International IberoAmerican Competition 1987 First Day Prove that if \(m\), \(n\), \(r\) are positive integers with \(r\) odd such that \[\left(2+\sqrt{3}\right)^r=1+m+n\sqrt{3}.\] then \(m\) is a perfect square. 証明 \(n\)を自然数とし, \((2+\sqrt{3})^n=a_n+b_n\sqrt{3}\)とおくと \[\begin{split}(2+\sqrt{3})^{n+1…