MathDB

4

Part of 2010 IMO Shortlist

Problems(2)

x_1 + x_2 + ... + x_n >= 0

Source: IMO Shortlist 2010, Algebra 4

7/17/2011
A sequence x1,x2,…x_1, x_2, \ldots is defined by x1=1x_1 = 1 and x2k=−xk,x2k−1=(−1)k+1xkx_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k for all k≥1.k \geq 1. Prove that ∀n≥1\forall n \geq 1 x1+x2+…+xn≥0.x_1 + x_2 + \ldots + x_n \geq 0.
Proposed by Gerhard Wöginger, Austria
algebraInequalitySequenceIMO Shortlistget the smallest
IMO Shortlist 2010 - Problem N4

Source:

7/17/2011
Let a,ba, b be integers, and let P(x)=ax3+bx.P(x) = ax^3+bx. For any positive integer nn we say that the pair (a,b)(a,b) is nn-good if n∣P(m)−P(k)n | P(m)-P(k) implies n∣m−kn | m - k for all integers m,k.m, k. We say that (a,b)(a,b) is very goodvery \ good if (a,b)(a,b) is nn-good for infinitely many positive integers n.n. * Find a pair (a,b)(a,b) which is 51-good, but not very good. * Show that all 2010-good pairs are very good.
Proposed by Okan Tekman, Turkey
modular arithmeticDivisibilitynumber theoryIMO Shortlist