A region is defined by the inequalities $x + y > 6$ and $x - y > - 4$
Consider the three statements:
$1 x > 1$
$2 y > 5$
$3 ( x + y ) ( x - y ) > - 24$
Which of the above statements is/are true for every point in the region?

In this question $x$ and $y$ are non-zero real numbers.
Which one of the following is sufficient to conclude that $x < y$ ?

$S$ is the complete set of values of $x$ which satisfy both the inequalities
$$x ^ { 2 } - 8 x + 12 < 0 \text { and } 2 x + 1 > 9$$
The set $S$ can also be represented as a single inequality.
Which one of the following single inequalities represents the set $S$ ?
A $\left( x ^ { 2 } - 8 x + 12 \right) ( 2 x + 1 ) < 0$
B $\left( x ^ { 2 } - 8 x + 12 \right) ( 2 x + 1 ) > 0$
C $x ^ { 2 } - 10 x + 24 < 0$
D $x ^ { 2 } - 10 x + 24 > 0$
E $\quad x ^ { 2 } - 6 x + 8 < 0$
F $\quad x ^ { 2 } - 6 x + 8 > 0$
G $x < 2$
H $x > 6$

The set of solutions to the inequality $x ^ { 2 } + b x + c < 0$ is the interval $p < x < q$ where $b , c , p$ and $q$ are real constants with $c < 0$.
In terms of $p , q$ and $c$, what is the set of solutions to the inequality $x ^ { 2 } + b c x + c ^ { 3 } < 0$ ?
A $\frac { p } { c } < x < \frac { q } { c }$
B $\frac { q } { c } < x < \frac { p } { c }$
C $p c < x < q c$
D $q c < x < p c$
E $p c ^ { 2 } < x < q c ^ { 2 }$
F $q c ^ { 2 } < x < p c ^ { 2 }$

$x$ and $y$ satisfy $| 2 - x | \leq 6$ and $| y + 2 | \leq 4$.
What is the greatest possible value of $| x y |$ ?
A 16
B 24
C 32
D 40
E 48
F There is no greatest possible value.

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 $p$ ?

$a , b$ and $c$ are real numbers with $a < b < c < 0$
Which of the following statements must be true?
I $a c < a b < a ^ { 2 }$
II $b ( c + a ) > 0$
III $\frac { c } { b } > \frac { a } { b }$

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 $k \geq 4$

Find the complete set of values of $x$ for which
$$(x+4)(x+3)(1-x) > 0 \text{ and } (x+2)(x-2) < 0$$
A $1 < x < 2$
B $-2 < x < 1$
C $-2 < x < 2$
D $x < -2$ or $x > 1$
E $x < -4$ or $x > 2$
F $x < -4$ or $-3 < x < 1$
G $-4 < x < -2$ or $x > 1$

The real numbers $a , b , c$ and $d$ satisfy both
$$0 < a + b < c + d$$
and
$$0 < a + c < b + d$$
Which of the following inequalities must be true? I $a < d$ II $b < c$ III $a + b + c + d > 0$
A none of them
B I only
C II only
D III only
E I and II only F I and III only G II and III only H I, II and III

A region $R$ in the ( $x , y$ )-plane is defined by the simultaneous inequalities
$$\begin{array} { r } y - x < 3 \\ y - x ^ { 2 } < 1 \end{array}$$
Which of the following statements is/are true for every point in $R$ ?
I $- 1 < x < 2$ II $\quad ( y - x ) \left( y - x ^ { 2 } \right) < 3$ III $y < 5$
A none of them
B I only
C II only
D III only
E I and II only F I and III only G II and III only H I, II and III

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

Consider the statement () about a real number $x$ : () There exists a real number $y$ such that $x - x y + y$ is negative.
For how many real values of $x$ is (*) true?
A no values of $x$
B exactly one value of $x$
C exactly two values of $x$
D all except exactly two values of $x$
E all except exactly one value of $x$ F all values of $x$

Consider the two inequalities:
$$\begin{aligned} & | x + 5 | < | x + 11 | \\ & | x + 11 | < | x + 1 | \end{aligned}$$
Which one of the following is correct?
A There is no real number for which both inequalities are true.
B There is exactly one real number for which both inequalities are true.
C The real numbers for which both inequalities are true form an interval of length 1 .
D The real numbers for which both inequalities are true form an interval of length 2 .
E The real numbers for which both inequalities are true form an interval of length 3 .
F The real numbers for which both inequalities are true form an interval of length 4 .
G The real numbers for which both inequalities are true form an interval of length 5 .

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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3
Consider a coordinate plane with origin O. For two points $\mathrm{S}(x_1,\ y_1)$, $\mathrm{T}(x_2,\ y_2)$ on the coordinate plane, we define that point S is sufficiently far from point T if $$|x_1 - x_2| \geqq 1 \quad \text{or} \quad |y_1 - y_2| \geqq 1$$ holds.
Let $D$ be the square region defined by the inequalities $0 \leqq x \leqq 3$, $0 \leqq y \leqq 3$, and consider two of its vertices $\mathrm{A}(3,\ 0)$, $\mathrm{B}(3,\ 3)$. Furthermore, let P be a point satisfying both of the following conditions (i) and (ii).

• [(i)] Point P is in region $D$, and lies on the parabola $y = x^2$.
• [(ii)] Point P is sufficiently far from each of the 3 points O, A, B.

Let $a$ be the $x$-coordinate of point P.

• [(1)] Find the range of possible values of $a$.
• [(2)] Find the area $f(a)$ of the region where a point Q satisfying both of the following conditions (iii) and (iv) can exist.
  • [(iii)] Point Q is in region $D$.
  • [(iv)] Point Q is sufficiently far from each of the 4 points O, A, B, P.
• [(3)] Suppose $a$ varies over the range found in (1). Find the value of $a$ that minimizes $f(a)$ from (2).

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$$(2x-1)\left(4x^{2}-1\right)<0$$
Which of the following open intervals is the solution set of the inequality in real numbers?
A) $\left(-\infty, \frac{-1}{2}\right)$
B) $\left(\frac{-1}{2}, 0\right)$
C) $\left(\frac{-1}{2}, \frac{1}{2}\right)$
D) $\left(\frac{1}{4}, \frac{1}{2}\right)$
E) $\left(\frac{1}{2}, \infty\right)$

For given positive real numbers $a$, $c$ and negative real number $b$, $$a^{2}b > abc + c^{2}$$ Given that the inequality is satisfied, which of the following is necessarily true?
A) $a = |b|$
B) $a = c$
C) $c > |b|$
D) $a < c$
E) $c < a$

$$\frac { - 5 } { 4 } < x < \frac { 7 } { 3 }$$
What is the sum of the integers $x$ that satisfy this inequality?
A) - 2
B) - 1
C) 0
D) 1
E) 2

For real numbers $x , y$ and $z$
$$\begin{aligned} & y > 0 \\ & x - y > z \end{aligned}$$
Given this, which of the following is always true?
A) $x > z$
B) $x > y$
C) $z > y$
D) $x > 0$
E) $z > 0$

The parabola $f(x)$ and the line $d$ are shown in the Cartesian coordinate plane above.
Accordingly, which of the following systems of inequalities has the shaded region as its solution set?
A) $\left.\begin{array}{l} y - x^{2} + 2x \leq 0 \\ y - x + 2 \geq 0 \end{array}\right\}$
B) $\left.\begin{array}{l} y - x^{2} + 2x \geq 0 \\ 2y - x + 2 \geq 0 \end{array}\right\}$
C) $\left.\begin{array}{l} y - x^{2} + 4x \leq 0 \\ 2y - x + 2 \leq 0 \end{array}\right\}$
D) $\left.\begin{array}{l} y + x^{2} - 4x \leq 0 \\ 2y - x + 4 \leq 0 \end{array}\right\}$
E) $\left.\begin{array}{l} y + x^{2} - 4x \leq 0 \\ 2y - x + 2 \geq 0 \end{array}\right\}$

$$0 \leq \log_{2}(x-5) \leq 2$$
How many integers $x$ satisfy these inequalities?
A) 2
B) 3
C) 4
D) 5
E) 6

$$\sqrt { 2 } < x < \sqrt { 3 }$$
Given this, which of the following can x be?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 7 } { 4 }$
E) $\frac { 6 } { 5 }$

For integers x and y, $x + 2y = 11$. Given that,
I. x is an odd number. II. x is greater than y. III. Both x and y are positive.
Which of the following statements are always true?
A) Only I B) Only III C) I and II D) I and III E) II and III