For $\mathbf { H } , \mathbf { I }$ in question (1), and for $\mathbf { J }$, $\mathbf { K }$ in question (2), choose the appropriate answer from among (0) $\sim$ (3) at the bottom of this page.
For $\mathbf { L } \sim \mathbf { R }$ in question (3), enter the appropriate number.
Consider the following three possible conditions on two real numbers $x$ and $y$:
$p : x$ and $y$ satisfy the equation $( x + y ) ^ { 2 } = a \left( x ^ { 2 } + y ^ { 2 } \right) + b x y$, where $a$ and $b$ are real constants.
$$\begin{aligned} & q : x = 0 \text { and } y = 0 . \\ & r : x = 0 \text { or } y = 0 . \end{aligned}$$
(1) Suppose that in condition $p$, $a = b = 1$. Then $p$ is $\mathbf { H }$ for $q$, and $p$ is $\mathbf { I }$ for $r$.
(2) Suppose that in condition $p$, $a = b = 2$. Then $p$ is $\mathbf { J }$ for $q$, and $p$ is $\mathbf { K }$ for $r$.
(3) If in condition $p$ we set $a = 2$, we can transform the equation in $p$ into
$$\left( x + \frac { b - \mathbf { L } } { \mathbf { L } } y \right) ^ { 2 } + \left( \mathbf { N } - \frac { ( b - \mathbf { Q} ) ^ { 2 } } { \mathbf { P } } \right) y ^ { 2 } = 0 .$$
Hence $p$ is a necessary and sufficient condition for $q$ if and only if $b$ satisfies $\mathbf { Q } < b < \mathbf { R }$.
(0) a necessary and sufficient condition
(1) a necessary condition but not a sufficient condition
(2) a sufficient condition but not a necessary condition
(3) neither a necessary condition nor a sufficient condition

Consider the two functions
$$\begin{aligned} & f ( x ) = x ^ { 2 } + 2 a x + 4 a - 3 \\ & g ( x ) = 2 x + 1 \end{aligned}$$
We are to find the condition on $a$ for which $f ( x ) \geqq g ( x )$ for all $x$ and also find the range of values of the minimum of $f ( x )$ under this condition.
We must find the condition under which
$$x ^ { 2 } + \mathbf { A } ( a - \mathbf { A } ) x + \mathbf { A C } a - \mathbf { A D } \geq 0$$
for all $x$.
For each of $\mathbf { E } \sim \mathbf{ H }$ in the following questions, choose the correct answer from among (0) $\sim$ (7) below each question.
(1) The required condition is that $a$ satisfy the quadratic inequality $\mathbf{E}$. Hence $a$ is in the range $\mathbf { F }$. (0) $a ^ { 2 } - 5 a + 4 \geqq 0$
(1) $a ^ { 2 } - 6 a + 5 \geqq 0$
(2) $a ^ { 2 } - 5 a + 4 \leqq 0$
(3) $a ^ { 2 } - 6 a + 5 \leqq 0$
(4) $a \leqq 1$ or $5 \leqq a$
(5) $1 \leqq a \leqq 5$ (6) $1 \leqq a \leqq 4$ (7) $a \leqq 1$ or $4 \leqq a$
(2) Let $m$ be the minimum value of $f ( x )$. Then, since $m = \mathbf { G }$, the range of values which $m$ can take under the condition in (1) is $\mathbf { H }$. (0) $a ^ { 2 } + 4 a - 3$
(1) $4 a ^ { 2 } + 4 a - 3$
(2) $- a ^ { 2 } + 4 a - 3$
(3) $2 a ^ { 2 } - 4 a + 3$
(4) $- 5 \leqq m \leqq 1$
(5) $- 8 \leqq m \leqq 1$ (6) $- 8 \leqq m \leqq - 1$ (7) $- 5 \leqq m \leqq - 1$

Q1 Let $a$ and $b$ be rational numbers and let $p$ be a real number. Consider the quadratic equation
$$x ^ { 2 } + a x + b = 0 \tag{1}$$
which has a solution $x = \frac { \sqrt { 5 } + 3 } { \sqrt { 5 } + 2 }$, and consider the inequality
$$x + 1 < 2 x + p + 3 . \tag{2}$$
(1) First, we are to find the values of $a$ and $b$.
When we rationalize the denominator of $x = \frac { \sqrt { 5 } + 3 } { \sqrt { 5 } + 2 }$, we have
$$x = \sqrt { \mathbf { A } } - \mathbf { B }$$
Since this is a solution of equation (1), by substituting this in (1) we have
$$- a + b + \mathbf { C } + ( a - \mathbf { D } ) \sqrt { \mathbf { E } } = 0 .$$
Hence we see that
$$a = \mathbf { F } \text { and } b = \mathbf { G H } .$$
(2) Next, we are to find the smallest integer $p$ such that both solutions of equation (1) satisfy inequality (2).
When we solve inequality (2), we have
$$x > - p - 1 .$$
Since both solutions of equation (1) satisfy this, we see that
$$p > \sqrt { \mathbf { J } } - \mathbf { K } .$$
Hence the smallest integer $p$ is $\mathbf { L }$.

Let us consider the three quadratic functions
$$f ( x ) = - x ^ { 2 } - 2 x + 1 , \quad g ( x ) = - x ^ { 2 } + 4 x , \quad h ( x ) = 2 x ^ { 2 } + a x + b$$
(1) When we denote the discriminant of the quadratic equation $h ( x ) - f ( x ) = 0$ by $D _ { 1 }$ and the discriminant of the quadratic equation $h ( x ) - g ( x ) = 0$ by $D _ { 2 }$, we have
$$D _ { 1 } = \mathbf { N } , \quad D _ { 2 } = \mathbf { O }$$
(for N and O, choose the correct answers from among choices (0) $\sim$ (5) below). (0) $a ^ { 2 } + 4 a - 3 b + 7$
(1) $a ^ { 2 } - 8 a - 12 b + 16$
(2) $a ^ { 2 } + 4 a - 12 b + 16$
(3) $a ^ { 2 } + 8 a + 12 b + 16$
(4) $a ^ { 2 } - 4 a + 12 b + 16$
(5) $a ^ { 2 } - 8 a - 3 b + 7$
(2) The values of $a$ and $b$ such that both of the two equations $f ( x ) = h ( x )$ and $g ( x ) = h ( x )$ have only one real solution are
$$a = \mathbf { P } , \quad b = \frac { \mathbf { Q } } { \mathbf{4} } .$$
In this case, the solution of $f ( x ) = h ( x )$ is $x = - \frac { \mathbf { S } } { \mathbf{T} }$ and the solution of $g ( x ) = h ( x )$ is $x = \frac { \mathbf { U } } { \mathbf{4} }$.
(3) Let $b = 3$. Then the range of the values of $a$ such that both $f ( x ) < h ( x )$ and $g ( x ) < h ( x )$ hold for any $x$ is $\square$ W (for $\square$ W, choose the correct answer from among choices (0) $\sim$ (5) below). (0) $- 2 - 2 \sqrt { 6 } < a < 10$
(1) $a < - 2 - 2 \sqrt { 6 } , 10 < a$
(2) $a < - 1 - \sqrt { 6 } , 10 < a$
(3) $- 2 < a < - 1 + \sqrt { 6 }$
(4) $- 2 < a < - 2 + 2 \sqrt { 6 }$
(5) $- 1 - \sqrt { 6 } < a < 10$

Q2 Let $a$ be a real number. For the two quadratic functions in $x$
$$\begin{aligned} & f ( x ) = x ^ { 2 } + 2 a x + a ^ { 2 } - a , \\ & g ( x ) = 4 - x ^ { 2 } , \end{aligned}$$
answer the following questions.
(1) The range of the values of $a$ such that the equation $f ( x ) = g ( x )$ has two different solutions is
$$- \mathbf { K } < a < \mathbf { L } .$$
(2) In the case of (1), the parabolas $y = f ( x )$ and $y = g ( x )$ intersect at two points. We are to find the range of the values of $a$ such that both of the $y$ coordinates of these points of intersection are positive.
First, let $h ( x ) = f ( x ) - g ( x )$. Since the solutions of the equation $f ( x ) = g ( x )$ are the $x$ coordinates of the points of intersection of parabolas $y = f ( x )$ and $y = g ( x )$, the solutions of $h ( x ) = 0$ have to be between $- \mathbf { M }$ and $\mathbf { N }$. Accordingly, we have
$$\begin{aligned} & h \left( - \mathbf { M } \right) = a ^ { 2 } - \mathbf { O } a + \mathbf { P } > 0 , \quad \cdots \cdots \cdots (1)\\ & h ( \mathbf { N } ) = a ^ { 2 } + \mathbf { Q } a + \mathbf { R } > 0 . \quad \cdots \cdots \cdots (2) \end{aligned}$$
Also, from the position of the axis of the parabola $y = h ( x )$ we have that
$$- \mathbf { S } < a < \mathbf { T } . \quad \cdots \cdots \cdots (3)$$
Therefore, from (1), (2), (3) and (4) we obtain
$$- \mathbf { U } < a < \mathbf{V}.$$

Let $f ( x )$ be a quadratic polynomial function with real coefficients such that $f ( x ) = 0$ has no real roots. Select the correct options.
(1) $f ( 0 ) > 0$
(2) $f ( 1 ) f ( 2 ) > 0$
(3) If $f ( x ) - 1 = 0$ has real roots, then $f ( x ) - 2 = 0$ has real roots
(4) If $f ( x ) - 1 = 0$ has a double root, then $f ( x ) - \frac { 1 } { 2 } = 0$ has no real roots
(5) If $f ( x ) - 1 = 0$ has two distinct real roots, then $f ( x ) - \frac { 1 } { 2 } = 0$ has real roots

Let $f ( x ) , g ( x )$ both be real-coefficient polynomials, where $g ( x )$ is a quadratic with positive leading coefficient. It is known that the remainder when $( g ( x ) ) ^ { 2 }$ is divided by $f ( x )$ is $g ( x )$ , and the graph of $y = f ( x )$ has no intersection with the $x$-axis. Select the option that cannot be the $y$-coordinate of the vertex of the graph of $y = g ( x )$.
(1) $\frac { \sqrt { 2 } } { 2 }$
(2) 1
(3) $\sqrt { 2 }$
(4) 2
(5) $\pi$

Let $b$ and $c$ be real numbers. The quadratic equation $x ^ { 2 } + b x + c = 0$ has real roots, but the quadratic equation $x ^ { 2 } + ( b + 2 ) x + c = 0$ has no real roots. Select the correct options.
(1) $c < 0$
(2) $b < 0$
(3) $x ^ { 2 } + ( b + 1 ) x + c = 0$ has real roots
(4) $x ^ { 2 } + ( b + 2 ) x - c = 0$ has real roots
(5) $x ^ { 2 } + ( b - 2 ) x + c = 0$ has real roots

$$y = x ^ { 2 } - 2 ( a + 1 ) x + a ^ { 2 } - 1$$
If the parabola is tangent to the line $y = 1$, what is a?
A) $\frac { -3 } { 2 }$
B) $\frac { -3 } { 4 }$
C) 0
D) 1
E) 2

Let k be a positive real number. If the roots of the equation
$$2 x ^ { 2 } + k x - 1 = 0$$
have a difference of 2, what is k?
A) 1
B) 2
C) $\sqrt { 2 }$
D) $2 \sqrt { 2 }$
E) $\sqrt { 3 }$

Let a be a real number. One root of the equation
$$a x ^ { 2 } - 18 x + 18 = 0$$
is 2 times the other. Accordingly, what is a?
A) 2
B) 3
C) 4
D) 5
E) 6

Let $m$ and $n$ be two non-zero and distinct real numbers,
$$x ^ { 2 } + ( m + 1 ) x + n - m = 0$$
one of the roots of the equation is the number $m - n$.
Accordingly, what is the ratio $\frac { \mathbf { n } } { \mathbf { m } }$?
A) 2 B) 3 C) 4 D) 5 E) 6

For the equation $x^2 - 2x + c = 0$, the discriminant is also a root of this equation. What is the product of the possible values of the real number $c$?
A) 1
B) 2
C) 4
D) $\frac{1}{2}$
E) $\frac{1}{4}$

Let $a$ and $b$ be real numbers. In the rectangular coordinate plane, the parabola $y = x^{2} + ax + b$ is tangent to the $x$-axis and to the line $y = x$.
What is the product $a \cdot b$?
A) $\dfrac{1}{2}$ B) $\dfrac{1}{4}$ C) $\dfrac{1}{8}$ D) $\dfrac{1}{16}$ E) $\dfrac{1}{32}$

Let $a$ and $b$ be positive real numbers. The equations
$$\begin{aligned} & x^{2} + ax + b = 0 \\ & ax^{2} + (b + 3)x + a = 0 \end{aligned}$$
are given. Given that the solution set of each of these equations has exactly 1 element, what is the product of the different values that the sum $a + b$ can take?
A) 24 B) 32 C) 45 D) 72 E) 120