad <span class='latex-bold'>(B)</span>\ -2000 \qquad <span class='latex-bold'>(C)</span>\ 2000 \qquad <span class='latex-bold'>(D)</span>\ 2010 \qquad <span class='latex-bold'>(E)</span>\ 2012