JMO 2000

Consider the points $O(0,0)$ and $A(0,1/ 2)$ 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 $$OP_1=P_1P_2=\ldots =P_{n-1}P_n=P_nA=1$$

Let $3n$ 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?

Given five points $A,B,C,D,E$ in a plane, no three of which are collinear, prove the inequality $$AB+BC+CA+DE\le AD+AE+BD+BE+CD+CE$$

Given a natural number $n\ge 3$, prove that there exists a set $A_n$ with the following two properties:

1. $A_n$ consists of $n$ distinct natural numbers
2. For any $a\in A$, the remainder of the product of all elements of $A_n\backslash \{a\}$ divided by $a$ is $1$.

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.