64th Putnam 2003

A1.  Given n, how many ways can we write n as a sum of one or more positive integers a1 ≤ a2 ≤ ... ≤ ak with ak - a1 = 0 or 1.
A2.  a1, a2, ... , an, b1, ... , bn are non-negative reals. Show that (∏ ai)1/n + (∏bi)1/n ≤ (∏(ai+bi))1/n.
A3.  Find the minimum of |sin x + cos x + tan x + cot x + sec x + cosec x| for real x.
A4.  a, b, c, A, B, C are reals with a, A non-zero such that |ax2 + bx + c| ≤ |Ax2 + Bx + C| for all real x. Show that |b2 - 4ac| ≤ |B2 - 4AC|.
A5.  An n-path is a lattice path starting at (0,0) made up of n upsteps (x,y) → (x+1,y+1) and n downsteps (x,y) → (x-1,y-1). A downramp of length m is an upstep followed by m downsteps ending on the line y = 0. Find a bijection between the (n-1)-paths and the n-paths which have no downramps of even length.
A6.  Is it possible to partition {0, 1, 2, 3, ... } into two parts such that n = x + y with x ≠ y has the same number of solutions in each part for each n?
B1.  Do their exist polynomials a(x), b(x), c(y), d(y) such that 1 + xy + x2y2 ≡ a(x)c(y) + b(x)d(y)?
B2.  Given a sequence of n terms, a1, a2, ... , an the derived sequence is the sequence (a1+a2)/2, (a2+a3)/2, ... , (an-1+an)/2 of n-1 terms. Thus the (n-1)th derivative has a single term. Show that if the original sequence is 1, 1/2, 1/3, ... , 1/n and the (n-1)th derivative is x, then x < 2/n.
B3.  Show that ∏i=1n lcm(1, 2, 3, ... , [n/i]) = n!.
B4.  az4 + bz3 + cz2 + dz + e has integer coefficients (with a ≠ 0) and roots r1, r2, r3, r4 with r1+r2 rational and r3+r4 ≠ r1+r2. Show that r1r2 is rational.
B5.  ABC is an equilateral triangle with circumcenter O. P is a point inside the circumcircle. Show that there is a triangle with side lengths |PA|, |PB|, |PC| and that its area depends only on |PO|.
B6.  Show that ∫0101 |f(x) + f(y)| dx dy ≥ ∫01 |f(x)| dx for any continuous real-valued function on [0,1].


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


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© John Scholes
9 Dec 2003
Last corrected/updated 9 Dec 03