SE-SKOLORNAS 2013

For $r> 0$ denote by $B_r$ the set of points at distance at most $r$ length units from the origin. If $P_r$ is the set of the points in $B_r$ whit integer coordinates, show that the equation $$xy^3z + 2x^3z^3-3x^5y = 0$$ has an odd number of solutions $(x, y, z)$ in $P_r$ .

The paper folding art origami is usually performed with square sheets of paper. Someone folds the sheet once along a line through the center of the sheet in orde to get a nonagon. Let $p$ be the perimeter of the nonagon minus the length of the fold, i.e. the total length of the eight sides that are not folds, and denote by s the original side length of the square. Express the area of the nonagon in terms of $p$ and $s$ .

Determine all primes $p$ and all non-negative integers $m$ and $n$ , such that $$1 + p^n = m^3.$$

A robotic lawnmower is located in the middle of a large lawn. Due a manufacturing defect, the robot can only move straight ahead and turn in directions that are multiples of $60^o$ . A fence must be set up so that it delimits the entire part of the lawn that the robot can get to, by traveling along a curve with length no more than $10$ meters from its starting position, given that it is facing north when it starts. How long must the fence be?

Let $n \geq 2$ be a positive integer. Show that there are exactly $2^{n-3}n(n-1)$ $n$ -tuples of integers $(a_1,a_2,\dots,a_n)$ , which satisfy the conditions: (i) $a_1=0$ ; (ii) for each $m$ , $2 \leq m \leq n$ , there is an index in $m$ , $1 \leq i_m <m$ , such that $\left|a_{i_m}-a_m\right|\leq 1$ ; (iii) the $n$ -tuple $(a_1,a_2,\dots,a_n)$ contains exactly $n-1$ different numbers.

Let $a, b, c$ , be real numbers such that $$a^2b^2 + 18 abc > 4b^3+4a^3c+27c^2 .$$ Prove that $a^2>3b$ .