63rd Czech and Slovak Mathematical Olympiad

We will compute the requested number $P$ of all internal segments for the cylinder formed by gluing the rectangle $x\times y$ along the opposite sides of length $y$.

This cylinder has two bases of perimeter $x$ and its lateral sides are of length $y$. We will use an obvious formula $P=P_0-P_1-P_2$, where $P_0$ denotes the total number of segments, while $P_1$ and $P_2$ denote the numbers of segments which lie on the lateral surface or on one of the two bases, respectively. Note that the vertices of the unit squares are situated on the surface of the cylinder in such a way that exactly $y+1$ of them lie on the same of the $x$ lateral sides and, in the same time, exactly $x$ vertices lie on the same boundary circle of the two bases. These facts lead immediately to the following formulae $$\begin{array}{rl}{P}_0& =\left(\genfrac{}{}{0ex}{}{x(y+1)}{2}\right)=\frac{x(y+1)(xy+x-1)}{2},\\ P_1& =x\cdot \left(\genfrac{}{}{0ex}{}{y+1}{2}\right)=\frac{x(y+1)y}{2},\\ P_2& =2\cdot \left(\genfrac{}{}{0ex}{}{x}{2}\right)=x(x-1).\end{array}$$ Consequently, $$\begin{array}{rl}P& =P_0-P_1-P_2=\frac{x(y+1)(xy+x-1)}{2}-\frac{x(y+1)y}{2}-x(x-1)\\ & =\frac{x(x-1)(y^2+2y-1)}{2}.\end{array}$$ In view of symmetry, the number $Q$ of internal segments for the other cylinder (with base's perimeter $y$ and lateral sides of length $x$) is given by $$Q=\frac{y(y-1)(x^2+2x-1)}{2}.$$ To decide which of the inequalities $P>Q$ or $Q>P$ holds in the case when $x>y$, we factorize the difference $P-Q$ (since $P=Q$ if $x=y$, the polynomial $P-Q$ must be divisible by $x-y$): $$\begin{array}{rl}2(P-Q)& =(x^2-x)(y^2+2y-1)-(y^2-y)(x^2+2x-1)\\ & =(x^2{y}^2-x{y}^2+2x^{2}y-2xy-x^2+x)\\ & -(x^2{y}^2-x^{2}y+2x{y}^2-2xy-y^2+y)\\ & =3xy(x-y)-(x-y)(x+y)+(x-y)\\ & =(x-y)(3xy-x-y+1).\end{array}$$ Thus $x>y$ implies that $P>Q$ if we show that the same condition $x>y$ implies that $3xy-x-y+1>0$. The last is almost evident: it follows from $y\ge 2$ that $3xy\ge 6x$ and hence $$3xy-x-y+1\ge 5x-y+1>4x+1>0.$$ Answer. In the case when $x>y$, the number of internal segments is larger for the cylinder with bases which perimeter has length $x$.