4th Putnam 1941

A1.  Prove that (a - x)6 - 3a(a - x)5 + 5/2 a2(a - x)4 - 1/2 a4(a - x)2 < 0 for 0 < x < a.
A2.  Define f(x) = ∫0xi=0n-1 (x - t)i / i! dt. Find the nth derivative f (n)(x).
A3.  A circle radius a rolls in the plane along the x-axis the envelope of a diameter is the curve C. Show that we can find a point on the circumference of a circle radius a/2, also rolling along the x-axis, which traces out the curve C.
A4.  The real polynomial x3 + px2 + qx + r has real roots a ≤ b ≤ c. Prove that f ' has a root in the interval [b/2 + c/2, b/3 + 2c/3]. What can we say about f if the root is at one of the endpoints?
A5.  The line L is parallel to the plane y = z and meets the parabola y2 = 2x, z = 0 and the parabola 3x = z2, y = 0. Prove that if L moves freely subject to these constraints then it generates the surface x = (y - z)(y/2 - z/3).
A6.  f is defined for the non-negative reals and takes positive real values. The centroid of the area lying under the curve y = f(x) between x = 0 and x = a has x-coordinate g(a). Prove that for some positive constant k, f(x) = k g'(x)/(x - g(x))2 e ∫ 1/(t - g(t)) dt.
A7.  Do either (1) or (2):

(1)   Let A be the 3 x 3 matrix

1+x2-y2-z2     2(xy+z)    2(zx-y)

  2(xy-z)    1+y2-z2-x2   2(yz+x)

  2(zx+y)     2(yz-x)   1+z2-x2-y2

Show that det A = (1 + x2 + y2 + z2)3.

(2)   A solid is formed by rotating about the x-axis the first quadrant of the ellipse x2/a2 + y2b2 = 1. Prove that this solid can rest in stable equilibrium on its vertex (corresponding to x = a, y = 0 on the ellipse) iff a/b ≤ √(8/5).

B1.  A particle moves in the plane so that its angular velocity about the point (1, 0) equals minus its angular velocity about the point (-1, 0). Show that its trajectory satisfies the differential equation y' x(x2 + y2 - 1) = y(x2 + y2 + 1). Verify that this has as solutions the rectangular hyperbolae with center at the origin and passing through (±1, 0).
B2.  Find:
(1) limn→∞1≤i≤n 1/√(n2 + i2);
(2) limn→∞1≤i≤n 1/√(n2 + i);
(3) limn→∞1≤i≤n2 1/√(n2 + i);
B3.  Let y1 and y2 be any two linearly independent solutions of the differential equation y'' + p(x) y' + q(x) y = 0. Let z = y1y2. Find the differential equation satisfied by z.
B4.  Given an ellipse center O, take two perpendicular diameters AOB and COD. Take the diameter A'OB' parallel to the tangents to the ellipse at A and B (this is said to be conjugate to the diameter AOB). Similarly, take C'OD' conjugate to COD. Prove that the rectangular hyperbola through A'B'C'D' passes through the foci of the ellipse.
B5.  A wheel radius r is traveling along a road without slipping with angular velocity ω > √(g/r). A particle is thrown off the rim of the wheel. Show that it can reach a maximum height above the road of (rω + g/ω)2/(2g). [Ignore air resistance.]
B6.  f is a real valued function on [0, 1], continuous on (0, 1). Prove that ∫x=0x=1y=xy=1z=xz=y f(x) f(y) f(z) dz dy dx = 1/6 ( ∫x=0x=1 f(x) dx )3.
B7.  Do either (1) or (2):

(1)   f is a real-valued function defined on the reals with a continuous second derivative and satisfies f(x + y) f(x - y) = f(x)2 + f(y)2 - 1 for all x, y. Show that for some constant k we have f ''(x) = ± k2 f(x). Deduce that f(x) is one of ±cos kx, ±cosh kx.

(2)   ai and bi are constants. Let A be the (n+1) x (n+1) matrix Aij, defined as follows: Ai1 = 1; A1j = xj-1 for j ≤ n; A1 (n+1) = p(x); Aij = ai-1j-1 for i > 1, j ≤ n; Ai (n+1) = bi-1 for i > 1. We use the identity det A = 0 to define the polynomial p(x). Now given any polynomial f(x), replace bi by f(bi) and p(x) by q(x), so that det A = 0 now defines a polynomial q(x). Prove that f( p(x) ) is a multiple of ∏ (x - ai) plus q(x).

The Putnam fellow (winner) was Richard Arens, his (winning) mark is unknown. 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
28 Sep 1999
Last corrected/updated 19 Jan 04