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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.
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A2. Define f(x) = ∫0x ∑i=0n-1 (x - t)i / i! dt. Find the nth derivative f (n)(x).
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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.
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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?
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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).
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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.
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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).
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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).
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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);
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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.
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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.
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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.]
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B6. f is a real valued function on [0, 1], continuous on (0, 1). Prove that ∫x=0x=1 ∫y=xy=1 ∫z=xz=y f(x) f(y) f(z) dz dy dx = 1/6 ( ∫x=0x=1 f(x) dx )3.
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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).
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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.
