Let $a$ be a constant. Consider the quadratic function in $x$
$$y = 2 x ^ { 2 } + a x + 3 .$$
Suppose that the vertex of the graph of (1) is in the first (upper right-hand) quadrant.
(1) The range of values which $a$ can take is
$$\mathbf { A B } \sqrt { \mathbf { C } } < a < \mathbf { D } ,$$
and the least integer $a$ satisfying this inequality is $\mathbf{EF}$.
(2) Let $a = \mathrm { EF }$ in (1). Let
$$y = 2 x ^ { 2 } + p x + q$$
be the equation of the graph which is obtained by translating the graph of (1) by $- \frac { 1 } { n }$ in the $x$-direction and by $\frac { 6 } { n ^ { 2 } }$ in the $y$-direction. Then
$$p = \frac { \mathbf { G } } { n } - \mathbf { H } , \quad q = \frac { \mathbf { I } } { n ^ { 2 } } - \frac { \mathbf { J } } { n } + \mathbf { K } .$$
(3) The total number of natural numbers $n$ for which $p$ in (2) is an integer is $\mathbf { L }$. Among these $n$, consider the ones such that the value of $q$ is also an integer. Then $\mathbf { M }$ is the value of the $n$ for which $q$ takes the minimum value $\mathbf { N }$.