MathDB

3

Part of 2015 Romanian Master of Mathematics

Problems(1)

Obtaining n!, after operations

Source: 7th RMM 2015, Problem 3

2/28/2015
A finite list of rational numbers is written on a blackboard. In an operation, we choose any two numbers aa, bb, erase them, and write down one of the numbers a+b,  a−b,  b−a,  a×b,  a/b (if b≠0),  b/a (if a≠0). a + b, \; a - b, \; b - a, \; a \times b, \; a/b \text{ (if $b \neq 0$)}, \; b/a \text{ (if $a \neq 0$)}. Prove that, for every integer n>100n > 100, there are only finitely many integers k≥0k \ge 0, such that, starting from the list k+1,  k+2,  …,  k+n, k + 1, \; k + 2, \; \dots, \; k + n, it is possible to obtain, after n−1n - 1 operations, the value n!n!.
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