Question 4
Let $f:\mathbb N\to\mathbb Z$ satisfy $f(0)=2$, $f(1)=503$ and
$$f(n+2)=503f(n+1)-1996f(n)\qquad(n\in\mathbb N).$$
For any positive integer $k$, choose integers $s_1,\ldots,s_k$ with $s_i\ge k$ for every $i$. For each $s_i$, choose any prime divisor $p(s_i)$ of $f(2^{s_i})$. Prove that for every positive integer $t\le k$,
$$2^t\mid\sum_{i=1}^k p(s_i)\quad\Longleftrightarrow\quad2^t\mid k.$$
Here $\mathbb N$ includes $0$; the $s_i$ need not be distinct.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
$$f(n+2)=503f(n+1)-1996f(n)\qquad(n\in\mathbb N).$$
For any positive integer $k$, choose integers $s_1,\ldots,s_k$ with $s_i\ge k$ for every $i$. For each $s_i$, choose any prime divisor $p(s_i)$ of $f(2^{s_i})$. Prove that for every positive integer $t\le k$,
$$2^t\mid\sum_{i=1}^k p(s_i)\quad\Longleftrightarrow\quad2^t\mid k.$$
Here $\mathbb N$ includes $0$; the $s_i$ need not be distinct.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.