2018 IMO Problems/Problem 2

Find all numbers $n \ge 3$ for which there exists real numbers $a_1, a_2, ..., a_{n+2}$ satisfying $a_{n+1} = a_1, a_{n+2} = a_2$ and $a_{i}a_{i+1} + 1 = a_{i+2}$ for $i = 1, 2, ..., n.$

Solution

We find at least one series of real numbers for $n = 3,$ for each $n = 3k$ and we prove that if $n = 3k \pm 1,$ then the series does not exist.

Case 1

Let $n = 3.$ We get system of equations $\begin{cases} a_1 a_2 + 1 = a_3 \\a_2 a_3 + 1 = a_1 \\a_3 a_1 + 1 = a_2 \end{cases}$

We subtract the first equation from the second and get: $a_2 (a_3 – a_1) = (a_1 – a_3).$ So $a_2 = – 1 \implies a_1 = 2, a_3 = – 1.$

Case 1'

Let $n = 3k, k={1,2,...}.$ Real numbers $a_1 =a_4 =...=2, a_2 = a_3 = a_5=...=-1$ satisfying $a_{n+1} = a_1, a_{n+2} = a_2$ and $a_{i}a_{i+1} + 1 = a_{i+2}$ .

Case 2

Let $n = 4.$ We get system of equations $\begin{cases} a_1 a_2 + 1 = a_3 \\a_2 a_3 + 1 = a_4 \\a_3 a_4 + 1 = a_1\\a_4 a_1 + 1 = a_2 \end{cases}$ We multiply each equation by the number on the right-hand side and get: $\begin{cases} a_1 a_2 a_3 + a_3 = a_3^2 \\a_2 a_3 a_4 + a_4 = a_4^2 \\a_3 a_4 a_1 + a_1 = a_1^2 \\a_4 a_1 a_2 + a_2 = a_2^2 \end{cases}$ We multiply each equation by a number that precedes a pair of product numbers in a given sequence $a_1, a_2, a_3, a_4, a_1, a_2.$ So we multiply the equation with product $a_1 a_2$ by $a_4$ , we multiply the equation with product $a_4 a_1$ by $a_3$ etc. We get:

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2018 IMO Problems/Problem 2

Solution