Describe how you would design and run a live production system to ensure zero downtime during deployments. Include strategies for high availability, rolling and blue-green deployments, handling distributed systems problems (like consistency and fault tolerance), and monitoring and alerting.

You have 10,000 servers and need to send a file of size 1 MB to each server, starting from a laptop with only 1 MB bandwidth between each server. Describe a solution to achieve the shortest possible time for this file distribution, using a tree structure and logarithmic time complexity.

How would you design the infrastructure for a retail web application? Explain how a web server, a database, firewalls, and other components would connect together.

How would you implement a caching proxy server? Describe the architecture, key components, and the steps involved in building it.