# Putnam exam 2003

 Questions to the 64rd annual Putnam exam: Source: http://www.math.niu.edu/~rusin/problems-math/ Questions from the 64th Putnam exam, Dec 6 2003. A1 Let n be a fixed positive integer. How many ways are there to write n as a sum of positive integers, n = a_1 + a_2 + ... + a_k, with k an arbitrary positive integer and a_1 \le a_2 \le .. a_k \le a_1 + 1? For example, with n=4, there are four ways: 4, 2+2, 1+1+2, 1+1+1+1. A2 Let a1, a2, ..., a_n and b1, b2, ..., b_n be nonnegative real numbers. Show that (a1 a2 ... a_n)^{1/n} + (b1 b2 ... b_n)^{1/n} <= ( (a1+b1) (a2+b2) ... (a_n + b_n) )^{1/n} A3 Find the minimum value of | sin x + cos x + tan x + cot x + sec x + csc x | for real numbers x. A4 Suppose that a,b,c,A,B= \int_0^1 |f(x)| dx.