Baltic Way 1990 · Problem 2

Algebra

The squares of a squared paper are enumerated as follows:

Enumeration diagram is part of the statement.

Devise a polynomial $p(m, n)$ of two variables $m, n$ such that for any positive integers $m$ and $n$ the number written in the square with coordinates $(m, n)$ will be equal to $p(m, n)$ .

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Review

Topics

Sequences and recurrences

Solutions

Solution

Solution: Since the square with the coordinates $(m, n)$ is the $n$ th on the $(n+m-1)$ -th diagonal, it contains the number

p(m, n) = \sum_{i=1}^{n+m-2} i + n = \frac{(n+m-1)(n+m-2)}{2} + n

Elsewhere

AoPS · Baltic Way 1990