IT-CESENATICO 2005

Let $ABC$ be a right angled triangle with hypotenuse $AC$ , and let $H$ be the foot of the altitude from $B$ to $AC$ . Knowing that there is a right-angled triangle with side-lengths $AB, BC, BH$ , determine all the possible values of $\frac{AH}{CH}$

Prove that among any $18$ consecutive positive integers not exceeding $2005$ there is at least one divisible by the sum of its digits.

In each cell of a $4 \times 4$ table a digit $1$ or $2$ is written. Suppose that the sum of the digits in each of the four $3 \times 3$ sub-tables is divisible by $4$ , but the sum of the digits in the entire table is not divisible by $4$ . Find the greatest and the smallest possible value of the sum of the $16$ digits.

Determine all $n \geq 3$ for which there are $n$ positive integers $a_1, \cdots , a_n$ any two of which have a common divisor greater than $1$ , but any three of which are coprime. Assuming that, moreover, the numbers $a_i$ are less than $5000$ , find the greatest possible $n$ .

Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and

$$a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }a_n+h \text{ otherwise }.\end{array}$$

For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$ ?

Two circles $\gamma_1, \gamma_2$ in a plane, with centers $A$ and $B$ respectively, intersect at $C$ and $D$ . Suppose that the circumcircle of $ABC$ intersects $\gamma_1$ in $E$ and $\gamma_2$ in $F$ , where the arc $EF$ not containing $C$ lies outside $\gamma_1$ and $\gamma_2$ . Prove that this arc $EF$ is bisected by the line $CD$ .