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2

Part of 1952 Czech and Slovak Olympiad III A

Problems(1)

Triangular table

Source: Czech and Slovak Olympiad 1952, National Round, Problem 2

4/11/2020
Consider a triangular table of positive integers \begin{matrix} & & & a_{11} & a_{12} & a_{13} & & & \\ & & a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & & \\ & a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} & a_{37} & \\ \iddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} The first row consists of odd numbers only. For i>1,j≥1i>1,j\ge1 we have aij=ai−1,j−2+ai−1,j−1+ai−1,j.a_{ij}=a_{i-1,j-2}+a_{i-1,j-1}+a_{i-1,j}. If we get out of range with the second index, we consider such aa to be zero (eg. a22=0+a11+a12a_{22}=0+a_{11}+a_{12} and a37=a25+0+0a_{37}=a_{25}+0+0). Show that for every i>1i>1 there is j∈{1,…,2i+1}j\in\{1,\ldots,2i+1\} such that aija_{ij} is even.
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