5. For ALL APPLICANTS.
A triangular triple is a triple of positive integers ( $a , b , c$ ) such that we can construct a triangle with sides of length $a , b$ and $c$. This means that the sum of any two of the numbers is strictly greater than the third; so if $a \leqslant b \leqslant c$, then it is equivalent to requiring $a + b > c$. For example, ( $3,3,3$ ) and ( $4,5,3$ ) are triangular triples, but ( $1,3,2$ ) and ( $3,3,6$ ) are not. For any positive integer $P$, we define $f ( P )$ to be the number of triangular triples such that the perimeter $a + b + c$ is equal to $P$. Triples with the same numbers, but in a different order, are counted as being distinct. So $f ( 12 ) = 10$, because there are 10 triangular triples with perimeter 12, shown below:
| $( 3,4,5 )$ | $( 3,5,4 )$ | $( 4,3,5 )$ | $( 4,5,3 )$ | $( 5,3,4 )$ | $( 5,4,3 )$ |
| $( 2,5,5 )$ | $( 5,2,5 )$ | $( 5,5,2 )$ | | | |
| $( 4,4,4 )$ | | | | | |
(i) Write down the values of $f ( 3 ) , f ( 4 ) , f ( 5 )$ and $f ( 6 )$.
(ii) If ( $a , b , c$ ) is a triangular triple, show that ( $a + 1 , b + 1 , c + 1$ ) is also a triangular triple.
(iii) If ( $x , y , z$ ) is a triangular triple, with $x + y + z$ equal to an even number greater than or equal to 6 , show that each of $x , y , z$ is at least 2 and that $( x - 1 , y - 1 , z - 1 )$ is also a triangular triple.
(iv) Using the previous two parts, prove that for any positive integer $k \geqslant 3$,
$$f ( 2 k - 3 ) = f ( 2 k )$$
(v) We will now consider the case where $P \geqslant 6$ is even, and we will write $P = 2 S$.
(a) Show that in this case ( $a , b , c$ ) is a triangular triple with $a + b + c = P$ if and only if each of $a , b , c$ is strictly smaller than $S$.
(b) For any $a$ such that $2 \leqslant a \leqslant S - 1$, show that the number of possible values of $b$ such that $( a , b , P - a - b )$ is a triangular triple is $a - 1$. Hence find an expression for $f ( P )$ for any even $P \geqslant 6$.
(vi) Find $f ( 21 )$.
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