Sunday Afternoon Maths VIII

Here's this week's collection. Answers & extensions here. Why not discuss the problems on Twitter using #SundayAfternoonMaths or on Reddit.

Two

Find a such that a+(a+A)^-1 = 2, where A = (a+A)^-1.

ie. a + /(a+1/(a+1/(a+1/(a+1/(a+1/(a+...)))))) = 2.

Find b such that b+(b+B)^0.5 = 2, where B = (b+B)^0.5.

ie. b + √(b+√(b+√(b+√(b+√(b+√(b+...)))))) = 2.

Find c such that c+(c+C)² = 2, where C = (c+C)².

In terms of k, find d such that d+(d+D)^k = 2, where D = (d+D)^k.

Reverse Bases Again

Find three digits a, b and c such that abc in base 10 is equal to cba in base 9?

Adding bases

Let a_b denote a in base b.

Find bases A, B and C less than 10 such that 12_A+34_B=56_C

Complex Squares

For which complex numbers, z, are Re(z²) and Im(z²) both positive?

Rebounds

Source: Starting Points by C.S. Banwell, K.D. Saunders & D.G. Tahta

In a 4x3 rectangle, a ball is fired from the top left corner at 45°.

It bounces around a rectangle until it hits a corner. Which corner does it end in?

Which corner will it end in for rectangles of other sizes?

I made a Flash applet to help with investigating this problem.