Track / Stage 5 / 206 of 400 #806 of 1964

Problem 806

AIME late

Number theory Difficulty 5.4 Find the answer

(1988 US Mathcounts Math Competition) In a game, answering an easy question scores 3 points, and answering a hard question scores 7 points. Among the set of integers that cannot be the total score of a contestant, find the maximum value.

untranslated text remains in its original format and line breaks are preserved.

Your answer

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

Official solution

2. Let the set of natural numbers that cannot be used as total scores be $S$ . Among the first 12 natural numbers, only $1,2,4,5,8,11$ belong to $S$ .

Since $7=2 \times 3+1,3 \times 5=7 \times 2+1$ , when the sum of several 3s and 7s $n \geqslant 12$ , one can replace a 7 with two 3s, or replace five 3s with two 7s, to increase the sum by 1. Therefore, natural numbers greater than 12 do not belong to $S$ . So
$S=\{1,2,4,5,8,11\} \text {. }$

The maximum value sought is 11.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.