Pentru \( n\ge 2, n\in\mathbb{N} \), notam cu \( M_{2,n}(\{-1,1\}) \) multimea matricelor de tip \( (2,n) \) cu elemente in multimea \( \{-1,1\} \). Daca \( A \in M_{2,n}(\{-1,1\}) \), notam cu \( m_A \)numarul minorilor nenuli de ordin 2 ai matricei \( A \). Sa se calculeze \( \max\{m_A|\ A \in M_{2,n}(\{-1,1\})\} \).
Laura Nastasescu, Bucuresti
Concursul GM "Nicolae Teodorescu" 2009, problema 1
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Laurentiu Tucaa
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Concursul GM "Nicolae Teodorescu" 2009, problema 1
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Marius Mainea
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Maximul cautat este \( \max\{a\cdot b|a,b\in \mathbb{N^\ast}, a+b=n\} \) adica
\( \left{\begin{array}{cc} \frac{n^2}{4}, \mbox{~daca n par~}\\\frac{n^2-1}{4}, \mbox{~daca n impar\end{array} \)
P.S. \( a \) este numarul coloanelor de tipul \( \left(\begin{array}{cc}1\\1\end{array}\right) \) si \( \left(\begin{array}{cc}-1\\-1\end{array}\right) \), iar \( b=n-a \).
\( \left{\begin{array}{cc} \frac{n^2}{4}, \mbox{~daca n par~}\\\frac{n^2-1}{4}, \mbox{~daca n impar\end{array} \)
P.S. \( a \) este numarul coloanelor de tipul \( \left(\begin{array}{cc}1\\1\end{array}\right) \) si \( \left(\begin{array}{cc}-1\\-1\end{array}\right) \), iar \( b=n-a \).