11th Putnam 1951

A1.  A is a skew-symmetric real 4 x 4 matrix. Show that det A ≥ 0.
A2.  k is a positive real and P1, P2, ... , Pn are points in the plane. What is the locus of P such that ∑ PPi2 = k? State in geometric terms the conditions on k for such points P to exist.
A3.  Find ∑0 (-1)n/(3n + 1).
A4.  Sketch the curve y4 - x4 - 96y2 + 100x2 = 0.
A5.  Show that a line in the plane with rational slope contains either no lattice points or an infinite number. Show that given any line L of rational slope we can find δ > 0, such that no lattice point is a distance k from L where 0 < k < δ.
A6.  Let C be a parabola. Take points P, Q on C such that (1) PQ is perpendicular to the tangent at P, (2) the area enclosed by the parabola and PQ is as small as possible. What is the position of the chord PQ?
A7.  Show that if ∑ an converges, then so does ∑ an/n.
B1.  R is the reals. f, g : R2 → R have continuous partial derivatives of all orders. What conditions must they satisfy for the differential equation f(x, y) dx + g(x, y) dy = 0 to have an integrating factor h(xy)?
B2.  R is the reals. Find an example of functions f, g : R → R, which are differentiable, not identically zero, and satisfy (f/g)' = f '/g' .
B3.  Show that ln(1 + 1/x) > 1/(1 + x) for x > 0.
B4.  Can we find four distinct concentric circles all touching an ellipse?
B5.  T is a torus, center O. The plane P contains O and touches T. Prove that P ∪ T is two circles.
B6.  The real polynomial p(x) = x3 + ax2 + bx + c has three real roots α < β < γ. Show that √(a2 - 3b) < (γ - α) ≤ 2 √(a2/3 - b).
B7.  In 4-space let S be the 3-sphere radius r: w2 + x2 + y2 + z2 = r2. What is the 3-dimensional volume of S? What is the 4-dimensional volume of its interior?

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.

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© John Scholes
20 Oct 1999