Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 11I / c / Solution

Write $\beta (s)=({\beta }_1(s),{\beta }_2(s))$, $\gamma (t)=({\gamma }_1(t),{\gamma }_2(t))$, andThe endpoint conditions implyLet $P(x)=\max (-1,\min (1,x))$ and define the continuous self-map of $I^2$By Brouwer, $F$ has a fixed point $(s,t)$. If $s$ is interior, its fixed-point equation gives $h_2=0$. At $s=-1$ or $s=1$, the projection equation and the corresponding displayed boundary sign again force $h_2=0$. The identical argument in the second coordinate gives $h_1=0$. HencesoThe two paths intersect.