\section*{2. For ALL APPLICANTS.}
Suppose that $a , b , c$ are integers such that

$$a \sqrt { 2 } + b = c \sqrt { 3 }$$

(i) By squaring both sides of the equation, show that $a = b = c = 0$.\\[0pt]
[You may assume that $\sqrt { 2 } , \sqrt { 3 }$ and $\sqrt { 2 / 3 }$ are all irrational numbers. An irrational number is one which cannot be written in the form $p / q$ where $p$ and $q$ are integers.]\\
(ii) Suppose now that $m , n , M , N$ are integers such that the distance from the point $( m , n )$ to $( \sqrt { 2 } , \sqrt { 3 } )$ equals the distance from $( M , N )$ to $( \sqrt { 2 } , \sqrt { 3 } )$.

Show that $m = M$ and $n = N$.\\
Given real numbers $a , b$ and a positive number $r$, let $N ( a , b , r )$ be the number of integer pairs $x , y$ such that the distance between the points $( x , y )$ and $( a , b )$ is less than or equal to $r$. For example, we see that $N ( 1.2,0,1.5 ) = 7$ in the diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{f229588a-5602-44a5-acb0-6efec65f41af-08_771_759_1231_644}\\
(iii) Explain why $N ( 0.5,0.5 , r )$ is a multiple of 4 for any value of $r$.\\
(iv) Let $k$ be any positive integer. Explain why there is a positive number $r$ such that

$$N ( \sqrt { 2 } , \sqrt { 3 } , r ) = k$$

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