### 60th Putnam 1999

 A1.  Find polynomials a(x), b(x), c(x) such that |a(x)| - |b(x)| + c(x) = -1 for x < -1, 3x + 2 for -1 ≤ x ≤ 0, -2x + 2 for x > 0. A2.  Show that for some fixed positive n we can always express a polynomial with real coefficients which is nowhere negative as a sum of the squares of n polynomials. A3.  Let 1/(1 - 2x - x2) = s0 + s1x + s2x2 + ... . Prove that for some f(n) we have sn2 + sn+12 = sf(n). A4.  Let aij = i2j/(3i(j 3i + i 3j)). Find ∑ aij where the sum is taken over all pairs of integers (i, j) with i, j > 0. A5.  Find a constant k such that for any polynomial f(x) of degree 1999, we have |f(0)| ≤ k ∫-11 |f(x)| dx. A6.  u1 = 1, u2 = 2, u3 = 24, un = (6 un-12un-3 - 8 un-1un-22)/(un-2un-3). Show that un is always a multiple of n. B1.  The triangle ABC has AC = 1, ∠ACB = 90o, and ∠BAC = φ. D is the point between A and B such that AD = 1. E is the point between B and C such that ∠EDC = φ. The perpendicular to BC at E meets AB at F. Find limφ→0 EF. B2.  p(x) is a polynomial of degree n. q(x) is a polynomial of degree 2. p(x) = p''(x)q(x) and the roots of p(x) are not all equal. Show that the roots of p(x) are all distinct. B3.  Let R be the reals. Define f :[0, 1) x [0, 1) → R by f(x, y) = ∑ xmyn, where the sum is taken over all pairs of positive integers (m, n) satisfying m ≥ n/2, n ≥ m/2. Find lim(x, y)→(1, 1) (1 - xy2)(1 - x2y)f(x, y). B4.  Let R be the reals. f :R → R is three times differentiable, and f(x), f '(x), f ''(x), f '''(x) are all positive for all x. Also f(x) ≥ f '''(x) for all x. Show that f '(x) < 2 f(x) for all x. B5.  n is an integer greater than 2 and φ = 2π/n. A is the n x n matrix (aij), where aij = cos( (i + j)φ) for i ≠ j, 1 + cos(2 j φ) for i = j. Find det A. B6.  X is a finite set of integers greater than 1 such that for any positive integer n, we can find m ∈ X such that m divides n or is relatively prime to n. Show that X contains a prime or two elements whose greatest common divisor is prime.

To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems.

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