MathDB

1

Part of 1998 IMO Shortlist

Problems(2)

IMO ShortList 1998, algebra problem 1

Source: IMO ShortList 1998, algebra problem 1

10/22/2004
Let a1,a2,…,ana_{1},a_{2},\ldots ,a_{n} be positive real numbers such that a1+a2+⋯+an<1a_{1}+a_{2}+\cdots +a_{n}<1. Prove that a1a2⋯an[1−(a1+a2+⋯+an)](a1+a2+⋯+an)(1−a1)(1−a2)⋯(1−an)≤1nn+1. \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}.
inequalitiesalgebraIMO Shortlistn-variable inequalityArithmetic Mean-Geometric Mean
IMO ShortList 1998, combinatorics theory problem 1

Source: IMO ShortList 1998, combinatorics theory problem 1

10/22/2004
A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number xx in the array can be changed into either ⌈x⌉\lceil x\rceil or ⌊x⌋\lfloor x\rfloor so that the row-sums and column-sums remain unchanged. (Note that ⌈x⌉\lceil x\rceil is the least integer greater than or equal to xx, while ⌊x⌋\lfloor x\rfloor is the greatest integer less than or equal to xx.)
algorithmmaxtrixroundingIMO Shortlist