SMO 2000

Triangle $ABC$ is inscribed in a circle with center $O$ . Let $D$ and $E$ be points on the respective sides $AB$ and $AC$ so that $DE$ is perpendicular to $AO$ . Show that the four points $B,D,E$ and $C$ lie on a circle.

Show that $240$ divides all numbers of the form $p^4 - q^4$ , where p and q are prime numbers strictly greater than $5$ . Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$ , with $p$ and $q$ prime numbers strictly greater than $5$ .

Is there a positive integer with at most four digits whose value is increased by exactly $60\%$ when the first digit is moved to the end of the number? For example, when the first digit of $1234$ is moved to the end of the number, the result is the integer $2341$ .

In a party of $1000$ people, the number of people who have shaken hands with at most $962$ people is less than or equal to $37$ . Show that one can find $27$ people in the party who have all shaken hands with each other.