# Understand the problem

Azambuja writes a rational number on a blackboard. One operation is to delete and replace it by ; or by ; or by if . The final goal of Azambuja is to write the number after performing a finite number of operations. Show that if the initial number written is , then Azambuja cannot reach his goal.

##### Source of the problem

Brazilian national mathematical olympiad 2018

##### Topic

Invariance

##### Difficulty Level

Easy

##### Suggested Book

Problem Solving Strategies by Arthur Engel

# Start with hints

Do you really need a hint? Just try it yourself!

It is always a good idea to try using the invariance principle in such problems.

Make a change of variables to see patterns.

Note that the operation is restricted to rational numbers. Hence, writing could help.

Let us denote by the number on the board at the th step. We shall use the new variable (just to simplify the denominator). Clearly, is either or . Writing , this means that is either or . Thus, we need to find out if is reachable starting from . However, (odd, odd) pairs can produce only other (odd,odd) pairs under this operation, and is an (even, even) pair. Hence cannot be reached starting from 0.

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