A 3-star is a collection of three distinct points, with each pair connected by a straight line segment. How many 3-stars are there in the first diagram in total? Explain your answer.
For $n \geq 4$, an $n$-star is a collection of $n$ distinct points, with each pair connected by a straight line segment. For $n = 4,5,6,7,8$, how many $n$-stars are there in the first diagram in total? Explain your answers.
A new diagram is formed of three separate collections of points named $A , B$, and $C$ respectively. $A$ contains 3 points, $B$ contains 4 points, and $C$ contains 5 points. Each point in $A$ is connected with a straight line segment to each point in $B$. Each point in $B$ is connected with a straight line segment to each point in $C$. Each point in $C$ is connected with a straight line segment to each point in $A$. No other pairs of points are connected with a straight line segment. How many 3-stars are there in this new diagram in total? Justify your answer carefully.
For $n \geq 2$, an $n$-loop is a sequence of $n$ distinct points with each point connected to the next in sequence with a straight line segment, and with the last connected to the first with a straight line segment. For which values of $n \geq 4$ does the new diagram described in part (iv) contain at least one $n$-loop? Justify your answer.
Find the coefficients of $( a + b x ) \cdot ( c + d x )$ in terms of $a , b , c , d$, and explain why $f ( x ) \cdot g ( x ) = g ( x ) \cdot f ( x )$ for all linear polynomials $f ( x )$ and $g ( x )$.
A strictly increasing linear function $f ( x )$ has the property that if $y > x$ then $f ( y ) > f ( x )$. A student claims that if $f ( x )$ and $g ( x )$ are both strictly increasing linear functions, then so is $f ( x ) \cdot g ( x )$. Is the student correct? If so, prove the student's claim. Otherwise, find a counterexample.
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 $h ( x )$.
Given that $f ( x )$ and $g ( x )$ are linear polynomials and $f ( x ) \cdot g ( x ) = 0$, describe all possibilities for the pair $f ( x )$ and $g ( x )$.
Given that $f ( x )$ and $g ( x )$ and $h ( x )$ are linear polynomials and $$f ( x ) \cdot g ( x ) \cdot h ( x ) = 0$$ prove that at least one of the following statements must be true; (I) $f ( x ) \cdot g ( x ) = 0$, (II) $g ( x ) \cdot h ( x ) = 0$, (III) $f ( x ) \cdot h ( x ) = 0$. For each of the three statements, give examples of polynomials for which that statement is true and the other two statements are false.
Now we consider infinite sets of whole numbers (for example, the set of all the positive whole numbers). Give examples to demonstrate that an infinitely large set of whole numbers might have zero, exactly one, or more than one target(s). Justify your answers, making it clear which is which.
Suppose that we want to find a nice set of two numbers that are the same colour. By considering the possibilities for the colours of the numbers 1, 2, and 3, prove such a set always exists.
Suppose that we want to find a nice set of four numbers that are all the same colour. By considering the possibilities for the colours of the numbers from 1 to 27 inclusive, prove such a set always exists. [Hint: consider $\{ 1,2,3 \} , \{ 4,5,6 \} , \ldots , \{ 25,26,27 \}$.]
For $n = 3$, there are 16 good lists, so $G ( 3 ) = 16$. List all of them, starting with good lists with $t ( 1 ) = 1$, then good lists with $t ( 1 ) = 2$, and then good lists with $t ( 1 ) = 3$.
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 $k$.