Question 6
Let $m$ and $n$ be positive integers with $m\geq n$.
There are $m$ cupcakes of different flavors arranged around a circle
and $n$ people who like cupcakes.
Each person assigns a nonnegative real number score to each cupcake,
depending on how much they like the cupcake.
Suppose that for each person $P$, it is possible to partition the circle
of $m$ cupcakes into $n$ groups of consecutive cupcakes
so that the sum of $P$'s scores of the cupcakes in each group is at least $1$.
Prove that it is possible to distribute the $m$ cupcakes to the $n$ people
so that each person $P$ receives cupcakes of total score at least $1$ with respect to $P$.
Proposed by Cheng-Yin Chang and Hung-Hsun Yu.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
There are $m$ cupcakes of different flavors arranged around a circle
and $n$ people who like cupcakes.
Each person assigns a nonnegative real number score to each cupcake,
depending on how much they like the cupcake.
Suppose that for each person $P$, it is possible to partition the circle
of $m$ cupcakes into $n$ groups of consecutive cupcakes
so that the sum of $P$'s scores of the cupcakes in each group is at least $1$.
Prove that it is possible to distribute the $m$ cupcakes to the $n$ people
so that each person $P$ receives cupcakes of total score at least $1$ with respect to $P$.
Proposed by Cheng-Yin Chang and Hung-Hsun Yu.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.