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KSAT Benchmark

2017.30

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

f(x)={1xt,if xt1,0,if xt>1.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 kk, define

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

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

List all such α\alpha in increasing order as α1,α2,,αm\alpha_1, \alpha_2, \dots, \alpha_m (where mm is a positive integer), and assume

i=1mαi=45.\sum_{i=1}^m \alpha_i = 45.

Find the value of

kπ2i=1mg(αi).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)=0P(x) = 0 and Q(x)=0Q(x) = 0 have 77 and 99 distinct real roots, respectively.

Define the set

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

This set AA is infinite. Consider the subset

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

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

Determine the maximum possible value of n(B)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