# 1962 AHSME Problems/Problem 27

## Problem

Let represent the operation on two numbers, and , which selects the larger of the two numbers, with . Let represent the operator which selects the smaller of the two numbers, with . Which of the following three rules is (are) correct?

## Solution

The first rule must be correct as both sides of the equation pick the larger out of a and b.

The second rule must also be correct as both sides would end up picking the largest out of a, b, and c.

WLOG, lets assume b < c.

The third rule is a little more complex. To see if it works, let’s split the possibilities into three cases. a < b < c, b < a < c, and b < c < a. For the first case, the equation simplifies to a = a, which is correct. In the second case, the equation simplifies to a = a, which is correct. For the last case, the equation simplifies to c = c, which is also correct. Since all three cases work, this rule also works.

Thus, our answer is

(Solution by someonenumber011)