2013 APMO #2

For $2k$ real numbers $a_1, a_2, ..., a_k$, $b_1, b_2, ..., b_k$ define a sequence of numbers $X_n$ by \[X_n = \sum_{i=1}^k [a_in + b_i] \quad (n=1,2,...).\]If the sequence $X_N$ forms an arithmetic progression, show that $\textstyle\sum_{i=1}^k a_i$ must be an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

Let $\displaystyle\sum_{i=1}^{k}a_i=A,\displaystyle\sum_{i=1}^{k}b_i=B$, and $X_n = \sum_{i=1}^k [a_in + b_i] = Sn+T$. 

By the definition of the floor function, we have $$\sum (a_in+b_i) \geq \sum [a_in + b_i] \geq \sum (a_in + b_i -1)$$ 

$$\Rightarrow An +B \geq Sn+T \geq An+B - k$$

$$\Rightarrow A+ \frac B n \geq S +\frac Tn \geq A + \frac Bn - \frac kn$$

Taking $n \rightarrow \infty$ implies $S = A$, and as $S$ is an integer (common difference from an integer arithmetic sequence), so is $A$. $\square$


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