Competition problem — Combinatorics

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Combinatorics Difficulty 4.5 AIME Find the answer United States

Problem:

Ken is the best sugar cube retailer in the nation. Trevor, who loves sugar, is coming over to make an order. Ken knows Trevor cannot afford more than $127$ sugar cubes, but might ask for any number of cubes less than or equal to that. Ken prepares seven cups of cubes, with which he can satisfy any order Trevor might make. How many cubes are in the cup with the most sugar?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

The only way to fill seven cups to satisfy the above condition is to use a binary scheme, so the cups must contain $1, 2, 4, 8, 16, 32$ , and $64$ cubes of sugar.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.