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Found problems: 2

2023 Romanian Master of Mathematics Shortlist, A2
Tags: algebra , interpolation , polynomial
Fix an integer $n \geq 2$ and let $a_1, \ldots, a_n$ be integers, where $a_1 = 1$. Let $$ f(x) = \sum_{m=1}^n a_mm^x. $$ Suppose that $f(x) = 0$ for some $K$ consecutive positive integer values of $x$. In terms of $n$, determine the maximum possible value of $K$.

2021 Romanian Master of Mathematics Shortlist, A2
Tags: polynomial , interpolation , algebra
Let $n$ be a positive integer and let $x_1,\ldots,x_n,y_1,\ldots,y_n$ be integers satisfying the following condition: the numbers $x_1,\ldots,x_n$ are pairwise distinct and for every positive integer $m$ there exists a polynomial $P_m$ with integer coefficients such that $P_m(x_i) - y_i$, $i=1,\ldots,n$, are all divisible by $m$. Prove that there exists a polynomial $P$ with integer coefficients such that $P(x_i) = y_i$ for all $i=1,\ldots,n$.