We could form a similar rectangle from two identical rectangles. What could be the ratio of the sides of the rectangles?
Let the sides of the rectangles used for tiling be $a$ and $b$. The area of the large rectangle is 30 times the area of the small rectangle, so its sides are $a \sqrt{30}$ and $b \sqrt{30}$. Imagine that we have tiled the large rectangle with the small rectangles and cut out one of the sides of length $a \sqrt{30}$. On this side, small rectangles fit (see the figure), and they completely fill the entire segment. If $k$ small rectangles fit along the side of length $a$, and $l$ fit along the side of length $b$ on this segment, then

$$
k a + l b = a \sqrt{30}
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_ff87628e0430059892efg-1.jpg?height=307&width=878&top_left_y=459&top_left_x=621)

Similarly, if we cut out one of the sides of length $b \sqrt{30}$, we can find non-negative integers $m, n$ such that

$$
m a + n b = b \sqrt{30}
$$

Thus, if the tiling described in the problem is possible, there must exist non-negative integers $k, l, m, n$ such that both (1) and (2) hold.

In the following, we will examine what the ratio $a / b$ must be for the above conditions to be satisfied. From (1) and (2),

$$
(a \sqrt{30} - k a)(b \sqrt{30} - n b) = l m a b,
$$

which, when divided by $a b \neq 0$ and rearranged, gives

$$
\sqrt{30}(k + n) = 30 + k n - l m
$$

Since $\sqrt{30}$ is an irrational number, and $k + n$ and $30 + k n - l m$ are integers, (3) can only hold if both sides of (3) are zero, i.e., $k = n = 0$, and $l \cdot m = 30$. Therefore, the possible values of $m$ (and $l$) are the positive divisors of 30, and according to (2), $a / b = \sqrt{30} / m$. Thus, $a / b$ can only be one of the following ratios:

$$
\sqrt{30}: 1, \sqrt{30}: 2, \sqrt{30}: 3, \sqrt{30}: 5, \sqrt{30}: 6, \sqrt{30}: 10, \sqrt{30}: 15, \sqrt{30}: 30
$$

These are the ratios for which the given condition is satisfied. However, rectangles with these side ratios can be tiled to form similar rectangles, as shown in the second figure. Therefore, the ratio of the sides of the rectangles can be any of the eight values listed in (4).
![](https://cdn.mathpix.com/cropped/2024_05_02_ff87628e0430059892efg-1.jpg?height=628&width=1502&top_left_y=1760&top_left_x=298)

Kurusa Árpád (Szeged, Ságvári E. Gyak. Gimn., IV. o. t.)

Note. The evaluation of the solutions was based on the following criteria. Correct: a flawless solution. Partial: those whose solution is based on the assumption that the tiled rectangles can only be in the same orientation, but the proof is incomplete.

1 point was given to those who did not consider any other possibility than the tiled rectangles being in the same orientation. Incorrect: those who did not find any valid ratio, or who obtained a false result due to a conceptual or calculation error.