Who of these cyclist travel takes less time? There are two cyclist. They start off with equal velocity $v_0$. 
The first one bikes a straight path, while the other bikes through a valley, something like this:

We assume friction doesn't affect either cyclist, so only gravitational potential energy plays a role here. Who should arrive first?
I think its about to find the angles of the second way where the time its a minimun.
Someone said I should use the conservation of mechanical energy, but I don't understand how could add the two different velocities (one simply contant, while the other is related to the gravitational potential).
 A: First, you should add some measures to your trajectories:

Then you start working out the potential and kinetic energies:

From that, you can get the velocity as a function of the height under that straight path:

From my intuition, I would say that the speed of the cyclist will look like this:

There is more distance to cover, but the speed is always faster than $v_0$. And as @Bernhard wrote, you the horizontal velocity is always greater than $v_0$. So the latter path is faster.
You could go on calculating the acceleration along the path and get the time needed to go from one bend to the next one. Then sum those times up and you get the total time. Should be faster than the $L/v_0$ that you get on path 1.
A: If there is no friction (and no pedaling), you can easily determine that the velocity of the cyclist on Path 2 is always higher than the velocity of the cyclist on Path 1. This is because any potential energy is converted in kinetic energy.
You also know that this will at least for a part be reflected in the horizontal velocity component. Otherwise, the cyclist would be in free fall, which is obviously not true. As the horizontal velocity of cyclist 2 is idential to, or higher than the velocity of cyclist 1, taking Path 2 will always be quicker, within these assumptions.
