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4 problems match “absolute value”. Clear

Alternating sum of minimum values ●●●○○ 50 ptsFor a positive integer $n$ , define $f_n(x)=\left(\sum_{i=1}^{n}|x-i|\right)^2-\sum_{i=1}^{n}(x-i)^2$ on the interval $[0,n+1]$ . Let $a_n$ be the minimum value of $f_n(x)$ . Find $\sum_{n=1}^{11}(-1)^{n+1}a_n$ .
Counting z with integer parts of f(z) ●●●●○ 100 ptsLet $f : \mathbb{C} \to \mathbb{C}$ be defined by $f(z) = z^2 + iz + 1$ . How many complex numbers $z$ are there such that $\text{Im}(z) > 0$ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $10$ ?
Largest finite count of small elements ●●●●○ 100 pts $S$ is a set of complex numbers such that if $u, v \in S$ , then $uv \in S$ and $u^{2}+v^{2} \in S$ . Suppose that the number $N$ of elements of $S$ with absolute value at most $1$ is finite. What is the largest possible value of $N$ ?
Maximum points removing stones from piles ●●●●● 200 ptsThere are 100 piles, each containing 400 stones. On each move, Petya selects two piles, removes one stone from each, and earns as many points as the absolute difference between the numbers of stones in these two piles (before the removal). Petya must remove all the stones. What is the maximum total number of points he can earn?