2009 Indonesia MO

Part of Indonesia MO

Subcontests

(4)

1

Existence of a good pair

A pair of integers

(m,n)

is called good if m\mid n^2 \plus{} n \ \text{and} \ n\mid m^2 \plus{} m Given 2 positive integers

a,b > 1

which are relatively prime, prove that there exists a good pair

(m,n)

a\mid m

b\mid n

a\nmid n

b\nmid m

1

4-cycle in a graph

In an island, there exist 7 towns and a railway system which connected some of the towns. Every railway segment connects 2 towns, and in every town there exists at least 3 railway segments that connects the town to another towns. Prove that there exists a route that visits 4 different towns once and go back to the original town. (Example: A\minus{}B\minus{}C\minus{}D\minus{}A)

2

The minimum value of a function

Find the lowest possible values from the function f(x) \equal{} x^{2008} \minus{} 2x^{2007} \plus{} 3x^{2006} \minus{} 4x^{2005} \plus{} 5x^{2004} \minus{} \cdots \minus{} 2006x^3 \plus{} 2007x^2 \minus{} 2008x \plus{} 2009 for any real numbers

x

Inequalities in floor function

x

\lfloor x\rfloor

be the largest integer that is not more than

x

. Given a sequence of positive integers

a_1,a_2,a_3,\ldots

a_1>1

and \left\lfloor\frac{a_1\plus{}1}{a_2}\right\rfloor\equal{}\left\lfloor\frac{a_2\plus{}1}{a_3}\right\rfloor\equal{}\left\lfloor\frac{a_3\plus{}1}{a_4}\right\rfloor\equal{}\cdots Prove that \left\lfloor\frac{a_n\plus{}1}{a_{n\plus{}1}}\right\rfloor\leq1 holds for every positive integer

n

2

Divisility by 7

Find all positive integers

n\in\{1,2,3,\ldots,2009\}

such that 4n^6 \plus{} n^3 \plus{} 5 is divisible by

7

Probabilities in a drawer

In a drawer, there are at most

2009

balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is

\frac12

. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?