A Fairly Medium Problem

Probability Level 2

Let S S be a set of n n distinct real numbers. Let A S A_S be the set of numbers that occur as averages of two distinct elements of S S . For a given n ≥ 2 n \geq 2 , what is the smallest possible number of elements of A S A_S ?

2 n − 3 2n-3 6 n − 5 6n-5 5 n − 3 5n-3 2 n − 7 2n-7

Vishruth Bharath by Vishruth Bharath , 17, USA

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1 solution

Vishruth Bharath
Jan 27, 2018

Let x 1 < x 2 < ⋯ < x n x_1 < x_2 < \dots < x_n represent the elements of S S . Then, x 1 + x 2 2 < x 1 + x 3 2 < ⋯ < x 1 + x n 2 < x 2 + x n 2 < x 3 + x n 2 < ⋯ < x n − 1 + x n 2 \frac{x_1 + x_2}{2} < \frac{x_1 + x_3}{2} < \dots < \frac{x_1 + x_n}{2}<\frac{x_2 + x_n}{2}<\frac{x_3 + x_n}{2}<\dots<\frac{x_{n-1} + x_n}{2} represents 2 n − 3 2n-3 distinct elements of A S A_S , so A S A_S has at least 2 n − 3 \color{#3D99F6}\boxed{2n-3} distinct elements.

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