Soal Kompetisi Matematika :3-rd Mediterranean Mathematical Competition 2000

1. Let F = {1, 2, . . . , 100} and let G be any 10-element subset of F. Prove that there exist two disjoint nonempty subsets S and T of G with the same sum of elements.

2. Suppose that in the exterior of a convex quadrilateral ABCD equilateral triangles XAB, Y BC,ZCD,WDA with centroids S1, S2, S3, S4 respectively are constructed. Prove that S1S3⏊ S2S4 if and only if AC ⏊ BD.

3. Let c1, . . . , cn, b1, . . . , bn (n ≥ 2) be positive real numbers. Prove that the equation

has a unique solution (x1, . . . , xn) if and only if

4. Let P,Q,R, S be the midpoints of the sides BC,CD,DA,AB of a convex quadrilateral, respectively. Prove that

4(AP2 + BQ2 + CR2 + DS2) ≤ 5(AB2 + BC2 + CD2 + DA2).

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Posted on Mei 1, 2011, in Matematika. Bookmark the permalink. Tinggalkan komentar.

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