Kent Mark is running for class president. Assume that there are a total of n ca− ndidates running, where n is a natu− ral number. After the votes are tallied, Kent Mark is told only the fraction of votes that he recieved. Suppose he recieved less than (1/n) of the votes. Show that he cannot have won the election.


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Question Number 116061 by MWSuSon last updated on 30/Sep/20
$Kent Mark is running for class president. Assume that there are a total of n ca− ndidates running, where n is a natu− ral number. After the votes are tallied, Kent Mark is told only the fraction of votes that he recieved. Suppose he recieved less than (1/n) of the votes. Show that he cannot have won the election.$
$$\mathrm{Kent}\:\mathrm{Mark}\:\mathrm{is}\:\mathrm{running}\:\mathrm{for}\:\mathrm{class}\:\mathrm{president}. \\ $$$$\mathrm{Assume}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{a}\:\mathrm{total}\:\mathrm{of}\:\mathrm{n}\:\mathrm{ca}− \\ $$$$\mathrm{ndidates}\:\mathrm{running},\:\mathrm{where}\:\mathrm{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{natu}− \\ $$$$\mathrm{ral}\:\mathrm{number}. \\ $$$$\mathrm{After}\:\mathrm{the}\:\mathrm{votes}\:\mathrm{are}\:\mathrm{tallied},\:\mathrm{Kent}\:\mathrm{Mark} \\ $$$$\mathrm{is}\:\mathrm{told}\:\mathrm{only}\:\mathrm{the}\:\mathrm{fraction}\:\mathrm{of}\:\mathrm{votes}\:\mathrm{that}\:\mathrm{he} \\ $$$$\mathrm{recieved}. \\ $$$$\mathrm{Suppose}\:\mathrm{he}\:\mathrm{recieved}\:\mathrm{less}\:\mathrm{than}\:\frac{\mathrm{1}}{\mathrm{n}}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{votes}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{he}\:\mathrm{cannot}\:\mathrm{have}\:\mathrm{won} \\ $$$$\mathrm{the}\:\mathrm{election}. \\ $$
	Answered by 1549442205PVT last updated on 01/Oct/20
	$We assume that sum of number of votes in that the election be S Suppose Kent Mark win the election then all other candidates ,each of them gets number of votes be a fraction less than (1/n)that means he received less than (1/n)S votes.Hence,sum of number of votes in that election less tan n.(1/n)S=S This contradiction shows that Kent Mark cannot have won in that election(Q.E.D)$
	$$\mathrm{We}\:\mathrm{assume}\:\mathrm{that}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{number}\:\mathrm{of}\:\mathrm{votes} \\ $$$$\mathrm{in}\:\mathrm{that}\:\mathrm{the}\:\mathrm{election}\:\mathrm{be}\:\mathrm{S} \\ $$$$\mathrm{Suppose}\:\mathrm{Kent}\:\mathrm{Mark}\:\mathrm{win}\:\mathrm{the}\:\mathrm{election} \\ $$$$\mathrm{then}\:\mathrm{all}\:\:\mathrm{other}\:\mathrm{candidates}\:,\mathrm{each}\:\mathrm{of}\:\mathrm{them} \\ $$$$\mathrm{gets}\:\mathrm{number}\:\mathrm{of}\:\mathrm{votes}\:\mathrm{be}\:\mathrm{a}\:\mathrm{fraction}\:\mathrm{less}\: \\ $$$$\mathrm{than}\:\frac{\mathrm{1}}{\mathrm{n}}\mathrm{that}\:\mathrm{means}\:\mathrm{he}\:\mathrm{received}\:\mathrm{less} \\ $$$$\mathrm{than}\:\frac{\mathrm{1}}{\mathrm{n}}\mathrm{S}\:\mathrm{votes}.\mathrm{Hence},\mathrm{sum}\:\mathrm{of}\:\mathrm{number}\:\mathrm{of}\:\mathrm{votes} \\ $$$$\mathrm{in}\:\mathrm{that}\:\mathrm{election}\:\mathrm{less}\:\mathrm{tan}\:\:\mathrm{n}.\frac{\mathrm{1}}{\mathrm{n}}\mathrm{S}=\mathrm{S} \\ $$$$\mathrm{This}\:\:\mathrm{contradiction}\:\mathrm{shows}\:\mathrm{that} \\ $$$$\mathrm{Kent}\:\mathrm{Mark}\:\mathrm{cannot}\:\mathrm{have}\:\mathrm{won}\:\mathrm{in}\: \\ $$$$\mathrm{that}\:\mathrm{election}\left(\boldsymbol{\mathrm{Q}}.\boldsymbol{\mathrm{E}}.\boldsymbol{\mathrm{D}}\right) \\ $$
	Commented by MWSuSon last updated on 03/Oct/20

	$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir}. \\ $$
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