HMMT February

Algebra Difficulty 5.0 AIME, harder Find the answer

(Maximal Determinant) In a $17 \times 17$ matrix $M$ , all entries are $\pm 1$ . The maximum possible value of $|\operatorname{det} M|$ is $N$ . Estimate $N$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Answer: $327680 \cdot 2^{16}$ This is Hadamard's maximal determinant problem. There's an upper bound of $n^{\frac{1}{2} n}$ which empirically seems to give reasonably good estimates, but in fact this is open for general $n$ .

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