**Find all function such that ,**

**form the three sides of a non-degenerate triangle. (IMO 2009 Q5)**

Answer: .

Let the condition be .

*It’s obvious that works, and that or don’t work. With the term which is linear in the functional inequality, we expect only linear solutions to exist.*

*Because the triangle condition implies strict inequalities, we wish to use the discreteness of integers to reduce it. *

. The two inequalities reduce to

If , we have and is a period of the function . Since is defined on , that implies that is bounded. Suppose that .

. This is absurd. Thus .

*We did this because there is one unbounded term!*

Using the same trick as above,

.

Or, is an involution. This automatically shows that it’s bijective.

We already know a lot about the function now… If we could show any extra condition the conclusion should be within reach. Monotonicity, or all work. Let’s see what happens if . Since . WLOG as otherwise one can replace with . We have . Now, . Reiterate this inequality, we have

.

*What does this imply? , and we thus have that grows ‘more slowly than . We wish to make to create a contradiction. But how? We want to move forward with as the length of our ‘pace’$. Thus if we exhaust a complete system of remainder modulo , we will be able to show that for large enough. That *should *be able to finish the problem off: as will be bad, so we only have to make sure is ‘large enough’ to ensure a contradiction is reached.*

Let . Letting , we have (upon reducing into ,

Let exhaust the set , we see that for large enough we indeed have .

We now claim that as . This is because is injective, and thus there are only finitely many such that for any fixed . Thus, if we let be such a number that , there exists such that . We let . Contradiction. Thus . We conclude that .

Comment: Easy problem. We used proof by contradiction because the problem’s condition is ultimately an *inequality *for which it is easier to assume certain inequalities to compare the values. One should be able to solve it rather quickly: Andrew Elvey Price only took half an hour to finish it up.

The following problem is basically a spin-off the previous one, and is left to the reader as an exercise.

**Find all function such that **

**1) **

**2) **

** **

**form the three sides of a non-degenerate triangle. (AUS&UNK 2016 IMO Final Training)**

**Determine whether there exists such that for all pairwise different ,**

** **

**are the three sides of an acute-angled triangle.(Renhui Jiuzhang 2015 Prac Exam 6 Q2)**

**Find all function such that , pairwise different, **** form the three sides of a non-degenerate triangle if and only if form the three sides of a non-degenerate triangle. (China TST 2016 Q18)**

Solution to be published gradually….