Use the pigeonhole principle to solve each of the following problems: 1. In an anonymous survey of a group of 20 men and 20 women, the men reported a total of 81 sexual encounters with women in the group, and all women reported having at most 4 sexual encounters with men in the group. Is everyone telling the truth? 2. A group of n agents all start at the same location and each one takes a 2m + 1, then some pair of agents finishes their walk at the same location. 3. A group of n agents all start at the same location and each one takes an m-walk on the line, where a m-walk is a sequence of exactly m steps and each step moves the agent one unit to the left or one unit to the right. (Different agents might take different walks.) Prove that, if n> m +1, then some pair of agents finishes their walk at the same location. Use the pigeonhole principle to solve each of the following problems: 1. In an anonymous survey of a group of 20 men and 20 women, the men reported a total of 81 sexual encounters with women in the group, and all women reported having at most 4 sexual encounters with men in the group. Is everyone telling the truth? 2. A group of n agents all start at the same location and each one takes a 2m + 1, then some pair of agents finishes their walk at the same location. 3. A group of n agents all start at the same location and each one takes an m-walk on the line, where a m-walk is a sequence of exactly m steps and each step moves the agent one unit to the left or one unit to the right. (Different agents might take different walks.) Prove that, if n> m +1, then some pair of agents finishes their walk at the same location