SE-SKOLORNAS 1982

How many solutions does $$x^2 - [x^2] = \left(x - [x]\right)^2$$ have satisfying $1 \leq x \leq n$ ?

Show that $$abc \geq (a+b-c)(b+c-a)(c+a-b)$$ for positive reals $a$ , $b$ , $c$ .

Show that there is a point $P$ inside the quadrilateral $ABCD$ such that the triangles $PAB$ , $PBC$ , $PCD$ , $PDA$ have equal area. Show that $P$ must lie on one of the diagonals.

$ABC$ is a triangle with $AB = 33$ , $AC = 21$ and $BC = m$ , an integer. There are points $D$ , $E$ on the sides $AB$ , $AC$ respectively such that $AD = DE = EC = n$ , an integer. Find $m$ .

Each point in a $12 \times 12$ array is colored red, white or blue. Show that it is always possible to find $4$ points of the same color forming a rectangle with sides parallel to the sides of the array.

Show that $$(2a-1) \sin x + (1-a) \sin(1-a)x \geq 0$$ for $0 \leq a \leq 1$ and $0 \leq x \leq \pi$ .