MathDB

Dominos

Source: Saudi Arabia IMO TST Day I Problem 2

7/22/2014

Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominoes is a list of distinct dominoes in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which

(i, j)

and

(j, i)

do not both appear for any

i

and

j

. Let

D_n

be the set of all dominoes whose coordinates are no larger than

n

. Find the length of the longest proper sequence of dominoes that can be formed using the dominoes of

D_n

.

analytic geometrygraph theorycombinatorics unsolvedcombinatorics

Set S of positive reals

Source: Saudi Arabia IMO TST Day II Problem 2

7/22/2014

Let

S

be a set of positive real numbers with five elements such that for any distinct

a,~ b,~ c

in

S

, the number

ab + bc + ca

is rational. Prove that for any

a

and

b

in

S

,

\tfrac{a}{b}

is a rational number.

algebra unsolvedalgebra

Floor functions in f(x)

Source: Saudi Arabia IMO TST Day III Problem 2

7/22/2014

Determine all functions

f:[0,\infty)\rightarrow\mathbb{R}

such that

f(0)=0

and f(x)=1+5f\left(\left\lfloor{\frac{x}{2}\right\rfloor}\right)-6f\left(\left\lfloor{\frac{x}{4}\right\rfloor}\right) for all

x>0

.

functionfloor functioninductionlogarithmsalgebra unsolvedalgebra

Players in a tournament

Source: Saudi Arabia IMO TST Day IV Problem 2

7/22/2014

In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded

1

point, the loser got

0

points, and each of the two players earned

\tfrac{1}{2}

point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned in games against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of his points against the other nine of the ten). What was the total number of players in the tournament?

AMCAIMEcombinatorics unsolvedcombinatorics

2

Problems(4)