2001 IMO Shortlist Problems/A1

Problem

Let $T$ denote the set of all ordered triples $(p,q,r)$ of nonnegative integers. Find all functions $f:T \rightarrow \mathbb{R}$ such that

$f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\ 1 + \tfrac{1}{6}\{f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\ + f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\ + f(p,q + 1,r - 1) + f(p,q - 1,r + 1)\} & \text{otherwise.} \end{cases}$

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2001 IMO Shortlist Problems/A1

Problem

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