2009 Indonesia MO Problems/Problem 2

Problem

For any real $x$ , let $\lfloor x\rfloor$ be the largest integer that is not more than $x$ . Given a sequence of positive integers $a_1,a_2,a_3,\ldots$ such that $a_1>1$ and $\left\lfloor\frac{a_1+1}{a_2}\right\rfloor=\left\lfloor\frac{a_2+1}{a_3}\right\rfloor=\left\lfloor\frac{a_3+1}{a_4}\right\rfloor=\cdots$ Prove that $\left\lfloor\frac{a_n+1}{a_{n+1}}\right\rfloor\leq1$ holds for every positive integer $n$ .

Solution

By assuming $\left\lfloor\frac{a_n+1}{a_{n+1}}\right\rfloor >2$ we can conclude that: $\left\lfloor\frac{a_1+1}{a_2}\right\rfloor=\left\lfloor\frac{a_2+1}{a_3}\right\rfloor=\left\lfloor\frac{a_3+1}{a_4}\right\rfloor=\cdots >2$

Since $\left\lfloor\frac{a_1+1}{a_{2}}\right\rfloor >2$ , it is also true that $\frac{a_1+1}{a_{2}} > 2$ , implying $a_1 > 2a_2 - 1 > a_2$ .

After repeating this same process to all given fractions, we get: $a_1 > a_2 > a_3 > ...$ which is a impossible statement because $a_1, a_2, a_3, ...$ are all positive integers.

Therefore, $\left\lfloor\frac{a_n+1}{a_{n+1}}\right\rfloor\leq1$

~NounZero

2009 Indonesia MO (Problems)
Preceded by First Problem	1 • 2 • 3 • 4 • 5 • 6 • 7 • 8	Followed by Problem 3
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2009 Indonesia MO Problems/Problem 2

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