MathDB

2009 Postal Coaching

Part of Postal Coaching

Subcontests

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\sqrt{ m^2 - 4} < 2\sqrt{n} −m < \sqrt{ m^2 - 2} , diophantine

Find all pairs (m,n)(m, n) of positive integers mm and nn for which one has m2−4<2n−m<m2−2\sqrt{ m^2 - 4} < 2\sqrt{n} - m < \sqrt{ m^2 - 2}

n lamps numbered 1, 2, ..., n be connected in cyclic order , turn on off lamps

Let n>2n > 2 and nn lamps numbered 1,2,...,n1, 2, ..., n be connected in cyclic order: 11 to 2,22, 2 to 3,...,n−13, ..., n-1 to n,nn, n to 11. At the beginning all lamps are off. If the switch of a lamp is operated, the lamp and its 22 neighbors change status: off to on, on to off. Prove that if 33 does not divide nn, then (all the) 2n2^n configurations can be reached and if 33 divides nn, then 2n−22^{n-2} configurations can be reached.

(f(x+y)+f(x)) / (2x+f(y)) = (2y+f(x) ) / (f(x+y)+f(y))

Find all functions f:N→Nf : N \to N such that f(x+y)+f(x)2x+f(y)=2y+f(x)f(x+y)+f(y)\frac{f(x+y)+f(x)}{2x+f(y)}= \frac{2y+f(x)}{f(x+y)+f(y)} , for all x,yx, y in NN.
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