Given the $7$-element set $A=\left\{a,b,c,d,e,f,g\right\}$, find a collection $T$ of $3$-element subsets of $A$ such that each pair of elements from $A$ occurs exactly in one of the subsets of $T$.

$\begin{array}{r} \mathrm{FORTY} \\ \mathrm{TEN} \\ \mathrm{TEN} \\ \mathrm{SIXTY} \\ \end{array}$

Five men $A, B, C, D, E$ are wearing caps of black or white colour without each knowing the colour of his cap. It is known that a man wearing a black cap always speaks the truth while a man wearing a white cap always lies. If they make the following statements, find the colour of the cap worn by each of them:

$A:$I see three black and one white cap.

$B:$I see four white caps.

$C:$I see one black and three white caps.

$D:$I see four black caps.

[$f^M$ denotes the composite function$f\circ f\circ \cdots \circ f$ repeated $M$ times.]

Show that there exists a convex hexagon in the plane such that:

$\left(a\right)$ all its interior angles are equal

$\left(b\right)$ its sides are $1,2,3,4,5,6$ in some order.