You have a piece of land close to a river, running straight. You are required to cut off a rectangular portion of the land, with the river forming one of the sides of the rectangle so, your fence will have three sides to it. You only have 60 meters of fencing. The maximum area that you can enclose is \_\_\_\_

A stationary point of a function $f$ is a real number $r$ such that $f ^ { \prime } ( r ) = 0$. A polynomial need not have a stationary point (e.g. $x ^ { 3 } + x$ has none). Consider a polynomial $p ( x )$.
(a) If $p ( x )$ is of degree 2022, then $p ( x )$ must have at least one stationary point.
(b) If the number of distinct real roots of $p ( x )$ is 2021, then $p ( x )$ must have at least 2020 stationary points.
(c) If the number of distinct real roots of $p ( x )$ is 2021, then $p ( x )$ can have at most 2020 stationary points.
(d) If $r$ is a stationary point of $p ( x )$ AND $p ^ { \prime \prime } ( r ) = 0$, then the point $( r , p ( r ) )$ is neither a local maximum nor a local minimum point on the graph of $p ( x )$.

[14 points] For a positive integer $n$, let $f(x) = \sum_{i=0}^{n} x^i = 1 + x + x^2 + \cdots + x^n$. Find the number of local maxima of $f(x)$. Find the number of local minima of $f(x)$. For each maximum/minimum $(c, f(c))$, find the integer $k$ such that $k \leq c < k+1$.
Hints (use these or your own method): It may be helpful to (i) look at small $n$, (ii) use the definition of $f$ as well as a closed formula, and (iii) treat $x \geq 0$ and $x < 0$ separately.

Let $a$ be a point in the domain of a continuous real valued function $f$. One says that $f$ has a flex point at $a$ if we can find a small interval $(a - \epsilon , a + \epsilon)$ in the domain of $f$ such that the following happens: (i) for all $x$ in the open interval $(a - \epsilon , a)$ the sign of $f ^ { \prime \prime } ( x )$ is constant and, (ii) for all $x$ in the open interval $(a , a + \epsilon)$ the sign of $f ^ { \prime \prime } ( x )$ is constant and opposite to the sign of $f ^ { \prime \prime } ( x )$ in $(a - \epsilon , a)$.
Statements
(29) If $f$ is a cubic polynomial with a local maximum at $x = p$ and a local minimum at $x = q$, then $f$ has a unique flex point at $x = \frac { p + q } { 2 }$. (30) If $f ^ { \prime \prime } ( a ) = 0$ then $f$ must have a flex point at $a$. (31) Let $f ( x ) = x ^ { 2 }$ for $x \geq 0$ and $f ( x ) = - x ^ { 2 }$ for $x < 0$. Then $f$ has no flex points. (32) If $f ^ { \prime }$ is monotonic on an open interval $I$, then $f$ cannot have a flex point in $I$.

8. Let $x$ be a variable that takes real values, and let $f ( x ) = x ^ { 3 } - 3 x$. Which of the following statements is/are true?
(a) $f ( x )$ has a local maximum at $x = - \sqrt { 3 }$
(b) $f ( x )$ has a local maximum at $x = - 1$
(c) $f ( x )$ has a local minimum at $x = \sqrt { 3 }$
(d) $f ( x )$ has a global minimum at $x = 1$

For $a > 1$, consider the function $f ( x ) = 2 x ^ { 3 } - 3 ( a + 1 ) x ^ { 2 } + 6 a x - 4 a + 2$. Let $b$ be one real root of the equation $f ( x ) = 0$. The following is a process for comparing the magnitudes of two numbers $a$ and $b$. $f ^ { \prime } ( x ) =$ (가) and since $a > 1$, $f ( x )$ has a (나) at $x = 1$. Since $f ( 1 ) < 0$ and $f ( b ) = 0$, $a$ (다) $b$.
What are the correct expressions for (가), (나), and (다) in the above process? [3 points]

(가)	(나)	(다)
(1) $6 ( x - 1 ) ( x - a )$	local minimum	$>$
(2) $6 ( x - 1 ) ( x - a )$	local minimum	$<$
(3) $6 ( x - 1 ) ( x - a )$	local maximum	$>$
(4) $6 ( x - a ) ( x - 1 )$	local maximum	$<$
(5) $6 ( x - a ) ( x - 1 )$	local maximum	$>$

The function $y = f ( x )$ is continuous on all real numbers, and for all $x$ with $| x | \neq 1$, the derivative $f ^ { \prime } ( x )$ is
$$f ^ { \prime } ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & ( | x | < 1 ) \\ - 1 & ( | x | > 1 ) \end{array} \right.$$
Which of the following statements in are true? [3 points]

ㄱ. The function $y = f ( x )$ has an extremum at $x = - 1$. ㄴ. For all real numbers $x$, $f ( x ) = f ( - x )$. ㄷ. If $f ( 0 ) = 0$, then $f ( 1 ) > 0$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

A quartic function $f ( x )$ with positive leading coefficient satisfies the following conditions. $f ^ { \prime } ( x ) = 0$ has three distinct real roots $\alpha , \beta , \gamma ( \alpha < \beta < \gamma )$, and $f ( \alpha ) f ( \beta ) f ( \gamma ) < 0$. Which of the following in are correct? [3 points]
ㄱ. The function $f ( x )$ has a local maximum value at $x = \beta$. ㄴ. The equation $f ( x ) = 0$ has two distinct real roots. ㄷ. If $f ( \alpha ) > 0$, then the equation $f ( x ) = 0$ has a real root less than $\beta$.
(1) ᄀ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ㄴ,ㄷ
(5) ᄀ, ᄂ, ᄃ

The function $f ( x ) = x ^ { 3 } - 12 x$ has a local maximum value $b$ at $x = a$. Find the value of $a + b$. [3 points]

For a natural number $n$, when two points $\mathrm { P } _ { n - 1 } , \mathrm { P } _ { n }$ are on the graph of the function $y = x ^ { 2 }$, the point $\mathrm { P } _ { n + 1 }$ is determined according to the following rule.
(a) The coordinates of the two points $\mathrm { P } _ { 0 } , \mathrm { P } _ { 1 }$ are $(0,0)$ and $(1,1)$, respectively.
(b) The point $\mathrm { P } _ { n + 1 }$ is the intersection of the line passing through point $\mathrm { P } _ { n }$ and perpendicular to the line $\mathrm { P } _ { n - 1 } \mathrm { P } _ { n }$ and the graph of the function $y = x ^ { 2 }$. (where $\mathrm { P } _ { n }$ and $\mathrm { P } _ { n + 1 }$ are distinct points.) Let $l _ { n } = \overline { \mathrm { P } _ { n - 1 } \mathrm { P } _ { n } }$. What is the value of $\lim _ { n \rightarrow \infty } \frac { l _ { n } } { n }$? [3 points]
(1) $2 \sqrt { 3 }$
(2) $2 \sqrt { 2 }$
(3) 2
(4) $\sqrt { 3 }$
(5) $\sqrt { 2 }$

When the local minimum value of the function $f ( x ) = ( x - 1 ) ^ { 2 } ( x - 4 ) + a$ is 10, find the value of the constant $a$. [3 points]

There is a quartic function $f ( x )$ with leading coefficient 1, $f ( 0 ) = 3$, and $f ^ { \prime } ( 3 ) < 0$. For a real number $t$, define the set $S$ as $$S = \{ a \mid \text{the function } | f ( x ) - t | \text{ is not differentiable at } x = a \}$$ and let $g ( t )$ be the number of elements in set $S$. When the function $g ( t )$ is discontinuous only at $t = 3$ and $t = 19$, find the value of $f ( - 2 )$. [4 points]

For a real number $m$, let $f ( m )$ be the number of intersection points of the line passing through the point $( 0,2 )$ with slope $m$ and the curve $y = x ^ { 3 } - 3 x ^ { 2 } + 1$. What is the maximum value of the real number $a$ such that the function $f ( m )$ is continuous on the interval $( - \infty , a )$? [4 points]
(1) $- 3$
(2) $- \frac { 3 } { 4 }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { 15 } { 4 }$
(5) 6

For the function $f ( x ) = k x ^ { 2 } e ^ { - x } ( k > 0 )$ and a real number $t$, let $g ( t )$ be the smaller of the distance from the point $( t , f ( t ) )$ on the curve $y = f ( x )$ to the $x$-axis and the distance to the $y$-axis. What is the maximum value of $k$ such that the function $g ( t )$ is not differentiable at exactly one point? [4 points]
(1) $\frac { 1 } { e }$
(2) $\frac { 1 } { \sqrt { e } }$
(3) $\frac { e } { 2 }$
(4) $\sqrt { e }$
(5) $e$

On the coordinate plane, for a cubic function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x$ and a real number $t$, let P be the point where the tangent line to the curve $y = f ( x )$ at the point $( t , f ( t ) )$ intersects the $y$-axis. Let $g ( t )$ be the distance from the origin to point P. The function $f ( x )$ and the function $g ( t )$ satisfy the following conditions. (가) $f ( 1 ) = 2$ (나) The function $g ( t )$ is differentiable on the entire set of real numbers. What is the value of $f ( 3 )$? (Here, $a , b$ are constants.) [4 points]
(1) 21
(2) 24
(3) 27
(4) 30
(5) 33

The function $f ( x ) = 2 x ^ { 3 } - 12 x ^ { 2 } + a x - 4$ has a local maximum value $M$ at $x = 1$. Find the value of $a + M$. (Here, $a$ is a constant.) [3 points]

For a quadratic function $f ( x )$, the function $g ( x ) = f ( x ) e ^ { - x }$ satisfies the following conditions. (가) The points $( 1 , g ( 1 ) )$ and $( 4 , g ( 4 ) )$ are inflection points of the curve $y = g ( x )$. (나) The number of tangent lines drawn from the point $( 0 , k )$ to the curve $y = g ( x )$ is 3 when $k$ is in the range $- 1 < k < 0$. Find the value of $g ( - 2 ) \times g ( 4 )$. [4 points]

For all cubic functions $f ( x )$ satisfying the following conditions, what is the minimum value of $f ( 2 )$? [4 points] (가) The leading coefficient of $f ( x )$ is 1. (나) $f ( 0 ) = f ^ { \prime } ( 0 )$ (다) For all real numbers $x \geq - 1$, $f ( x ) \geq f ^ { \prime } ( x )$.
(1) 28
(2) 33
(3) 38
(4) 43
(5) 48

Two polynomial functions $f ( x )$ and $g ( x )$ satisfy $$g ( x ) = \left( x ^ { 3 } + 2 \right) f ( x )$$ for all real numbers $x$. If $g ( x )$ has a local minimum value of 24 at $x = 1$, find the value of $f ( 1 ) - f ^ { \prime } ( 1 )$. [4 points]

In the coordinate plane, point A has coordinates $( 1,0 )$, and for $\theta$ with $0 < \theta < \frac { \pi } { 2 }$, point B has coordinates $( \cos \theta , \sin \theta )$. For point C in the first quadrant such that quadrilateral OACB is a parallelogram, let $f ( \theta )$ be the area of quadrilateral OACB and $g ( \theta )$ be the square of the length of segment OC. What is the maximum value of $f ( \theta ) + g ( \theta )$? (Here, O is the origin.) [4 points]
(1) $2 + \sqrt { 5 }$
(2) $2 + \sqrt { 6 }$
(3) $2 + \sqrt { 7 }$
(4) $2 + 2 \sqrt { 2 }$
(5) 5

For all cubic functions $f ( x )$ satisfying $f ( 0 ) = 0$ and the following conditions, let $M$ be the maximum value and $m$ be the minimum value of $\frac { f ^ { \prime } ( 0 ) } { f ( 0 ) }$. What is the value of $M m$? [4 points] (가) The function $| f ( x ) |$ is not differentiable only at $x = - 1$. (나) The equation $f ( x ) = 0$ has at least one real root in the closed interval $[ 3,5 ]$.
(1) $\frac { 1 } { 15 }$
(2) $\frac { 1 } { 10 }$
(3) $\frac { 2 } { 15 }$
(4) $\frac { 1 } { 6 }$
(5) $\frac { 1 } { 5 }$

For a real number $0 < t < 41$, the curve $y = x ^ { 3 } + 2 x ^ { 2 } - 15 x + 5$ and the line $y = t$ intersect at three points. Let the point with the largest $x$-coordinate be $( f ( t ) , t )$ and the point with the smallest $x$-coordinate be $( g ( t ) , t )$. Let $h ( t ) = t \times \{ f ( t ) - g ( t ) \}$. What is the value of $h ^ { \prime } ( 5 )$? [4 points]
(1) $\frac { 79 } { 12 }$
(2) $\frac { 85 } { 12 }$
(3) $\frac { 91 } { 12 }$
(4) $\frac { 97 } { 12 }$
(5) $\frac { 103 } { 12 }$

As shown in the figure, there is a sector OAB with radius 1 and central angle $\frac { \pi } { 2 }$. Let H be the foot of the perpendicular from point P on arc AB to line segment OA, and let Q be the intersection of line segment PH and line segment AB. When $\angle \mathrm { POH } = \theta$, let $S ( \theta )$ be the area of triangle AQH. What is the value of $\lim _ { \theta \rightarrow 0 + } \frac { S ( \theta ) } { \theta ^ { 4 } }$? (Here, $0 < \theta < \frac { \pi } { 2 }$) [4 points]
(1) $\frac { 1 } { 8 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 1 } { 2 }$
(5) $\frac { 5 } { 8 }$

A quartic function $f ( x )$ with leading coefficient 1 satisfies the following conditions. (가) $f ^ { \prime } ( 0 ) = 0 , f ^ { \prime } ( 2 ) = 16$ (나) For some positive number $k$, $f ^ { \prime } ( x ) < 0$ on the two open intervals $( - \infty , 0 ) , ( 0 , k )$. Choose all correct statements from the following. [4 points]
$\langle$Statements$\rangle$ ㄱ. The equation $f ^ { \prime } ( x ) = 0$ has exactly one real root in the open interval $( 0,2 )$. ㄴ. The function $f ( x )$ has a local maximum value. ㄷ. If $f ( 0 ) = 0$, then $f ( x ) \geq - \frac { 1 } { 3 }$ for all real numbers $x$.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For two real numbers $a$ and $k$, two functions $f ( x )$ and $g ( x )$ are defined as $$\begin{aligned} & f ( x ) = \begin{cases} 0 & ( x \leq a ) \\ ( x - 1 ) ^ { 2 } ( 2 x + 1 ) & ( x > a ) \end{cases} \\ & g ( x ) = \begin{cases} 0 & ( x \leq k ) \\ 12 ( x - k ) & ( x > k ) \end{cases} \end{aligned}$$ and satisfy the following conditions. (가) The function $f ( x )$ is differentiable on the entire set of real numbers. (나) For all real numbers $x$, $f ( x ) \geq g ( x )$. When the minimum value of $k$ is $\frac { q } { p }$, find the value of $a + p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]