KSAT Benchmark

2017.30

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

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

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

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

The correct answer is 21.

1998.29

Suppose the two equations $P(x) = 0$ and $Q(x) = 0$ have $7$ and $9$ distinct real roots, respectively.

Define the set

$$A = \{(x,y)\mid P(x)Q(y)=0 \text{ and } Q(x)P(y)=0, x,y\in\mathbb{R}\}.$$

This set $A$ is infinite. Consider the subset

$$B = \{(x,y)\mid (x,y)\in A \text{ and } x = y\}.$$

Let $n(B)$ denote the number of elements of $B$. Note that $n(B)$ depends on the specific choice of $P(x)$ and $Q(x)$.

Determine the maximum possible value of $n(B)$.

Suppose the two equations $P(x) = 0$ and $Q(x) = 0$ have $7$ and $9$ distinct real roots, respectively.
Define the set

$$
A = \{(x,y)\mid P(x)Q(y)=0 \text{ and } Q(x)P(y)=0, x,y\in\mathbb{R}\}.
$$

This set $A$ is infinite. Consider the subset

$$
B = \{(x,y)\mid (x,y)\in A \text{ and } x = y\}.
$$

Let $n(B)$ denote the number of elements of $B$. Note that $n(B)$ depends on the specific choice of $P(x)$ and $Q(x)$.
Determine the maximum possible value of $n(B)$.

The answer is 15