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2018 PUMaC Live Round, 5.1

Let $w$ and $h$ be positive integers and define $N(w,h)$ to be the number of ways of arranging $wh$ people of distinct heights for a photoshoot in such a way that they form $w$ columns of $h$ people, with the people of each column sorted by height (i.e. shortest at the front, tallest at the back). Find the largest value of $N(w,h)$ that divides $1008$.

2018 PUMaC Live Round, Calculus 3

Tags: calculus , PuMAC , Live Round

Let $\mathcal{R}(f(x))$ denote the number of distinct real roots of $f(x)$. Compute $$\sum_{a=1}^{1009}\sum_{b=1010}^{2018}\mathcal{R}(x^{2018}-ax^{2016}+b).$$

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