6th Putnam 1946 Problem A5 Let T be a tangent plane to the ellipsoid x 2 /a 2 + y 2 /b 2 + z 2 /c 2 = 1. What is the smallest possible volume for the tetrahedral volume bounded by T and the planes x = 0, y = 0, z = 0? Solution
: √3 abc/2. The normal at (x 0 , y 0 , z 0 ) is (xx 0 /a 2 , yy 0 /b 2 , zz 0 /c 2 ). So the tangent plane is x x 0 /a 2 + y y 0 /b 2 + z z 0 /c 2 = 1. This cuts the three axes at x = a 2 /x 0 , y = b 2 /y 0 , z = c 2 /z 0 . We can regard two of these lengths as defining the base of the tetrahedron, and the third as forming its height. Hence its volume is a 2 b 2 c 2 /(6x 0 y 0 z 0 ). We wish to maximize x 0 y 0 z 0 . That is equivalent to maximising x 0 2 /a 2 y 0 2 /b 2 z 0 2 /c 2 . But we know that the sum of these three numbers is 1, so their maximum product is 1/27 (achieved when they are all equal - the arithmetic/geometric mean result). Hence x 0 y 0 z 0 has maximum value abc/(3√3). 6th Putnam 1946 © John Scholes jscholes@kalva.demon.co.uk 14 Dec 1999