Bariz - Problem

Source: Estonia IMO TST 2015 p3

Tags: number theory, rational

parmenides51

22.07.2020 15:13

Let $q$ be a fixed positive rational number. Call number $x$ charismatic if there exist a positive integer $n$ and integers $a_1, a_2, . . . , a_n$ such that $x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} ...(q + n)^{a_n}$. a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic. b) Is it true for every $q$ that, for every charismatic number $x$, the number $x + 1$ is charismatic, too?

For a) we claim that for $q=1$ every positive number is charismatic. For example if we want to represent $\frac{r}{s}$ where wlog $r<s$ we let $n=s-1$ and we let $a_{r-1}=1$, $a_{s-1}=-1$ and all other $a_i=0$. We handle the case $r>s$ analogously and we note that number $1$ is charsmatic for any $q$. This motivates us to find $q$ such that $x=2$ isn't charismatic. We can pick $q=\frac{1}{4}$ because that makes power of $2$ dividing numerator and denomitor in reducible form even no matter how we combine the factors, so $v_2$ of numerator of our product in irreducible form will be even so product can't be equal to $2$.