13th Putnam 1953 Problem B4 R is the reals. S is a surface in R 3 containing the point (1, 1, 1) such that the tangent plane at any point P ∈ S cuts the axes at three points whose orthocenter is P. Find the equation of S.
Consider a plane cutting the axes at a = (a, 0, 0), b = (0, b, 0), c = (0, 0, c). If the orthocentre is at p = (x, y, z), then we have ( p - a ).( b - c ) = ( p - b ).( a - c ) = 0. But a . b = b . c = c . a = 0, so we have p .( b - c ) = p .( a - c ) = 0. In other words the line from the origin (0, 0, 0) to (x, y, z) is normal to the plane. So the surface satisfies the condition that all its normals pass through the origin and it passes through (1, 1, 1). This implies that it is the sphere x 2 + y 2 + z 2 = 3. Note, however, that for points with one coordinate zero, the tangent plane will meet one axis at infinity, so we should arguable exclude all such points. That divides the sphere into 8 disconnected pieces. The piece containing (1, 1, 1) is that in the positive octant (x > 0, y > 0, z > 0). 13th Putnam 1953 © John Scholes jscholes@kalva.demon.co.uk 5 Mar 2002