KSAT Benchmark

Let $t$ be a real number. Define the function $f(x)$ by

$$f(x) = \begin{cases} 1 - |x - t|, & \text{if } |x - t| \le 1,\\ 0, & \text{if } |x - t| > 1. \end{cases}$$

For some odd integer $k$, define

$$g(t) = \int_{k}^{k + 8} f(x)\,\cos(\pi x)\,dx.$$

Suppose $g(t)$ has a local minimum at $t = \alpha$ with $g(\alpha) < 0$.

List all such $\alpha$ in increasing order as $\alpha_1, \alpha_2, \dots, \alpha_m$ (where $m$ is a positive integer), and assume

$$\sum_{i=1}^m \alpha_i = 45.$$

Find the value of

$$k - \pi^2 \sum_{i=1}^m g(\alpha_i).$$