On a particularly artistic fruit stall, a pile of oranges is arranged to form a truncated square pyramid. Each layer is a square, with the lengths of the side of successive layers reducing by one orange (as shown below). The bottom layer measures 2n Xx 2n oranges, and there are n layers. Prove that the number of oranges used is n(2n + 1)(7n+ 1). (i) (ii) How many complete layers can the person setting up the stall use for this arrangement, given their stock of 1000 oranges? How many oranges are left over?

I actually need help with number 37.

Solve the tquation 3(-x+5)=14ation %3D

et △ABC be a triangle in S^2 (the two-dimensional sphere). Let the dual point C'∈S^2 of C be defined by the following three conditions:(i) d(C' , A)=π/2(ii) d(C' , B)=π/2(iii) d(C' , C)≤π/2A' and B' are defined analogously. Thus we get a dual triangle △A' B' C'. More precisely, △ABC is a non-degenerate triangle, andit follows (you may assume) that △A' B' C' is too.(question) Are there triangles △ABC in S^2 identical to their own dual △A'B'C'?   I would be very thankful if you could provide some explanation with the steps, thank you in advance.

Give 4c-d=f, d=