SE-SKOLORNAS 2018

Let the $ABCD$ be a quadrilateral without parallel sides, inscribed in a circle. Let $P$ and $Q$ be the intersection points between the lines containing the quadrilateral opposite sides. Show that the bisectors to the angles at $P$ and $Q$ are parallel to the bisectors of the angles at the intersection point of the diagonals of the quadrilateral.

Find all functions $f: R \to R$ that satisfy $f (x) + 2f (\sqrt[3]{1-x^3}) = x^3$ for all real $x$ .

(Here $\sqrt[3]{x}$ is defined all over $R$ .)

Let m be a positive integer. An $m$ -pattern is a sequence of $m$ symbols of strict inequalities. An $m$ -pattern is said to be realized by a sequence of $m + 1$ real numbers when the numbers meet each of the inequalities in the given order. (For example, the $5$ -pattern $<, <,>, < ,>$ is realized by the sequence of numbers $1, 4, 7, -3, 1, 0$ .) Given $m$ , which is the least integer $n$ for which there exists any number sequence $x_1,... , x_n$ such that each $m$ -pattern is realized by a subsequence $x_{i_1},... , x_{i_{m + 1}}$ with $1 \le i_1 <... < i_{m + 1} \le n$ ?

Find the least positive integer $n$ with the property: Among arbitrarily $n$ selected consecutive positive integers, all smaller than $2018$ , there is at least one that is divisible by its sum of digits .

In a triangle $ABC$ , two lines are drawn that together trisect the angle at $A$ . These intersect the side $BC$ at points $P$ and $Q$ so that $P$ is closer to $B$ and $Q$ is closer to $C$ . Determine the smallest constant $k$ such that $| P Q | \le k (| BP | + | QC |)$ , for all such triangles. Determine if there are triangles for which equality applies.

For which positive integers $n$ can the polynomial $p(x) = 1 + x^n + x^{2n}$ is written as a product of two polynomials with integer coefficients (of degree $\ge 1$ )?