3rd Putnam 1940

A1.  p(x) is a polynomial with integer coefficients. For some positive integer c, none of p(1), p(2), ... , p(c) are divisible by c. Prove that p(b) is not zero for any integer b.
A2.  y = f(x) is continuous with continuous derivative. The arc PQ is concave to the chord PQ. X is chosen on the arc PQ to maximize PX + XQ. Prove that XP and XQ are equally inclined to the tangent at X.
A3.  α is a fixed real number. Find all functions f: R → R (where R is the reals) which are continuous, have a continuous derivative, and satisfy   ∫by fα(x) dx = ( ∫by f(x) dx )α for all y and some b.
A4.  p is a positive constant. Let R is the curve y2 = 4px. Let S be the mirror image of R in the y-axis (y2 = - 4px). R remains fixed and S rolls around it without slipping. O is the point of S initially at the origin. Find the equation for the locus of O as S rolls?
A5.  Prove that the set of points satisfying x4 - x2 = y4 - y2 = z4 - z2 is the union of 4 straight lines and 6 ellipses.
A6.  p(x) is a polynomial with real coefficients and derivative r(x) = p'(x). For some positive integers a, b, ra(x) divides pb(x). Prove that for some real numbers A and α and for some integer n, we have p(x) = A(x - α)n.
A7.  ai and bi are real, and ∑1 ai2 and ∑1 bi2 converge. Prove that ∑1 (ai - bi)p converges for p ≥ 2.
A8.  Show that the area of the triangle bounded by the lines aix + biy + ci = 0 (i = 1, 2, 3) is Δ2/|2(a2b3 - a3b2)(a3b1 - a1b3)(a1b2 - a2b1)|, where Δ is the 3 x 3 determinant with columns ai, bi, ci.
B1.  A stone is thrown from the ground with speed v at an angle θ to the horizontal. There is no friction and the ground is flat. Find the total distance it travels before hitting the ground. Show that the distance is greatest when sin θ ln (sec θ + tan θ) = 1.
B2.  C1, C2 are cylindrical surfaces with radii r1, r2 respectively. The axes of the two surfaces intersect at right angles and r1 > r2. Let S be the area of C1 which is enclosed within C2. Prove that S = 8r22A = 8r12C - 8(r12 - r22)B, where A = ∫01 (1 - x2)1/2(1 - k2x2)-1/2 dx, B = ∫01 (1 - x2)-1/2(1 - k2x2)-1/2 dx, and C = ∫01 (1 - x2)-1/2(1 - k2x2)1/2 dx, and k = r2/r1.
B3.  Let p be a positive real, let S be the parabola y2 = 4px, and let P be a point with coordinates (a, b). Show that there are 1, 2 or 3 normals from P to S according as 4(2p - a)2 + 27 pb2 >, = or < 0.
B4.  Let S be the surface ax2 + by2 + cz2 = 1 (a, b, c all non-zero), and let K be the sphere x2 + y2 + z2 = 1/a + 1/b + 1/c (known as the director sphere). Prove that if a point P lies on 3 mutually perpendicular planes, each of which is tangent to S, then P lies on K.

Comment. The original question also asked, apparently in error, for a proof that every point of K had this property, which is (a) false unless we allow planes which are asymptotes (or tangents at infinity), and (b) unreasonably hard - at least, I cannot see a neat proof.

B5.  Find all rational triples (a, b, c) for which a, b, c are the roots of x3 + ax2 + bx + c = 0.
B6.  The n x n matrix (mij) is defined as mij = aiaj for i ≠ j, and ai2 + k for i = j. Show that det(mij) is divisible by kn-1 and find its other factor.
B7.  Given n > 8, let a = √n and b = √(n+1). Which is greater ab or ba?

The Putnam fellow (winner) was Andrew M Gleason, 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
14 Sep 1999