A few months ago, I stumbled a pond a new problem solving idea. It is called the telescope method. I did know about it but I just used it subconsciously. I want to present a rather easy problem that can be solved with this method. In 1672 Huygens asked Leibnitz to solve this problem, which he did using the same method.
We define x to be a+b, y to be b+c and z to be a+c.
Now we have to prove, that x/y+y/x is bigger or equal to 2. We multiply the equation with x times y times 2. We get: x^2+y^2 is bigger or equal to 2xy. Now we subtract 2xy on both sides and we get (x-y)^2 is bigger or equal to 0. This is true. Therefor we proved that x/y+y/x is bigger or equal to 2.