*Challenge question #41 (04/01/2017)*

**Junior Question**

Two circles with centres B and C touch each other externally, and touch a given fixed circle with centre A internally.

Show that as the two smaller circles vary in size, the perimeter of triangle ABC remains constant.

**Senior Question**

Show that by adding one to the product of four consecutive integers, a perfect square is obtained.

For example, $2\times$ $3\times$ $4\times$ $5+$ $1=$ $121=$ $11^{2}$