ZA-SAMO 2014

Determine the last two digits of the product of the squares of all positive odd integers less than $2014$ .

In obtuse triangle $ABC$ , with the obtuse angle at $A$ , let $D$ , $E$ , $F$ be the feet of the altitudes through $A$ , $B$ , $C$ respectively. $DE$ is parallel to $CF$ , and $DF$ is parallel to the angle bisector of $\angle BAC$ . Find the angles of the triangle.

(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$ , the also $$ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).$$

(b) Show that there are no two positive integers $a$ and $b$ such that $$ab+\gcd(a,b)+\text{lcm}(a,b)=2014.$$

Let $n > 1$ be an integer. An $n \times n$ -square is divided into $n^2$ unit squares. Of these unit squares, $n$ are coloured green and $n$ are coloured blue, and all remaining ones are coloured white. Are there more such colourings for which there is exactly one green square in each row and exactly one blue square in each column; or colourings for which there is exactly one green square and exactly one blue square in each row?

Let $O$ be the centre of a two-dimensional coordinate system, and let $A_1, A_2, \ldots ,A_n$ be points in the first quadrant and $B_1, B_2, \ldots , B_m$ points in the second quadrant. We associate numbers $a_1, a_2, \ldots , a_n$ to the points $A_1, A_2, \ldots ,A_n$ and numbers $b_1, b_2, \ldots, b_m$ to the points $B_1, B_2, \ldots , B_m$ , respectively. It turns out that the area of triangle $OA_jB_k$ is always equal to the product $a_jb_k$ , for any $j$ and $k$ . Show that either all the $A_j$ or all the $B_k$ lie on a single line through $O$ .