MathDB

1

2010 Japan Mathematical Olympiad Finals Problem 3

2010

islands and

2009

bridges connecting them. Suppose that any bridges are connected by one bridge or not the endpoints are connected to 2 distinct islands and we can travel a few times by crossing bridges from each island to any other islands. Now a letter from each island was sent to some island, note that, some letter may sent to same island, then the following fact was proved that: In case of connecting island

A

and island

B

by bridge, the habitant of island

A

and that of island

B

are mutually connected by bridge or the same island (itself). Prove that at least one of the following statements (1) or (2) hold.
(1) There exists island for which a letter was sent to the same island.
(2) There exist 2 islands, connecting bridge, whose letter are exchanged each other.

5

1

2010 Japan Mathematical Olympiad Finals Problem 5

Given a convex

2010

polygonal whose any 3 diagonals have no intersection points except vertices. Consider closed broken lines which have

2010

diagonals (not including edges) and they pass through each vertex exactly one time. Find the possible maximum value of the number of self-crossing. Note that we call closed broken lines such that broken line

P_1P_2\cdots P_nP_{n+1}

has the property

P_1=P_{n+1}.

2

1

Proof of being existed n such that 2^k|n^n-m from JMO 2010

Let

k

be positive integer and

m

be odd number. Prove that there exists positive integer

n

such that n^n \minus{} m is divisible by

2^k

.

1

Proof of being Circumcenter of APQ from JMO 2010

Given an acute-angled triangle

ABC

such that

AB\neq AC

. Draw the perpendicular

AH

from

A

to

BC

. Suppose that if we take points

P,\ Q

in such a way that three points

A,\ B,\ P

and three points

A,\ C,\ Q

are collinear in this order respectively, then we have four points

B,\ C,\ P,\ Q

are concyclic and HP \equal{} HQ. Prove that

H

is the circumcenter of

\triangle{APQ}

.

4

1

Inequality from JMO 2010

Let

x,\ y,\ z

be positive real numbers. Prove that \frac {1 \plus{} yz \plus{} zx}{(1 \plus{} x \plus{} y)^2} \plus{} \frac {1 \plus{} zx \plus{} xy}{(1 \plus{} y \plus{} z)^2} \plus{} \frac {1 \plus{} xy \plus{} yz}{(1 \plus{} z \plus{} x)^2}\geq 1

2010 Japan MO Finals

Subcontests

2010 Japan Mathematical Olympiad Finals Problem 3

2010 Japan Mathematical Olympiad Finals Problem 5

Proof of being existed n such that 2^k|n^n-m from JMO 2010

Proof of being Circumcenter of APQ from JMO 2010

Inequality from JMO 2010