CONO-SUR 2010

Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that:

• The sum of the fractions is equal to $2$ .
• The sum of the numerators of the fractions is equal to $1000$ .

In how many ways can Pedro do this?

On a line, $44$ points are marked and numbered $1, 2, 3,…,44$ from left to right. Various crickets jump around the line. Each starts at point $1$ , jumping on the marked points and ending up at point $44$ . In addition, each cricket jumps from a marked point to another marked point with a greater number. When all the crickets have finished jumping, it turns out that for pair $i, j$ with ${1}\leq{i}<{j}\leq{44}$ , there was a cricket that jumped directly from point $i$ to point $j$ , without visiting any of the points in between the two. Determine the smallest number of crickets such that this is possible.

Let us define cutting a convex polygon with $n$ sides by choosing a pair of consecutive sides $AB$ and $BC$ and substituting them by three segments $AM, MN$ , and $NC$ , where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$ . In other words, the triangle $MBN$ is removed and a convex polygon with $n+1$ sides is obtained. Let $P_6$ be a regular hexagon with area $1$ . $P_6$ is cut and the polygon $P_7$ is obtained. Then $P_7$ is cut in one of seven ways and polygon $P_8$ is obtained, and so on. Prove that, regardless of how the cuts are made, the area of $P_n$ is always greater than $2/3$ .

The incircle of triangle $ABC$ touches sides $BC$ , $AC$ , and $AB$ at $D, E$ , and $F$ respectively. Let $\omega_a, \omega_b$ and $\omega_c$ be the circumcircles of triangles $EAF, DBF$ , and $DCE$ , respectively. The lines $DE$ and $DF$ cut $\omega_a$ at $E_a\neq{E}$ and $F_a\neq{F}$ , respectively. Let $r_A$ be the line $E_{a}F_a$ . Let $r_B$ and $r_C$ be defined analogously. Show that the lines $r_A$ , $r_B$ , and $r_C$ determine a triangle with its vertices on the sides of triangle $ABC$ .

Determine if there exists an infinite sequence $a_0, a_1, a_2, a_3,...$ of nonegative integers that satisfies the following conditions: (i) All nonegative integers appear in the sequence exactly once. (ii) The succession $b_n=a_{n}+n,$ , $n\geq0$ , is formed by all prime numbers and each one appears exactly once.