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3

Part of 1982 Bundeswettbewerb Mathematik

Problems(2)

point inside convex polygon

Source:

12/3/2014
Suppose PP is a point inside a convex 2n2n-gon, such that PP does not lie on any diagonal. Show that PP lies inside an even number of triangles whose vertices are vertices of the polygon.
combinatorics unsolvedcombinatorics
Bundeswettbewerb Mathematik 1982 Problem 2.3

Source: Bundeswettbewerb Mathematik 1982 Round 2

9/22/2022
Given that a1,a2,...,ana_1, a_2, . . . , a_n are nonnegative real numbers with a1+⋯+an=1a_1 + \cdots + a_n = 1, prove that the expression a11+a2+a3+⋯+an  +  a21+a1+a3+⋯+an  +  ⋯  + an1+a1+a2+⋯+an−1 \frac{a_1}{1+a_2 +a_3 +\cdots +a_n }\; +\; \frac{a_2}{1+a_1 +a_3 +\cdots +a_n }\; +\; \cdots \; +\, \frac{a_n }{1+a_1 +a_2+\cdots +a_{n-1} } attains its minimum, and determine this minimum.
algebrainequalitiesminimum