GULF 2014 Problem 2

Ahmad and Salem play the following game. Ahmad writes two integers (not necessarily different) on a board. Salem writes their sum and product. Ahmad does the same thing: he writes the sum and product of the two numbers which Salem has just written. They continue in this manner, not stopping unless the two players write the same two numbers one after the other (for then they are stuck!). The order of the two numbers which each player writes is not important. Thus if Ahmad starts by writing $3$ and $-2$ , the first five moves (or steps) are as shown: (a) Step 1 (Ahmad) $3$ and $-2$ (b) Step 2 (Salem) $1$ and $-6$ (c) Step 3 (Ahmad) $-5$ and $-6$ (d) Step 4 (Salem) $-11$ and $30$ (e) Step 5 (Ahmad) $19$ and $-330$ (i) Describe all pairs of numbers that Ahmad could write, and ensure that Salem must write the same numbers, and so the game stops at step 2. (ii) What pair of integers should Ahmad write so that the game finishes at step 4? (iii) Describe all pairs of integers which Ahmad could write at step 1, so that the game will finish after finitely many steps. (iv) Ahmad and Salem decide to change the game. The first player writes three numbers on the board, $u, v$ and $w$ . The second player then writes the three numbers $u + v + w,uv + vw + wu$ and $uvw$ , and they proceed as before, taking turns, and using this new rule describing how to work out the next three numbers. If Ahmad goes first, determine all collections of three numbers which he can write down, ensuring that Salem has to write the same three numbers at the next step.