Where does the Amazon River really begin?
From an Earth Science standpoint, it's meaningless to ask about the "source" of a river in this way.  The "source" of a river is the rainfall within its drainage basin, which percolates down into groundwater, and the "river" appears where the land has eroded down below the water table.

The Amazon has an enormous drainage basin: 


and every point in it is its "source" in some way.

The trouble with your question is that it asks about the "source" of a river in a way that you first have to define what you consider the "source". Does a stream begin where its name begins? Where water first permanently flows above ground? Most of the time? Some of the time? Anywhere there's a big enough pile of wet leaves?

In Brazil, the part of the river above its confluence with the Rio Negro at Manaus has its own name -- the Solimões, so you could say the "Amazon" proper begins at Manaus. But this is thousands of miles from the river's farthest upper reaches.

The Wikipedia article on the Amazon River has the following to say about its "farthest source":


  The headwaters of the Apurímac River on Nevado Mismi had been considered for nearly a century as the Amazon’s most distant source, until a 2014 study found it to be the Cordillera Rumi Cruz at the headwaters of the Mantaro River in Peru. The Mantaro and Apurímac join, and with other tributaries form the Ucayali River, which in turn meets the Marañón River upstream of Iquitos, Peru, to form what countries other than Brazil consider to be the main stem of the Amazon. 


But because a tunnel associated with the Tablachaca Dam diverts water away from a big loop of the Mantaro River, effectively shortening it, all you'll ever get now is an argument about what's "really" the furthest source.