2nd Putnam 1939

A1.  Let C be the curve y2 = x3 (where x takes all non-negative real values). Let O be the origin, and A be the point where the gradient is 1. Find the length of the curve from O to A.
A2.  Let C be the curve y = x3 (where x takes all real values). The tangent at A meets the curve again at B. Prove that the gradient at B is 4 times the gradient at A.
A3.  The roots of x3 + a x2 + b x + c = 0 are α, β and γ. Find the cubic whose roots are α3, β3, γ3.
A4.  Given 4 lines in Euclidean 3-space:

L1:   x = 1, y = 0;
L2:   y = 1, z = 0;
L3:   x = 0, z = 1;
L4:   x = y, y = -6z.

Find the equations of the two lines which both meet all of the Li.

A5.  Do either (1) or (2)

(1)   x and y are functions of t. Solve x' = x + y - 3, y' = -2x + 3y + 1, given that x(0) = y(0) = 0.

(2)   A weightless rod is hinged at O so that it can rotate without friction in a vertical plane. A mass m is attached to the end of the rod A, which is balanced vertically above O. At time t = 0, the rod moves away from the vertical with negligible initial angular velocity. Prove that the mass first reaches the position under O at t = √(OA/g) ln (1 + √2).

A6.  Do either (1) or (2):

(1)   A circle radius r rolls around the inside of a circle radius 3r, so that a point on its circumference traces out a curvilinear triangle. Find the area inside this figure.

(2)   A frictionless shell is fired from the ground with speed v at an unknown angle to the vertical. It hits a plane at a height h. Show that the gun must be sited within a radius v/g (v2 - 2gh)1/2 of the point directly below the point of impact.

A7.  Do either (1) or (2):

(1)   Let Ca be the curve (y - a2)2 = x2(a2 - x2). Find the curve which touches all Ca for a > 0. Sketch the solution and at least two of the Ca.

(2)   Given that (1 - hx)-1(1 - kx)-1 = ∑i≥0  ai xi, prove that (1 + hkx)(1 - hkx)-1(1 - h2x)-1(1 - k2x)-1 = ∑i≥0  ai2 xi.

B1.  The points P(a,b) and Q(0,c) are on the curve y/c = cosh (x/c). The line through Q parallel to the normal at P cuts the x-axis at R. Prove that QR = b.
B2.  Evaluate   ∫13  ( (x - 1)(3 - x) )-1/2 dx   and   ∫1  (ex+1 + e3-x)-1 dx.
B3.  Given   an = (n2 + 1) 3n, find a recurrence relation an + p an+1 + q an+2 + r an+3 = 0. Hence evaluate n≥0 an xn.
B4.  The axis of a parabola is its axis of symmetry and its vertex is its point of intersection with its axis. Find: the equation of the parabola which touches y = 0 at (1,0) and x = 0 at (0,2); the equation of its axis; and its vertex.
B5.  Do either (1) or (2):

(1)   Prove that ∫1k [x] f '(x) dx = [k] f(k) - ∑1[k] f(n), where k > 1, and [z] denotes the greatest integer ≤ z. Find a similar expression for:   ∫1k [x2] f '(x) dx.

(2)   A particle moves freely in a straight line except for a resistive force proportional to its speed. Its speed falls from 1,000 ft/s to 900 ft/s over 1,200 ft. Find the time taken to the nearest 0.01 s. [No calculators or log tables allowed!]

B6.  Do either (1) or (2):

(1)   f is continuous on the closed interval [a, b] and twice differentiable on the open interval (a, b). Given x0 ∈ (a, b), prove that we can find ξ ∈ (a, b) such that ( (f(x0) - f(a))/(x0 - a) - (f(b) - f(a))/(b - a) )/(x0 - b) = f ''(ξ)/2.

(2)   AB and CD are identical uniform rods, each with mass m and length 2a. They are placed a distance b apart, so that ABCD is a rectangle. Calculate the gravitational attraction between them. What is the limiting value as a tends to zero?

B7.  Do either (1) or (2):

(1)   Let ai = ∑n=0 x3n+i/(3n+i)!   Prove that a03 + a13 + a23 - 3 a0a1a2 = 1.

(2)   Let O be the origin, λ a positive real number, C be the conic   ax2 + by2 + cx + dy + e = 0, and Cλ the conic   ax2 + by2 + λcx + λdy + λ2e = 0. Given a point P and a non-zero real number k, define the transformation D(P,k) as follows. Take coordinates (x',y') with P as the origin. Then D(P,k) takes (x',y') to (kx',ky'). Show that D(O,λ) and D(A,-λ) both take C into Cλ, where A is the point (-cλ/(a(1 + λ)), -dλ/(b(1 + λ)) ). Comment on the case λ = 1.

The Putnam fellow was Edward Kaplan; Richard Feynman was also in the first 5. To avoid possible copyright problems and sometimes to increase clarity, I have changed the wording, but not the substance, of 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
4 Sep 1999