Czech Republic

Consider an arbitrary line segment of the polyline and denote by $d$ its length and by $x$ and $y$ the lengths of its perpendicular projections on the sides of lengths 2 and 3, respectively. Cauchy-Schwarz inequality gives us $$(2x+3y{)}^2\le (2^2+3^2)(x^2+y^2)=13d^2,$$ which means $2x+3y\le d\cdot \sqrt{13}$. Denote by $X$ and $Y$ the total length of all the perpendicular projections of all the line segments on the sides of lengths 2 and 3, respectively. Summing up our estimations for each line segment gives us $2X+3Y\le 36\cdot \sqrt{13}<130$. But then either $2X<40$, or $3Y<90$. In the first case we would have $X<20$, so on the side of length 2 there is a point that is contained in fewer than 10 projections. A line perpendicular to this side at this point intersects the polyline at most 9 times. The other case is analogous.