5k and 10k races are pretty much all fueled from stored glycogen so you'll want to make sure that this is well topped off before the start. The same applies to most of the training sessions you'd likely do for such a race. Once you start getting up above the hour range (longer if you are fitter and have bigger reserves of glycogen), you'll probably need to eat some carbs to maintain anything better than an endurance pace.
Given enough time, your body will refill it's glycogen stores from pretty much whatever you eat but it happens faster eating carbs so if you are doing two workouts in a day, it's important to get some carbs in quickly after the first one.
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I run about 23 miles a week, do 5ks when available, and pushing myself for my first 10k. I add some sweet potato or squash if my runs feel dodgy.
I managed to take 2.5 mins off my 5k time this summer by training under 50g of carbs a day.
For those who are suggesting just to consume more fat (which is definitely a good solution here), I just want to add that if you want to primarily use fat for energy, I would recommend keeping your overall diet low-carb so that your body becomes more fat adapted. Of course unlimited fibrous veggies are always a go, but I would severely limit stuff like fruit and sweet potatoes if you go that route.
Here are some relevant medical studies:
I would recommend getting some coconut oil before exercise if you go this route, as the medium-chain triglycerides are efficiently metabolized for energy.
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Given the OP's racing background and if she trains properly, she should be able to finish a 10K in 50 minutes or even less. I stand by my earlier comment that a banana would be sufficient for a 50-minute run.
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Just my 2 cents but even before I went primal I stayed away from carbs before a 10k...they just seemed to bloat me. If I race more than an hour is when I will still hit a gel.....it helps tremendously in that case....you can also do it primally with Dates....essentially all sugar there and I have used them with success.
I run 5k races about twice a year. Being primal/paleo is perfect for these type of races. The sprint work really helps shave off your overall time. And the fact that its only 3 miles isn't too tough on your system compared to trianing for half and full marathons. My diet is essentially full paleo compliant, I still have no grains and I keep my carbs under 100grams per day. Most days under 75 grams. I don't even drink extra water or worry about hydration for running 5k's.
Hey Sonnenblume (and everyone else)! As a longtime runner and serious competitor, I've been impressed with how good I feel training on a high-fat diet. As far as racing goes, I plan on eating more carbs for a couple days before competition (I run everything from the mile through the 10k). Runnerbean said it first, we use glycogen (stored carbs in our muscles and liver) for shorter, more intense work. The 5k and 10k are fast, hard races, not short marathons. Slow and steady will not win them. I think experimenting with one's carb intake before less important races and time trials and noting the results would be a good place to start. Good luck!