Competition problem — Algebra — Maths Olympiad Prep

Library / /17 of 36

Algebra Difficulty 6.6 National Olympiad Prove it Italy

Problem:

Observing the temperatures recorded in Cesenatico in the last months of December and January, Stefano noticed a strange coincidence: on all the days of this period (excluding the first and the last) the minimum temperature was the sum of the minimum temperature of the previous day and of the following day.

Knowing that on December 3rd the minimum temperature was 5 degrees, and on January 31st it was 2 degrees, determine the minimum temperature on December 25th.

Solution

Solution:

The minimum temperature recorded on December 25th was -3 degrees.

To prove this, let us denote by $x$ the minimum temperature recorded on December 1st and by $y$ the minimum temperature recorded on December 2nd. Using the relation observed by Stefano, one can derive the minimum temperature on a given day, knowing those of the two preceding days: in this way one obtains that the minimum temperatures in the first days of December are those given in the following table

Giorno	1	2	3	4	5	6	7	8	9	10
Temp. minima	$x$	$y$	$y-x$	$-x$	$-y$	$x-y$	$x$	$y$	$y-x$	$-x$

It is thus easily seen that the sequence of temperatures repeats with a period of 6 days. Consequently the minimum temperature on December 3rd was

y-x

, while that of January 31st, which is the 62

\text{nd}

day of the period, coincides with that of the second day, that is

y

. From the information given we deduce that

y-x=5

and

y=2

, from which

x=-3

Now the minimum temperature on December 25th coincides with that of the first day, since $25-1$ is a multiple of 6, and was therefore -3 degrees.

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 translated into English from it; metadata (topic, difficulty) added by this project.