# Mock AIME 1 2006-2007 Problems/Problem 14

## Problem

Three points , , and are fixed such that lies on segment , closer to point . Let and where and are positive integers. Construct circle with a variable radius that is tangent to at . Let be the point such that circle is the incircle of . Construct as the midpoint of . Let denote the maximum value for fixed and where . If is an integer, find the sum of all possible values of .

## Solution

Please see below an attempted solution to understand why this problem doesn't have a solution:

Lemma:

Proof of lemma:

Construct at .

Case (i)

Case (ii)

Case (iii)

, proof done.

Now we try to find .

Let O be the centre of the incircle, and be the inradius.

Similarly,

Therefore,

Therefore, .

Therefore, all possible values of are 48, 47, 42, 35, and the answer is 48+47+42+35=172.

What's the problem with this solution?

When AM-GM was used, is when "=" is achieved. However, in this case, , so contradiction.

If the phrase "maximum value" in the original problem is changed to "least upper bound of", then the problem should have the solution above.

~Di Xu