Competition problem — Algebra — Maths Olympiad Prep

Library / /16 of 68

Algebra Difficulty 4.9 AIME Prove it Estonia

Some numbers are printed on a spool of paper, one below another. Starting from the third number, each number is the sum of the two preceding numbers. The number $2018$ occurs fourth, the number $2020$ occurs eighth. Which are the first and the second numbers?

Solution

Let the first and the second number be $x$ and $y$ , respectively. Computing the following numbers, we see that the fourth number is $x + 2y$ and the eighth one is $8x + 13y$ . The conditions of the problem imply the system of equations
$\begin{cases} x + 2y = 2018, \\ 8x + 13y = 2020, \end{cases}$
solving of which gives $x = -7398$ and $y = 4708$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.