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 1/2^n-1 ∑₁^[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 f_x₀ : R → R defined by f_x₀(x) = f(x₀, x) is continuous for every x₀ and g_y₀ : R → R defined by g_y₀(x) = f(x, y₀) is continuous for every y₀. Show that there is a sequence of continuous functions h_n : R² → R which tend to f pointwise.
B5. Let f_n(x) = ∑₀ⁿ xⁱ/i! . Show that f(n) > eⁿ/2. [Assume e^x - f(x) = 1/n! ∫₀^x (x - t)ⁿ e^t 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.