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4 numbers game

Source: 1st National Women's Contest of Mexican Mathematics Olympiad 2022 , levels 1+2 p2

September 17, 2022
combinatorics

Problem Statement

In the training of a state, the coach proposes a game. The coach writes four real numbers on the board in order from least to greatest: a<b<c<da < b < c < d.
Each Olympian draws the figure on the right in her notebook and arranges the numbers inside the corner shapes, however she wants, putting a number on each one. Once arranged, on each segment write the square of the difference of the numbers at its ends. Then, add the 44 numbers obtained. https://cdn.artofproblemsolving.com/attachments/9/a/ea348c637ae266c908e0b97e64605808b3b1d2.png For example, if Vania arranges them as in the figure on the right, then the result would be (c−b)2+(b−a)2+(a−d)2+(d−c)2. (c - b)^2 + (b- a)^2 + (a - d)^2 + (d - c)^2. https://cdn.artofproblemsolving.com/attachments/8/b/9c5375d66a4a6344b2bce333534fa7fac2ad6c.png The Olympians with the lowest result win. In what ways can you arrange the numbers to win? Give all the possible solutions.
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