1. Two candidates, Alice and Bob, are running for mayor of a small town. For transparency,...

 

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1. Two candidates, Alice and Bob, are running for mayor of a small town. For transparency, the ballots are read one at a time by the esteemed Vote Controller, a process which is observed by several members of the town. Shockingly, the vote ended in a tie: 30 votes to 30 votes. However, as one observer noted, after each vote was counted, Alice always had at least as many counted votes as Bob. In how many ways can this be done? Prove your answer is correct. 2. Let Ln be the n-th Lucas number, and let Fn be the n-th Fibonacci number. Prove by induction: For all n ≥ 2, LnFn+Fn-2. 1. Two candidates, Alice and Bob, are running for mayor of a small town. For transparency, the ballots are read one at a time by the esteemed Vote Controller, a process which is observed by several members of the town. Shockingly, the vote ended in a tie: 30 votes to 30 votes. However, as one observer noted, after each vote was counted, Alice always had at least as many counted votes as Bob. In how many ways can this be done? Prove your answer is correct. 2. Let Ln be the n-th Lucas number, and let Fn be the n-th Fibonacci number. Prove by induction: For all n ≥ 2, LnFn+Fn-2.

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