Olympiad Maths Prep

Track / Stage 5 / 62 of 400 #662 of 2000

Problem 662

AIME late
Combinatorics Difficulty 5.2 Find the answer

101.

In this task, we will assume that the criminal is a knight. (Our assumption is internally consistent: to commit a crime, one does not necessarily have to lie.) Suppose also that you are a knight (which is unknown to the jury), but you are not guilty of the crime. What would you state in court?

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

101. This problem is in a sense "dual" to problem 99 (and even somewhat easier). You need only state in court: "I am not guilty." Upon hearing your statement, the jurors would reason as follows. If you are a knight (which they do not know), then your statement is true. Consequently, you are not guilty. If, however, you are not a knight, then you are again not guilty, since the jurors' information indicates that the criminal is a knight.

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