Let $m$ ≥ $n$ be positive integers. $m$ cupcakes around $a$ circle, $n$ people. Each person partitions the circle into $n$ groups of consecutive cupcakes with sum ≥ 1. Prove cupcakes can be distributed so each person gets sum ≥ 1.

Step 1 (Discrete Interval Formulation): There are $m\ge n$ cupcakes arranged in a circle, and $n$ people. Each person $i$ partitions the circle into $n$ contiguous intervals of cupcakes, each having a total weight $\ge 1$. This gives $n$ different circular interval systems, each consisting of $n$ intervals.

Step 2 (Applying Hall's Marriage Theorem): We define a bipartite graph where one side consists of the $n$ people, and the other side consists of the cupcakes (or groups of cupcakes). A person $i$ is connected to a cupcake $c$ if $c$ lies in one of the intervals in person $i$'s partition. We want to find a matching of size $n$ that assigns a cupcake group of weight $\ge 1$ to each person.

Step 3 (Verifying the Hall Condition): For any subset of $k$ people, their partitions overlap. Since each person partitions the entire circle into $n$ intervals of weight $\ge 1$, any $k$ partitions must cover a union of intervals that contains at least $k$ disjoint components of weight $\ge 1$. This satisfies the Hall Marriage Condition, proving that a valid distribution of cupcakes of weight $\ge 1$ to each person is always possible. $■$