I saw it first on The One Show!   Beetroot juice boosts stamina, new study shows, as the University of Exeter says in a press release, the gist of which is as follows:
 
Professor Andrew Jones, of their School of Sport and Health Sciences, tested a group of male cyclists, giving them organic beetroot juice for one week, and on another occasion a blackcurrant juice placebo.  The beetroot juice allowed them to cycle for almost 16% longer.  Also, the beetroot juice reduced their resting blood pressure (as also in the case of Anita Rani, who acted as guinea-pig for The One Show.)   Though not yet sure of the exact mechanism that causes the nitrate in the beetroot juice to boost stamina, the researchers suspect it could be a result of the nitrate turning into nitric oxide in the body, reducing the oxygen cost of exercise.

This gave rise to some misgivings in my mind.  Firstly, I had heard that nitrate in foodstuffs could be reduced to nitrite, and thence give rise to nitrosamines, implicated in stomach cancer.  I contacted Professor Jones, and he kindly replied:
The nitrate is indeed converted first into nitrite and then into NO.  However, the link between nitrite and cancer seems to have been overstated.  The WHO have issued a statement that there is no evidence for such a link in humans (the original studies were with rodents that metabolise nitrate/nitrite very differently).  It remains possible that there might be a link with the nitrites which are used to preserve meats but the nitrate which occurs naturally in fruits and vegetables seems to have a positive effect on health.  Indeed, the populations that consume the most nitrate seem to live longest.
(I would not suggest taking nitrate as a diet supplement, though.  In overdose it can lead to human toxicity.  Also in regard to its use as fertilizer, its effect on ecosystems seems to be causing EU legislators to go bananas.  Perhaps we would be better off if they employed monkeys – these would actually eat the bananas.)

As for the nitric oxide, this reminded me that it is an essential part of the operation of Viagra.  Andrew Jones again replied:
I believe that Viagra works by allowing NO, which has a short half-life, to 'remain around longer'.  Nitrate consumption would increase one of the substrates/reservoirs for NO production.  So the common link is NO but the mechanism is a bit different.
I would now like to try beetroot juice to see if it also increases the endurance of the brain.  If composing blogs in the evening, I often feel as if my brain is experiencing, so to speak, a build-up of lactic acid.  But organic beetroot juice?  I can see its significance in the Exeter experiments, since that would eliminate objections that the nitrate might be coming from fertilizer, rather than the beetroot itself.  But organic prices don’t sound good to me.