TUYMAADA 1994

World Cup in America introduced a new point system. For a victory $3$ points are given, for a draw $1$ point and for defeat $0$ points. In the preliminary games, the teams are divided into groups of $4$ teams. In groups, teams play with each other, once, then in accordance with the points scored $a,b,c$ and $d$ ( $a>b>c>d$ ) teams take the first, second, third and fourth place in their groups. Give all possible options for the distribution points $a,b,c$ and $d$

The set of numbers $M=\{4k-3 | k\in N\}$ is considered. A number of of this set is called "simple" if it is impossible to put in the form of a product of numbers from $M$ other than $1$ . Show that in this set, the decomposition of numbers in the product of "simple" factors is ambiguous.

Point $M$ lies inside triangle $ABC$ . Prove that for any other point $N$ lying inside the triangle $ABC$ , at least one of the following three inequalities is fulfilled: $AN>AM, BN>BM, CN>CM$ .

Let a convex polyhedron be given with volume $V$ and full surface $S$ . Prove that inside a polyhedron it is possible to arrange a ball of radius $\frac{V}{S}$ .

Find the smallest natural number $n$ for which $sin \Big(\frac{1}{n+1934}\Big)<\frac{1}{1994}$ .

In three houses $A,B$ and $C$ , forming a right triangle with the legs $AC=30$ and $CB=40$ , live three beetles $a,b$ and $c$ , capable of moving at speeds of $2, 3$ and $4$ , respectively. Suppose that you simultaneously release these bugs from point $M$ and mark the time after which beetles reach their homes. Find on the plane such a point $M$ , where is the last time to reach the house a bug would be minimal.

Prove that there are infinitely many natural numbers $a,b,c,u$ and $v$ with greatest common divisor $1$ satisfying the system of equations: $a+b+c=u+v$ and $a^2+b^2+c^2=u^2+v^2$

Prove that in space there is a sphere containing exactly $1994$ points with integer coordinates.