2002 Japan MO Finals

Part of Japan MO Finals

Subcontests

(5)

1

rectangles in the coordinate plane

S

be a set of 2002 points in the coordinate plane, no two of which have the same x\minus{} or y\minus{}coordinate. For any two points

P,Q \in S

, consider the rectangle with one diagonal

PQ

and the sides parallel to the axes. Denote by

W_{PQ}

the number of points of

S

lying in the interior of this rectangle. Determine the maximum

N

such that, no matter how the points of

S

are distributed, there always exist points

P,Q

S

W_{PQ}\ge N

1

good coins

n \ge 3

coins on a circle. Consider a coin and the two coins adjacent to it; if there are an odd number of heads among the three, we call it good. An operation consists of turning over all good coins simultaneously. Initially, exactly one of the

n

coins is a head. The operation is repeatedly performed. (a) Prove that if

n

is odd, the coins will never be all-tails. (b) For which values of

n

is it possible to make the coins all-tails after several operations? Find, in terms of

n

, the number of operations needed for this to occur.

1

the product MP·MQ is independent of the position of P

Distinct points

A,M,B

with AM \equal{} MB are given on circle

(C_0)

in this order. Let

P

be a point on the arc

AB

M

(C_1)

is internally tangent to

(C_0)

P

AB

Q

. Prove that the product

MP\cdot MQ

is independent of the position of

P

1

A well know problem about sum-of-digit function

S(n)

the sum of decimal digits of a positive integer

n

. Show that there exist

2002

distinct positive integers

n_1, n_2, \cdots, n_{2002}

such that n_1 \plus{} S(n_1) \equal{} n_2 \plus{} S(n_2) \equal{} \cdots \equal{} n_{2002} \plus{} S(n_{2002}).

1

Cyclic inequality

n\geq 3

be natural numbers, and let

a_1,\ a_2,\ \cdots,\ a_n,\ \ b_1,\ b_2,\ \cdots,\ b_n

be positive numbers such that

a_1+a_2+\cdots +a_n=1,\ b_1^2+b_2^2+\cdots +b_n^2=1.

a_1(b_1+a_2)+a_2(b_2+a_3)+\cdots +a_n(b_n+a_1)<1.