Use the pigeonhole principle to solve each of the following problems: 4. Let V = {V1,..., Uk} be any set of vectors in RP. Suppose n agents each start at (0,0) and each takes a mV-walk where a mV-walk consists of a sequence of exactly m steps and each step moves the agent along a vector in V. Prove that, if n>( ), then some pair of agents finishes their walk at the same location. 5. Let S be a k-element subset of {1,...,n}. Prove that, if k> [n/2), then there exists X, Y ES such that x - y=1. Use the pigeonhole principle to solve each of the following problems: 4. Let V = {V1,..., Uk} be any set of vectors in RP. Suppose n agents each start at (0,0) and each takes a mV-walk where a mV-walk consists of a sequence of exactly m steps and each step moves the agent along a vector in V. Prove that, if n>( ), then some pair of agents finishes their walk at the same location. 5. Let S be a k-element subset of {1,...,n}. Prove that, if k> [n/2), then there exists X, Y ES such that x - y=1