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Combinatorics Difficulty 5.0 AIME, harder Find the answer United Kingdom

Christina has eight coins whose weights in grams are different positive integers. When Christina puts any two coins in one pan of her balance scales and any two in the other pan of the balance scales, the side containing the heaviest of those four coins is always the heavier side. What is the smallest possible weight of the heaviest of the eight coins?

(A) $8$ (B) $12$ (C) $34$ (D) $55$ (E) $256$

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