IMO Shortlist 1995 problem S3
Dodao/la:
arhiva2. travnja 2012. For an integer
![x \geq 1,](/media/m/a/d/d/add156dbf6fa9f58eb052b1df0d02e27.png)
let
![p(x)](/media/m/3/9/3/393afccb4b82415d2114a3ff957b444f.png)
be the least prime that does not divide
![x,](/media/m/a/9/6/a96327826d8ff11119b89ba22bc1e70a.png)
and define
![q(x)](/media/m/2/a/2/2a21746f3bd729798cef4a37762db366.png)
to be the product of all primes less than
![p(x).](/media/m/3/9/2/3929be3b2f1804de530c44ca9e7d80b0.png)
In particular,
![p(1) = 2.](/media/m/6/2/f/62f41d3b6aa4811d94f95fcc9ff88f15.png)
For
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
having
![p(x) = 2,](/media/m/5/5/2/552ce370343cca9a60dca1e67fe1bc93.png)
define
![q(x) = 1.](/media/m/1/8/1/1811e2a489be843dbfbc508701345f98.png)
Consider the sequence
![x_0, x_1, x_2, \ldots](/media/m/2/9/6/296bd78da969b701abdbfea8df0da66e.png)
defined by
![x_0 = 1](/media/m/1/8/4/1842a030917545833c1edd52c0c4afb5.png)
and
![x_{n+1} = \frac{x_n p(x_n)}{q(x_n)}](/media/m/3/7/0/370dae486f0598c5ee4878b7ecf57456.png)
for
![n \geq 0.](/media/m/d/7/1/d717529de6faf333978205f786a3d99f.png)
Find all
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
such that
%V0
For an integer $x \geq 1,$ let $p(x)$ be the least prime that does not divide $x,$ and define $q(x)$ to be the product of all primes less than $p(x).$ In particular, $p(1) = 2.$ For $x$ having $p(x) = 2,$ define $q(x) = 1.$ Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and
$$x_{n+1} = \frac{x_n p(x_n)}{q(x_n)}$$ for $n \geq 0.$ Find all $n$ such that $x^n = 1995.$
Izvor: Međunarodna matematička olimpijada, shortlist 1995