SE-SKOLORNAS 2010

Exists a triangle whose three altitudes have lengths $1, 2$ and $3$ respectively?

Consider the four lines $y = mx-k^2$ for different integer $k$ . Let $(x_i,y_i)$ , $i = 1,2,3,4$ be four different points , such that each belongs to two different lines and on each line pass through just the two of them. Lat $x_1\leq x_2\leq x_3\leq x_4$ . Show that $x_1 + x_4 =x_2+x_3$ and $y_1y_4 =y_2y_3$ .

Find all natural numbers $n \ge 1$ such that there is a polynomial $p(x)$ with integer coefficients for which $p (1) = p (2) = 0$ and where $p (n)$ is a prime number .

We create a sequence by setting $a_1 = 2010$ and requiring that $a_n-a_{n-1}\leq n$ and $a_n$ is also divisible by $n$ . Show that $a_{100},a_{101},a_{102},\dots$ form an arithmetic sequence.

Consider the number of triangles where the side lengths $a,b,c$ satisfy $(a + b + c) (a + b -c) = 2b^2$ . Determine the angles in the triangle for which the angle opposite to the side with the length $a$ is as big as possible.

An infinite number of squares on an infinitely square grid paper are painted red. Show that you can draw a number of squares on the paper, with sides along the grid lines, such that:

(1) no square in the grid belongs to more than one square (an edge, on the other hand, may belong to more than one square)

(2) each red square is located in one of the squares and the number of red squares in such square is at least $1/5$ and at most $4/5$ of the number of squares in the square.

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PS. I always post the original wording when I doubt about my (using Google) translation.