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Sperner's lemma says that if a triangle is triangulated, its vertices are labelled $1,2,3$, and label $i$ is forbidden on the edge opposite vertex $i$, then an odd number of small triangles have all three labels.

To prove it, count incidences with edges whose endpoint labels are $1$ and $2$. Along the outer $12$-edge, the labels begin at $1$ and end at $2$, so the number of $12$ transitions is odd; no other boundary edge contributes. An interior edge contributes twice. A small triangle contributes an odd number precisely when its three labels are $1,2,3$: a triangle using only $1,2$ contributes two, and every other non-tricoloured triangle contributes zero. The number of tricoloured triangles is therefore odd.

Now suppose the closed sets $A_i$ in the question existed. Let $d_i(x)$ be the distance from a point to a closed set $A_i$. Empty triple intersection makes $d_1+d_2+d_3>0$, sois a continuous function from the large triangle to itself, where $v_i$ is the vertex opposite ${\alpha }_i$. Since the $A_i$ cover the triangle, at least one $d_i(x)$ vanishes, so $r(x)$ always lies on the boundary. On ${\alpha }_i\subseteq A_i$, its $i$th barycentric coordinate vanishes, hence $r({\alpha }_i)\subseteq {\alpha }_i$. Thus $r$ is face-preserving on the boundary; its boundary restriction is homotopic there to the identity by the straight-line homotopy. This would give a map from the triangle into its boundary whose boundary degree is one, contradicting the no-retraction theorem, the standard topological consequence of Sperner's lemma. Therefore the three closed sets must have a common point.

An Inhomogeneous Poisson process with intensity $\lambda$ is a counting process with $N_0=0$, independent increments, andThe continuity and positivity of $\lambda$ make $\Lambda$ strictly increasing.

For $M_t=N_{g(t)}$ and $0\le s<t$,has a Poisson distribution with meanDisjoint increments remain independent, so $M$ is a homogeneous rate-one Poisson process. Taking $g={\Lambda }^{-1}$ gives $N_t=M_{\Lambda (t)}$, and henceThis is the time change of an inhomogeneous Poisson process.

For the bicycle model, let an arrival occur at time $s\le t$ and have velocity $V$. At time $t$ its position is $V(t-s)$, so it lies in the first $x$ miles exactly whenConditional on its arrival time, independently marking the bicycle by whether this event occurs is Poisson thinning. Therefore the desired count is Poisson with meanThus