APMO 2001

For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$ . Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a stump of $n$ . Let $T(n)$ be the sum of all stumps of $n$ . Prove that $n=S(n)+9T(n)$ .

Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).

Two equal-sized regular $n$ -gons intersect to form a $2n$ -gon $C$ . Prove that the sum of the sides of $C$ which form part of one $n$ -gon equals half the perimeter of $C$ .

Alternative formulation:

Let two equal regular $n$ -gons $S$ and $T$ be located in the plane such that their intersection $S\cap T$ is a $2n$ -gon (with $n\ge 3$ ). The sides of the polygon $S$ are coloured in red and the sides of $T$ in blue.

Prove that the sum of the lengths of the blue sides of the polygon $S\cap T$ is equal to the sum of the lengths of its red sides.

A point in the plane with a cartesian coordinate system is called a mixed point if one of its coordinates is rational and the other one is irrational. Find all polynomials with real coefficients such that their graphs do not contain any mixed point.

Find the greatest integer $n$ , such that there are $n+4$ points $A$ , $B$ , $C$ , $D$ , $X_1,\dots,~X_n$ in the plane with $AB\ne CD$ that satisfy the following condition: for each $i=1,2,\dots,n$ triangles $ABX_i$ and $CDX_i$ are equal.