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2

Part of 2005 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Equal angles in trapezoid where sum of the bases = diagonal

Source: XIV Rioplatense Mathematical Olympiad (2005), Level 3

7/26/2011
In trapezoid ABCDABCD, the sum of the lengths of the bases ABAB and CDCD is equal to the length of the diagonal BDBD. Let MM denote the midpoint of BCBC, and let EE denote the reflection of CC about the line DMDM. Prove that ∠AEB=∠ACD\angle AEB=\angle ACD.
geometrytrapezoidgeometric transformationreflectiongreatest common divisorparallelogramcircumcircle
Worst approximation to 100 by summing subsequences

Source: XIV Rioplatense Mathematical Olympiad (2005), Level 3

7/26/2011
Consider all finite sequences of positive real numbers each of whose terms is at most 33 and the sum of whose terms is more than 100100. For each such sequence, let SS denote the sum of the subsequence whose sum is the closest to 100100, and define the defect of this sequence to be the value ∣S−100∣|S-100|. Find the maximum possible value of the defect.
algebra unsolvedalgebra