Question 174
Let $a_{0}, a_{1}, \ldots, a_{99}, b_{0}, b_{1}, \ldots, b_{99}$ be strictly positive real numbers.
For $k=0,1, \ldots, 198$, set $S_{k} = \sum_{i=0}^{198} a_{i} b_{k-i}$, with $a_{j} = 0$ and $b_{j} = 0$ if $j \lt 0$ or $j \gt 99$.
Is it possible that the numbers $S_{0}, S_{1}, \ldots, S_{198}$ are all equal?
For $k=0,1, \ldots, 198$, set $S_{k} = \sum_{i=0}^{198} a_{i} b_{k-i}$, with $a_{j} = 0$ and $b_{j} = 0$ if $j \lt 0$ or $j \gt 99$.
Is it possible that the numbers $S_{0}, S_{1}, \ldots, S_{198}$ are all equal?