35th Putnam 1974

A1.  S is a subset of {1, 2, 3, ... , 16} which does not contain three integers which are relatively prime in pairs. How many elements can S have?
A2.  C is a vertical circle fixed to a horizontal line. P is a fixed point outside the circle and above the horizontal line. For a point Q on the circle, f(Q) ∈ (0, ∞] is the time taken for a particle to slide down the straight line from P to Q (under the influence of gravity). What point Q minimizes f(Q)?
A3.  Which odd primes p can be written in the form m2 + 16n2? In the form 4m2 + 4mn + 5n2 ,where m and n may be negative? [You may assume that p can be written in the form m2 + n2 iff p = 1 (mod 4).]
A4.  Find 1/2n-11[n/2] (n - 2i) nCi, where nCi is the binomial coefficient.
A5.  The parabola y = x2 rolls around the fixed parabola y = -x2. Find the locus of its focus (initially at x = 0, y = 1/4).
A6.  Let f(n) be the degree of the lowest order polynomial p(x) with integer coefficients and leading coefficient 1, such that n divides p(m) for all integral m. Describe f(n). Evaluate f(1000000).
B1.  P, Q, R, S, T are points on a circle radius 1. How should they be placed to maximise the sum of the perimeter and the five diagonals?
B2.  f(x) is a real valued function on the reals, and has a continuous derivative. f '(x)2 + f(x)3 → 0 as x → ∞. Show that f(x) and f '(x) → 0 as x → ∞.
B3.  Prove that (cos-1(1/3) )/π is irrational.
B4.  R is the reals. f : R2→R is such that fx0 : R → R defined by fx0(x) = f(x0, x) is continuous for every x0 and gy0 : R → R defined by gy0(x) = f(x, y0) is continuous for every y0. Show that there is a sequence of continuous functions hn : R2 → R which tend to f pointwise.
B5.  Let fn(x) = ∑0n xi/i! . Show that f(n) > en/2. [Assume ex - f(x) = 1/n! ∫0x (x - t)n et dt, and ∫0 tn e-t dt = n! ]
B6.  Let S be a set with 1000 elements. Find a, b, c, the number of subsets R of S such that |R| = 0, 1, 2 (mod 3) respectively. Find a, b, c if |S| = 1001.

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 82 (1975) 907-12. They are also available (with the solutions expanded) in: Gerald L Alexanderson et al, The William Lowell Putnam Mathematical Competition, 1965-1984. Out of print, but in some university libraries.

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© John Scholes
9 Oct 1999