feat: implement multiplayer mode

[diiabblo]

Summary

Implements real-time multiplayer trivia battles with room creation, live leaderboards, chat functionality, and prize pool distributions.

Changes

• Added multiplayer type definitions and room configurations
• Created useGameRooms, useGameRoom, and useRoomChat custom hooks
• Implemented multiplayer UI components (RoomCard, RoomList, PlayerList, RoomChat, CreateRoomModal, LiveLeaderboard)
• Added multiplayer constants with prize distribution logic
• Created dedicated multiplayer page with room browser
• Added comprehensive documentation

Features

• Room System: Create and join custom game rooms (2-10 players)
• Live Leaderboard: Real-time player rankings during games
• Chat Integration: In-room chat for player communication
• Prize Pools: Entry fees accumulate into distributed prizes
• Room Filters: Browse by status, players, category
• Real-Time Updates: Automatic polling for live synchronization
• Player States: Waiting, Ready, Playing, Finished tracking
• Room Configuration: Customizable players, questions, categories, entry fees

Room Configuration Options

• Room Name: Custom identifier
• Max Players: 2-10 participants
• Questions: 5-20 per game
• Category: Specific or mixed topics
• Entry Fee: 0-10,000 points
• Time Limits: Per-question constraints

Prize Distribution

2 Players:  1st: 70%, 2nd: 30%
3 Players:  1st: 50%, 2nd: 30%, 3rd: 20%
4+ Players: 1st: 50%, 2nd: 30%, 3rd: 20%

Scoring System

Answer Score = Base (100) + Time Bonus (0-50)
Time Bonus = (timeRemaining / timeLimit) * 50

Components Added

• RoomCard: Individual room display with details
• RoomList: Grid of available rooms
• PlayerList: Room participants with ready states
• RoomChat: Real-time messaging system
• CreateRoomModal: Room creation interface
• LiveLeaderboard: Real-time rankings during gameplay

Closes #5