Mock AIME 1 2006-2007 Problems/Problem 14
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 .
Please see below an attempted solution to understand why this problem doesn't have a solution:
Proof of lemma:
Construct at .
, proof done.
Now we try to find .
Let O be the centre of the incircle, and be the inradius.
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.