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6

Part of 2003 Putnam

Problems(2)

Putnam 2003 A6

Source:

6/23/2011
For a set SS of nonnegative integers, let rS(n)r_S(n) denote the number of ordered pairs (s1,s2)(s_1, s_2) such that s1∈Ss_1 \in S, s2∈Ss_2 \in S, s1≠s2s_1 \neq s_2, and s1+s2=ns_1 + s_2 = n. Is it possible to partition the nonnegative integers into two sets AA and BB in such a way that rA(n)=rB(n)r_A(n) = r_B(n) for all nn?
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Putnam 2003 B6

Source:

6/23/2011
Let f(x)f(x) be a continuous real-valued function defined on the interval [0,1][0, 1]. Show that ∫01∫01∣f(x)+f(y)∣dx  dy≥∫01∣f(x)∣dx\int_0^1\int_0^1|f(x)+f(y)|dx \; dy \ge \int_0^1 |f(x)|dx
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