Track / Stage 5 / 354 of 400 #954 of 1964

Problem 954

AIME late

Combinatorics Difficulty 5.9 Prove it

On the sides of a hexagon, six numbers were written, and at each vertex - a number equal to the sum of the two numbers on the adjacent sides. Then all the numbers on the sides and one number at a vertex were erased. Can the number that was at the vertex be restored?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Let $A B C D E F$ be the given hexagon, and we need to restore the number at vertex $A$ . Note that the sum of the numbers at vertices $A, C$ and $E$ is equal to the sum of the numbers at vertices $B, D$ and $F$ : both these sums are equal to the sum of the numbers originally on all sides of the hexagon. Therefore, to restore the number at vertex $A$ , we need to subtract the numbers at vertices $C$ and $E$ from the sum of the numbers at vertices $B, D$ and $F$ .

## Answer

It is possible.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.