34th Putnam 1973

A1.  ABC is a triangle. P, Q, R are points on the sides BC, CA, AB. Show that one of the triangles AQR, BRP, CPQ has area no greater than PQR. If BP ≤ PC, CQ ≤ QA, AR ≤ RB, show that the area of PQR is at least 1/4 of the area of ABC.
A2.  an = ±1/n and an+8 > 0 iff an > 0. Show that if four of a1, a2, ... , a8 are positive, then ∑ an converges. Is the converse true?
A3.  n is a positive integer. Prove that [√(4n + 1) ] = [ min (k + n/k) ], where the minimum is taken over all positive integers k.
A4.  How many real roots does 2x = 1 + x2 have?
A5.  An object's equations of motion are: x' = yz, y' = zx, z' = xy. Its coordinates at time t = 0 are (x0, y0, z0). If two of these coordinates are zero, show that the object is stationary for all t. If (x0, y0, z0) = (1, 1, 0), show that at time t, (x, y, z) = (sec t, sec t, tan t). If (x0, y0, z0) = (1, 1, -1), show that at time t, (x, y, z) = (1/(1 + t), 1/(1 + t), -1/(1 + t) ). If two of the coordinates x0, y0, z0 are non-zero, show that the object's distance from the origin d → ∞ at some finite time in the past or future.
A6.  Show that there are no seven lines in the plane such that there are at least six points on just three lines and at least four points on just two lines.
B1.  S is a finite collection of integers, not necessarily distinct. If any element of S is removed, then the remaining integers can be divided into two collections with the same size and the same sum. Show that all elements of S are equal.
B2.  The real and imaginary parts of z are rational, and z has unit modulus. Show that |z2n - 1| is rational for any integer n.
B3.  The prime p has the property that n2 - n + p is prime for all positive integers less than p. Show that there is exactly one integer triple (a, b, c) such that b2 - 4ac = 1 - 4p, 0 < a ≤ c, -a ≤ b < a.
B4.  f is defined on the closed interval [0, 1], f(0) = 0, and f has a continuous derivative with values in (0, 1]. By considering the inverse f -1 or otherwise, show that ( ∫01 f(x) dx )2 ≥ ∫01 f(x)3 dx. Give an example where we have equality.
B5.  If x is a solution of the quadratic ax2 + bx + c = 0, show that, for any n, we can find polynomials p and q with rational coefficients such that x = p(xn, a, b, c) / q(xn, a, b, c). Hence or otherwise find polynomials r, s with rational coefficients so that x = r (x3, x + 1/x) / s(x3, x + 1/x).
B6.  Show that sin2x sin 2x has two maxima in the interval [0, 2π], at π/3 and 4π/3. Let f(x) = the absolute value of sin2x sin34x sin38x ... sin32n-1x sin 2nx. Show that f(π/3) ≥ f(x). Let g(x) = sin2x sin24x sin28x ... sin22nx. Show that g(x) ≤ 3n/4n.

To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems. The original text and official solutions were published in American Mathematical Monthly 81 (1973) 1089-95. They are also available in: Gerald L Alexanderson et al, The William Lowell Putnam Mathematical Competition, 1965-1984. Out of print.

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© John Scholes
22 Aug 2001