7

Part of 1995 IMO Shortlist

Problems(2)

sqrt([AKON]) + sqrt([CLOM]) <= sqrt([ABCD])

Source: IMO Shortlist 1995, G7

Let ABCD be a convex quadrilateral and O a point inside it. Let the parallels to the lines BC, AB, DA, CD through the point O meet the sides AB, BC, CD, DA of the quadrilateral ABCD at the points E, F, G, H, respectively. Then, prove that \sqrt {\left|AHOE\right|} \plus{} \sqrt {\left|CFOG\right|}\leq\sqrt {\left|ABCD\right|}, where

\left|P_1P_2...P_n\right|

is an abbreviation for the non-directed area of an arbitrary polygon

P_1P_2...P_n

geometryinequalitiesIMO Shortlistsimilar trianglesgeometric inequalityquadrilateral

One taken from each of any n − 1 of the subsets

Source: IMO Shortlist 1995, N7

Does there exist an integer

n > 1

which satisfies the following condition? The set of positive integers can be partitioned into

n

nonempty subsets, such that an arbitrary sum of n \minus{} 1 integers, one taken from each of any n \minus{} 1 of the subsets, lies in the remaining subset.

combinatoricspartitionAdditive combinatoricsAdditive Number TheoryIMO Shortlist