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.

Topic
Combinatorics

Difficulty Level

7 out of 10

Suggested Book

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

Connected Program at Cheenta

I.S.I. & C.M.I. Entrance Program

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.

Similar Problem

A fraction involving primes

Understand the problemGiven that the number is an integer where and are prime positive numbers, determine $latex a$.Danube 2014 Number Theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!First settle the case where $latex...

A taste of affine transformations

Understand the problemLet be the area of the triangle . A non-regular convex polygon is called guayaco if exists a point in its interior such thatShow that, for every integer , a guayaco polygon of sides exists.Cono sur olympiad 2017 Geometry Easy Problem Solving...

Solving a congruence

Understand the problemProve that the number of ordered triples in the set of residues of $latex p$ such that , where and is prime is . Brazilian Olympiad Revenge 2010 Number Theory Medium Elementary Number Theory by David Burton Start with hintsDo you really need...

Inequality involving sides of a triangle

Understand the problemLet be the lengths of sides of a (possibly degenerate) triangle. Prove the inequalityLet be the lengths of sides of a triangle. Prove the inequalityCaucasus Mathematical Olympiad Inequalities Easy An Excursion in Mathematics Start with hintsDo...

Vectors of prime length

Understand the problemGiven a prime number and let be distinct vectors of length with integer coordinates in an Cartesian coordinate system. Suppose that for any , there exists an integer such that all three coordinates of is divisible by . Prove that .Kürschák...

Missing digits of 34!

Understand the problem34!=295232799cd96041408476186096435ab000000 Find $latex a,b,c,d$ (all single digits).BMO 2002 Number Theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!Get prepared to find the residue of 34! modulo...

An inequality involving unknown polynomials

Understand the problemFind all the polynomials of a degree with real non-negative coefficients such that , . Albanian BMO TST 2009 Algebra Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!This problem is all about...

Hidden triangular inequality (PRMO Problem 23, 2019)

Problem Let ABCD be a convex cyclic quadrilateral . Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least .If {PA,PB,PC,PD} = {3,4,6,8}.What is the maximum possible area of ABCD? TopicGeometry...

Bangladesh MO 2019 Problem 1 – Number Theory

A basic and beautiful application of Numebr Theory and Modular Arithmetic to the Bangladesh MO 2019 Problem 1.

Functional equation dependent on a constant

Understand the problem34!=295232799cd96041408476186096435ab000000 Find $latex a,b,c,d$ (all single digits).BMO 2002 Number Theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!Get prepared to find the residue of 34! modulo...