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Problem 2. Given a positive integer $n$, Marie plays a game where she starts with the number $1$ on a blackboard. As many times as she wants, she can choose an integer $j$ such that $1 \le j \le n$ and replace the number $V$ on the blackboard with the number $j\cdot R(\frac{V}{j})$. Here $R(x)$ denotes the nearest integer to $x$; if $x$ is exactly halfway between two consecutive integers, it is rounded up. For example, $R(1.3) = 1$ and $R(1.5) = R(1.8) = 2$.
a) Prove that for each given $n$, there is a positive integer $B$ such that Marie can never end up with a number larger than $B$ on the blackboard.
b) For any given $n$, let $f(n)$ be the maximum number obtainable on the blackboard after finitely many replacements. Show that there exists a positive integer $N$ such that for all $n \ge N$, we have that $2026$ divides $f(n)$.
Solution 1Solution 2Solution 3Solution 4
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