Three numbers from their pair sums

Three numbers \(a\), \(b\) and \(c\) satisfy \(a+b=23\), \(b+c=30\) and \(c+a=35\).

1. By adding the three equations, find \(a+b+c\), and hence find \(a\), \(b\) and \(c\).[3]
2. In general \(a+b=p\), \(b+c=q\) and \(c+a=r\). Show that \(a=\dfrac{p-q+r}{2}\), and write down similar expressions for \(b\) and \(c\).[2]
3. Explain why the three pair sums of three whole numbers always add up to an even number. Hence explain why no three whole numbers have pair sums \(10\), \(15\) and \(20\).[2]