The negation of the Boolean expression $x \leftrightarrow \sim y$ is equivalent to:
(1) $( \sim x \wedge y ) \vee ( \sim x \wedge \sim y )$
(2) $( x \wedge y ) \vee ( \sim x \wedge \sim y )$
(3) $( x \wedge \sim y ) \vee ( \sim x \wedge y )$
(4) $( x \wedge y ) \wedge ( \sim x \vee \sim y )$

The negation of the Boolean expression $p \vee ( \sim p \wedge q )$ is equivalent to :
(1) $p \wedge \sim q$
(2) $\sim p \wedge \sim q$
(3) $\sim p \vee \sim \mathrm { q }$
(4) $\sim p \vee q$

Consider the statement: ``For an integer n, if $\mathrm{n}^{3}-1$ is even, then n is odd''. The contrapositive statement of this statement is:
(1) For an integer n, if n is even, then $\mathrm{n}^{3}-1$ is odd.
(2) For an integer n, if $\mathrm{n}^{3}-1$ is not even, then n is not odd.
(3) For an integer n, if n is even, then $\mathrm{n}^{3}-1$ is even.
(4) For an integer n, if n is odd, then $\mathrm{n}^{3}-1$ is even.

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. [Figure]
(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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Prove that $$( f ( x ) \cdot g ( x ) ) \cdot h ( x ) = f ( x ) \cdot ( g ( x ) \cdot h ( x ) )$$ for all linear polynomials $f ( x )$ and $g ( x )$ and $h ( x )$.

Prove that for any nice set of six numbers, the total of those six numbers must be a multiple of 3.

For $k = 1 , \ldots , n$, let $F ( n , k )$ be the number of good lists of length $n$ which result in team $n$ booking room $k$. Explain why $F ( n , k )$ is a multiple of $k$.

Describe the relationship between $G ( n )$ and $F ( n , 1 ) , F ( n , 2 ) , \ldots , F ( n , n )$.

Explain why $F ( 4,1 ) = G ( 3 )$ and $F ( 4,4 ) = 4 \times G ( 3 )$.