KSAT Benchmark
2017.30
Let be a real number. Define the function by
For some odd integer , define
Suppose has a local minimum at with .
List all such in increasing order as (where is a positive integer), and assume
Find the value of
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 and have and distinct real roots, respectively.
Define the set
This set is infinite. Consider the subset
Let denote the number of elements of . Note that depends on the specific choice of and .
Determine the maximum possible value of .
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