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Understand the problem

Suppose that in a sports tournament featuring n players, each pair
plays one game and there is always a winner and a loser (no draws).
Show that the players can be arranged in an order P1, P2, . . . , Pn such
that player Pi has beaten Pi+1 for all i = 1, 2, . . . , n.

Source of the problem

I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2016. Subjective Problem no.1.

Combinatorics

7 out of 10

Start with hints

Do you really need a hint? Try it first!

we define a sequence $p_1$>$p_2$>$p_3$………..>$p_n$ . such that $p_i>p_{i+1}$ implies $p_i$ has beaten $p_{i+1}$

let assume that the statement is true for n=k  i.e $p_1> p_2>p_3………p_k$ now look into the case of $p_{k+1}$

if $p_{k+1}$ > $p_1$ or $p_k$>$p_{k+1}$ we are done (why?)

if $p_k$<$p_{k+1}$ or $p_{k+1}$<$p_1$ then  we can find out at least one pair $p_i$ >$p_{i+1}$ such that $p_i$> $p_{k+1}$ >$p_{i+1}$ , we can just find it by shifting $p_{k+1}$ term by term [ you can make some small experiment ]. other wise it contradict our initial assumption that $p_k$<$p_{k+1}$ or $p_{k+1}$<$p_1$ [ why?]   hence we are done

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