How do I prove I own a particular amount of BTC if I am not required to provide the private key for that BTC?

Private keys of BTC account are mainly used to sign transactions to move funds but also to sign text messages. If you are able to sign a message we provide, it means that you also control the funds in the given BTC address. Sending us a signature of a text message does not reveal or compromise your private key. Never send your private key to anyone!
 
You can generate a signature by using any wallet with signing capability (Bitcoind, Bitcoin-qt, Electrum, Mycelium, etc.).
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