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Omni-MATH / Let T denote the set of all ordered triples (p,q,r) of nonnegative integers. Find all functions f:T→R satisfying …

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problem

Let T T denote the set of all ordered triples (p,q,r) (p,q,r) of nonnegative integers. Find all functions f:T→R f: T \rightarrow \mathbb{R} satisfying \[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\ 1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\ + f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\ + f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases} \] for all nonnegative integers p p, q q, r r.
Plain-text mathematical notation (without MathML)
Let T denote the set of all ordered triples (p,q,r) of nonnegative integers. Find all functions f:T→R satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
 + f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers p, q, r.
Original LaTeX notation
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
 + f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers $ p$, $ q$, $ r$.

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Source and history

task_id
6f365cba-795d-595f-92cf-ae0a8045a293
task_key
test--6f365cba-795d-595f-92cf-ae0a8045a293
task_revision_id
1
config / split
/ test

Official source

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