2003 Japan MO Finals

Part of Japan MO Finals

Subcontests

(5)

1

color in three

Find the greatest possible integer

n

such that one can place

n

points in a plane with no three on a line, and color each of them either red, green, or yellow so that: (i) inside each triangle with all vertices red there is a green point. (ii) inside each triangle with all vertices green there is a yellow point. (iii) inside each triangle with all vertices yellow there is a red point.

1

weights of all expansions

p,q\geq 2

be coprime integers. A list of integers

(r,a_{1},a_{2},...,a_{n})

|a_{i}|\geq 2

i

is said to be an expansion of

p/q

\frac{p}{q}=r+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{...+\frac{1}{a_{n}}}}}

. Now define the weight of an expansion

(r,a_{1},a_{2},...,a_{n})

to be the product

(|a_{1}|-1)(|a_{2}|-1)...(|a_{n}|-1)

. Show that the sum of the weights of all expansions of

p/q

q

1

find greatest number

Find the greatest real number

k

such that, for any positive

a,b,c

a^{2}>bc

(a^{2}-bc)^{2}>k(b^{2}-ca)(c^{2}-ab)

1

digits reversed

We have two distinct positive integers

a,b

a|b

a,b

2n

decimal digits. The first

n

a

are identical to the last

n

b

, and vice versa. Determine

a,b

1

find the angle

P

\triangle ABC

BP,CP

AC,AB

Q,R

respectively. Given that

AR=RB=CP, CQ=PQ

\angle BRC