Why do these bicycle tires keep going flat?

My friend had her bike tires go flat, and now they won't hold air. I'm taking it to get quoted for her, since I can transport it better, but I really want to know what's wrong now. I'd figured it was the tubes and had a look at them, but when lightly inflated, I can't find anything remotely approaching a leak, and there isn't even any debris in the tires themselves indicating where it was punctured.

The only thing wrong with it that I can visually identify is that the rim tape has broken off on the front tire. It looks fine on the back, but the back goes flat also. The thing is, they still went flat even after being gently walked 2 blocks, and I don't see a pinch flat/snakebite anywhere, which is the only reason I've heard of crappy rim tape crippling tires.
 
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