, 2023

(a) $x$, $y$ and $z$ are real numbers, with $x \leq y \leq z$ and
$x + y = 4$
$y + z = 7$
Let $T = x + y + z$.
(i) Show that $x \leq 2$ and that $T = 11 - y$.
(ii) Find the minimum possible value of $T$, giving one example of values of $x$, $y$ and $z$ where this occurs.

(b) $a$, $b$, $c$, $d$ and $e$ are real numbers, with $a \leq b \leq c \leq d \leq e$ and
$a + b + c = 4$
$b + c + d = 5$
$c + d + e = 9$
Let $S = a + b + c + d + e$.
Find the minimum possible value of $S$, giving one example of values of $a$, $b$, $c$, $d$ and $e$ where this occurs.
Your solution must fully justify why no smaller value of $S$ is possible.