MathDB

2

Part of 2009 Indonesia MO

Problems(2)

The minimum value of a function

Source: Indonesian MO (INAMO) 2009, Day 2, Problem 6

8/8/2009
Find the lowest possible values from the function f(x) \equal{} x^{2008} \minus{} 2x^{2007} \plus{} 3x^{2006} \minus{} 4x^{2005} \plus{} 5x^{2004} \minus{} \cdots \minus{} 2006x^3 \plus{} 2007x^2 \minus{} 2008x \plus{} 2009 for any real numbers x x.
functioninequalities unsolvedinequalities
Inequalities in floor function

Source: Indonesian MO (INAMO) 2009, Day 1, Problem 2

8/8/2009
For any real x x, let ⌊x⌋ \lfloor x\rfloor be the largest integer that is not more than x x. Given a sequence of positive integers a1,a2,a3,… a_1,a_2,a_3,\ldots such that a1>1 a_1>1 and \left\lfloor\frac{a_1\plus{}1}{a_2}\right\rfloor\equal{}\left\lfloor\frac{a_2\plus{}1}{a_3}\right\rfloor\equal{}\left\lfloor\frac{a_3\plus{}1}{a_4}\right\rfloor\equal{}\cdots Prove that \left\lfloor\frac{a_n\plus{}1}{a_{n\plus{}1}}\right\rfloor\leq1 holds for every positive integer n n.
inequalitiesfunctionfloor functionnumber theory unsolvednumber theory