|A1. A solid in Euclidean 3-space extends from z = -h/2 to z = +h/2 and the area of the section z = k is a polynomial in k of degree at most 3. Show that the volume of the solid is h(B + 4M + T)/6, where B is the area of the bottom (z = -h/2), M is the area of the middle section (z = 0), and T is the area of the top (z = h/2). Derive the formulae for the volumes of a cone and a sphere.|
|A2. A solid has a cylindrical middle with a conical cap at each end. The height of each cap equals the length of the middle. For a given surface area, what shape maximizes the volume?|
|A3. A particle moves in the Euclidean plane. At time t (taking all real values) its coordinates are x = t3 - t and y = t4 + t. Show that its velocity has a maximum at t = 0, and that its path has an inflection at t = 0.|
|A4. A notch is cut in a cylindrical vertical tree trunk. The notch penetrates to the axis of the cylinder and is bounded by two half-planes. Each half-plane is bounded by a horizontal line passing through the axis of the cylinder. The angle between the two half-planes is θ. Prove that the volume of the notch is minimized (for given tree and θ) by taking the bounding planes at equal angles to the horizontal plane.|
A5. (1) Find limx->inf x2/ex.
(2) Find limk->0 1/k ∫0k (1 + sin 2x)1/x dx.
|A6. A swimmer is standing at a corner of a square swimming pool. She swims at a fixed speed and runs at a fixed speed (possibly different). No time is taken entering or leaving the pool. What path should she follow to reach the opposite corner of the pool in the shortest possible time?|
A7. Do either (1) or (2)
(1) S is a thin spherical shell of constant thickness and density with total mass M and center O. P is a point outside S. Prove that the gravitational attraction of S at P is the same as the gravitational attraction of a point mass M at O.
(2) K is the surface z = xy in Euclidean 3-space. Find all straight lines lying in S. Draw a diagram to illustrate them.
B1. Do either (1) or (2)
(1) Let A be matrix (aij), 1 ≤ i,j ≤ 4. Let d = det(A), and let Aij be the cofactor of aij, that is, the determinant of the 3 x 3 matrix formed from A by deleting aij and other elements in the same row and column. Let B be the 4 x 4 matrix (Aij) and let D be det B. Prove D = d3.
(2) Let P(x) be the quadratic Ax2 + Bx + C. Suppose that P(x) = x has unequal real roots. Show that the roots are also roots of P(P(x)) = x. Find a quadratic equation for the other two roots of this equation. Hence solve (y2 - 3y + 2)2 - 3(y2 - 3y + 2) + 2 - y = 0.
|B2. Find all solutions of the differential equation zz" - 2z'z' = 0 which pass through the point x=1, z=1.|
|B3. A horizontal disk diameter 3 inches rotates once every 15 seconds. An insect starts at the southernmost point of the disk facing due north. Always facing due north, it crawls over the disk at 1 inch per second. Where does it again reach the edge of the disk?|
|B4. The parabola P has focus a distance m from the directrix. The chord AB is normal to P at A. What is the minimum length for AB?|
|B5. Find the locus of the foot of the perpendicular from the center of a rectangular hyperbola to a tangent. Obtain its equation in polar coordinates and sketch it.|
|B6. What is the shortest distance between the plane Ax + By + Cz + 1 = 0 and the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1. You may find it convenient to use the notation h = (A2 + B2 + C2)-1/2, m = (a2A2 + b2B2 + c2C2)1/2. What is the algebraic condition for the plane not to intersect the ellipsoid?|
The Putnam fellow was Irving Kaplansky. To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems. The original text and solutions are available in: A M Gleason, R E Greenwood & L M Kelly, The William Lowell Putnam Mathematical Competition, Problems and Solutions, 1938-1964, MAA 1980. Out of print, but available in some university libraries.
© John Scholes
4 Sep 1999