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Show that there exists a point 0 ≤ y ≤ 1

Source: IMO LongList 1982 - P2

March 16, 2011
inequalitiescombinatorics proposedcombinatorics

Problem Statement

Given nn points X1,X2,…,XnX_1,X_2,\ldots, X_n in the interval 0≤Xi≤1,i=1,2,…,n0 \leq X_i \leq 1, i = 1, 2,\ldots, n, show that there is a point y,0≤y≤1y, 0 \leq y \leq 1, such that 1n∑i=1n∣y−Xi∣=12.\frac{1}{n} \sum_{i=1}^{n} | y - X_i | = \frac 12.
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