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6

Part of 2012 Putnam

Problems(2)

Putnam 2012 A6

Source:

12/3/2012
Let f(x,y)f(x,y) be a continuous, real-valued function on R2.\mathbb{R}^2. Suppose that, for every rectangular region RR of area 1,1, the double integral of f(x,y)f(x,y) over RR equals 0.0. Must f(x,y)f(x,y) be identically 0?0?
Putnamfunctioncalculusintegrationrectanglecollege contestsreal analysis
Putnam 2012 B6

Source:

12/3/2012
Let pp be an odd prime number such that p≡2(mod3).p\equiv 2\pmod{3}. Define a permutation π\pi of the residue classes modulo pp by π(x)≡x3(modp).\pi(x)\equiv x^3\pmod{p}. Show that π\pi is an even permutation if and only if p≡3(mod4).p\equiv 3\pmod{4}.
Putnammodular arithmeticEulerquadraticsalgebrapolynomiallinear algebra