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smallest of a_k (1-a_{2002-k}) ($0 \le k le 2002$) is <= 1/4

Source: Austrian Regional Competition For Advanced Students 2002 p4

January 11, 2020
algebrainequalitiesProduct

Problem Statement

Let a0,a1,...,a2002a_0, a_1, ..., a_{2002} be real numbers. a) Show that the smallest of the values ak(1−a2002−k)a_k (1-a_{2002-k}) (0≤k≤20020 \le k \le 2002) the following applies: it is smaller or equal to 1/41/4. b) Does this statement always apply to the smallest of the values ak(1−a2003−k)a_k (1-a_{2003-k}) (1≤k≤20021 \le k \le 2002) ? c) Show for positive real numbers a0,a1,...,a2002a_0, a_1, ..., a_{2002} : the smallest of the values ak(1−a2003−k)a_k (1-a_{2003-k}) (1≤k≤20021 \le k \le 2002) is less than or equal to 1/41/4.
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