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Putnam 1941 A7

Source: Putnam 1941

February 23, 2022
Putnamlinear algebramatrixconicsellipse

Problem Statement

Do either (1) or (2): (1) Prove that the determinant of the matrix (1+a2−b2−c22(ab+c)2(ac−b)2(ab−c)1−a2+b2−c22(bc+a)2(ac+b)2(bc−a)1−a2−b2+c2)\begin{pmatrix} 1+a^2 -b^2 -c^2 & 2(ab+c) & 2(ac-b)\\ 2(ab-c) & 1-a^2 +b^2 -c^2 & 2(bc+a)\\ 2(ac+b)& 2(bc-a) & 1-a^2 -b^2 +c^2 \end{pmatrix} is given by (1+a2+b2+c2)3(1+a^2 +b^2 +c^2)^{3}.
(2) A solid is formed by rotating the first quadrant of the ellipse x2a2+y2b2=1\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 around the xx-axis. Prove that this solid can rest in stable equilibrium on its vertex if and only if ab≤85\frac{a}{b}\leq \sqrt{\frac{8}{5}}.
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