How to manage API Keys in "front-end only" application?

Suppose I am writing a front-end only "mashup" application to run in a browser and invoke some Web APIs that require API Keys.

Since I don't want to expose the API keys to the browsers I need a proxy server to store the API keys and invoke the Web APIs for behalf of my application.

However it looks like a chicken-egg problem ! In order to invoke this proxy I need some "secret" too. Now I am wondering how to manage API Keys and invoke Web APIs from the front-end only application