[Optional 4-5: Inequalities] (10 points)
Given functions $f(x) = -x^2 + ax + 4$ and $g(x) = |x + 1| + |x - 1|$.
(2) If the solution set of the inequality $f(x) \geq g(x)$ contains $[-1, 1]$, find the range of values for $a$.

11. If the range of the function $f ( x ) = a \cdot \left( \frac { 1 } { 3 } \right) ^ { x } \left( \frac { 1 } { 2 } \leq x \leq 1 \right)$ is a subset of the range of the function $g ( x ) = \frac { x ^ { 2 } - 1 } { x ^ { 2 } + x + 1 } ( x \in \mathbb{R} )$, then the range of positive number $a$ is
A. $(0,2]$
B. $(0,1]$
C. $( 0,2 \sqrt { 3 } ]$
D. $( 0 , \sqrt { 3 } ]$

If the equation $\cos ^ { 4 } \theta + \sin ^ { 4 } \theta + \lambda = 0$ has real solutions for $\theta$ then $\lambda$ lies in interval
(1) $\left( - \frac { 5 } { 4 } , - 1 \right)$
(2) $\left[ - 1 , - \frac { 1 } { 2 } \right]$
(3) $\left( - \frac { 1 } { 2 } , - \frac { 1 } { 4 } \right]$
(4) $\left[ - \frac { 3 } { 2 } , - \frac { 5 } { 4 } \right]$

Let $S$ be the set of positive integral values of $a$ for which $\frac { a x ^ { 2 } + 2 a + 1 x + 9 a + 4 } { x ^ { 2 } - 8 x + 32 } < 0 , \quad \forall x \in \mathbb { R }$. Then, the number of elements in $S$ is:
(1) 1
(2) 0
(3) $\infty$
(4) 3

Let $a$ be a constant. Consider the quadratic inequality
$$x ^ { 2 } - 2 ( a + 2 ) x + 25 > 0 . \tag{1}$$
The left-hand side of inequality (1) can be transformed into
$$( x - a - \mathbf { A } ) ^ { 2 } - a ^ { 2 } - \mathbf { B } a + \mathbf { C D } .$$
Hence, we have the following results.
(1) The condition under which inequality (1) holds for all real numbers $x$ is
$$\mathbf { E F } < a < \mathbf { G } .$$
(2) The condition under which inequality (1) holds for all real numbers $x$ satisfying $x \geqq - 1$ is
$$\mathbf { H I J } < a < \mathbf { K } .$$

It is given that the equation $\sqrt { x + p } + \sqrt { x } = p$ has at least one real solution for $x$, where $p$ is a real constant.
What is the complete set of possible values for $p$ ?

Consider the following inequality:
$( * ) : \quad a | x | + 1 \leq | x - 2 |$
where $a$ is a real constant.
Which one of the following describes the complete set of values of $a$ such that (*) is true for all real $x$ ?

Find the complete set of values of $k$ for which the line $y = x - 2$ crosses or touches the curve $y = x ^ { 2 } + k x + 2$
A $- 1 \leq k \leq 3$
B $- 3 \leq k \leq 5$
C $- 4 \leq k \leq 4$
D $k \leq - 1$ or $k \geq 3$
E $k \leq - 3$ or $k \geq 5$ F $k \leq - 4$ or $k \geq 4$

Consider the following statement: () For all real numbers $x$, if $x < k$ then $x ^ { 2 } < k$ What is the complete set of values of $k$ for which () is true?
A no real numbers
B $k > 0$
C $k < 1$
D $k \leq 1$
E $\quad 0 < k < 1$ F $0 < k \leq 1$ G all real numbers

1

Let $a, b, c, p$ be real numbers. Suppose that the set of real numbers $x$ satisfying all of the inequalities $$ax^2 + bx + c > 0, \quad bx^2 + cx + a > 0, \quad cx^2 + ax + b > 0$$ coincides with the set of real numbers $x$ satisfying $x > p$.

• [(1)] Show that $a, b, c$ are all non-negative.
• [(2)] Show that at least one of $a, b, c$ is $0$.
• [(3)] Show that $p = 0$.

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When points P, Q, R in a plane are not collinear, we denote the area of the triangle with these three vertices by $\triangle\mathrm{PQR}$. When P, Q, R are collinear, we set $\triangle\mathrm{PQR} = 0$.
Let A, B, C be three points in a plane with $\triangle\mathrm{ABC} = 1$. A point X in this plane satisfies $$2 \leqq \triangle\mathrm{ABX} + \triangle\mathrm{BCX} + \triangle\mathrm{CAX} \leqq 3.$$
Find the area of the region in which X can move.
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