I figure it isn't going to happen within 10 years. Since it isn't even started yet.
When the Americans decided to put a man on the moon, they did it within ten years. In an era where manned space flight was in its infancy
We're now in an era where manned space flight is more or less routine, and where rocket technology and other space technologies have vastly improved.
I'm quite positive it's possible to put a man on Mars within ten years, if only we really want to. If the US would put only a quarter of the money that now goes towards destruction of other people (i.e. the military) towards this mission, it can be done - the amount that goes to their military is so huge, a space mission will look cheap in comparison. It'll need a multiple of spacecraft, one for the crew (with life support for the duration of the trip and a while on Mars), several robotic freighters to carry further supplies needed to set up a permanent base. Indeed it'd be a one-way mission, until the Mars colonists produce the fuel needed for the return trip.
What's missing is the political will. The Americans got to the moon as they thought they were being outcompeted by the Russians, and that this was the only non-selfdestructive way to show the Russians their might (invading Russia would've amounted to suicide of course). This political will, this competition is missing now.
So that indeed gets us back to China. Russia has lost most of it's capabilities, and won't be back on track soon. The Chinese however develop quickly, and start to seriously catch up with the Americans. The moment the Chinese take over and say "we're going to put a man on Mars, just to show the world how great we are", the Americans will say "we're going to be there first!". As soon as that happens, less than a decade later, there will be people on the moon.