# Introduction

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2" locked="off"]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.

# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2" locked="off"]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.

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.22.4" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.22.4"]

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

[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.22.4" open="off"]Combinatorics

[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.23.3" open="off"]

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.22.4" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.22.4"]

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

[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.22.4"]

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 $p_{k+1}$

[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.23.3"]

if $p_{k+1}$ > $p_1$ or $p_k$>$p_{k+1}$ we are done (why?)[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.22.4"] 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

# Connected Program at Cheenta

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.