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Baltic Way competition problems

Algebra Difficulty 5.1 AIME, harder Prove it Baltic Way

Problem:
The squares of a squared paper are enumerated as follows:
Figure 1
Devise a polynomial p(m,n)p(m, n) of two variables m,nm, n such that for any positive integers mm and nn the number written in the square with coordinates (m,n)(m, n) will be equal to p(m,n)p(m, n).

Solution

Solution:
Since the square with the coordinates (m,n)(m, n) is the nnth on the (n+m−1)(n+m-1)-th diagonal, it contains the number
p(m,n)=∑i=1n+m−2i+n=(n+m−1)(n+m−2)2+n p(m, n) = \sum_{i=1}^{n+m-2} i + n = \frac{(n+m-1)(n+m-2)}{2} + n

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