Question 38

Consider the $4$ rectangles in the bottom right hand corner.

Length $a$ must be twice that of length $b$, since the $240$ m$^{2}$ and $120$ m$^{2}$ rectangles have the same height.

Therefore the unknown area $x=90\times 2=180$.

We can repeat this process to find the remaining unknown areas.

Let the integers in the sequence be $u_1$, $u_2$, $u_3$, $u_4$, $u_5$.

Let $Si$ be the sum of the first $i$ integers, so

$S_1=u_1$
$S_2=u_1+u_2$
$S_3=u_1+u_2+u_3$
$S_3=u_1+u_2+u_3+u_4$
$S_3=u_1+u_2+u_3+u_4+u_5$

If any of the values $S_j$ is a multiple of $5$, then we are done, as $u_1+u_2+....+u_j$ is a multiple of $5$.

If not, then each sum $S_i$ must leave a remainder of $1$, $2$, $3$, or $4$ when divided by $5$.

By the Pigeonhole Principle, there must be two values of $S_i$ with the same remainder when divided by $5$. We shall call these values $S_j$ and $S_k$, where $j\lt k$.

In this case, $S_k-S_j=u_{j+1}+u_{j+2}+....+u_k$ is a multiple of $5$.