2000 Japan MO Finals

Part of Japan MO Finals

Subcontests

(5)

1

Minimum number of 'good' intersection points

Finitely many lines are given in a plane. We call an intersection point a point that belongs to at least two of the given lines, and a good intersection point a point that belongs to exactly two lines. Assuming there at least two intersection points, find the minimum number of good intersection points.

1

Inequality for 5 points in the plane

Given five points

A,B,C,D,E

in a plane, no three of which are collinear, prove the inequality

AB+BC+CA+DE\le AD+AE+BD+BE+CD+CE

1

Reversing sequence of cards by shuffling

3n

cards, denoted by distinct letters

a_1,a_2,\ldots ,a_{3n}

, be put in line in this order from left to right. After each shuffle, the sequence

a_1,a_2,\ldots ,a_{3n}

is replaced by the sequence

a_3,a_6,\ldots ,a_{3n},a_2,a_5,\ldots ,a_{3n-1},a_1,a_4,\ldots ,a_{3n-2}

. Is it possible to replace the sequence of cards

1,2,\ldots ,192

by the reverse sequence

192,191,\ldots ,1

by a finite number of shuffles?

1

Sequence of rational points P_i forming unit distances

Consider the points

O(0,0)

A(0,1/ 2)

on the coordinate plane. Prove that there is no finite sequence of rational points

P_1,P_2,\ldots ,P_n

in the plane such that

OP_1=P_1P_2=\ldots =P_{n-1}P_n=P_nA=1

1

Japan olympiad

Given a natural number

n\ge 3

, prove that there exists a set

A_n

with the following two properties: 1)

A_n

n

distinct natural numbers 2) For any

a\in A

, the remainder of the product of all elements of

A_n\backslash \{a\}

a

1