1. Liverpool played 38 games and obtained 84 points in a particular season. You obtain no points for losing a game, one point for drawing a game and three points if you win a game. If Liverpool won twice as many games as they lost, how many games did they win, draw and lose?
2. Consider a series of integers a1 < a2 < ··· < a43 < a44 that are all positive and all of the numbers are taken to be less than 125. There are 43 differences between adjacent numbers in this series defined as, di = ai+1 - ai. Can you prove that some value of the differences (which must also be positive integers) must occur at least 10 times?
3. Jane is walking her dog, Spot. She sees her friend, Dick, walking toward her along the same long, straight road. Both Dick and Jane are walking at 3 mph. When Dick and Jane are 600 feet apart, Spot runs from Dick to Jane, turns and runs back to Dick, and then back and forth between them at a constant speed of 8 mph. Dick and Jane both continue walking toward each other at a constant 3 mph. Neglecting the time lost each time Spot reverses direction, how far has Spot run in the time it takes Dick and Jane to meet?