The sides of a triangle and the measurements of two medians, arranged in ascending order, are as follows: $13,16,17,18$, 22. What is the length of the third median?
We need to find which three of the given measurements are the sides of the triangle in question. The sides of the triangle are $a, b, c$, and the corresponding medians are $s_{a}, s_{b}, s_{c}$. The following relationships hold:

$$
s_{a}^{2}=\frac{-a^{2}+2 b^{2}+2 c^{2}}{4}, \quad s_{b}^{2}=\frac{2 a^{2}-b^{2}+2 c^{2}}{4}, \quad s_{c}^{2}=\frac{2 a^{2}+2 b^{2}-c^{2}}{4}
$$

Furthermore, we use the fact that all given measurements are integers.

Let's choose the labeling such that $s_{a}$ and $s_{b}$ are integers, so their squares are also integers. Thus,

$$
s_{a}^{2}+s_{b}^{2}=\frac{a^{2}+b^{2}}{4}+c^{2}
$$

is an integer, so $a^{2}+b^{2}$ is divisible by 4. Therefore, both $a$ and $b$ must be even, because the sum of the squares of an even and an odd number would be odd, and the sum of the squares of two odd numbers would leave a remainder of 2 when divided by 4, since $(2 k+1)^{2}=4 k(k+1)+1$. From the fact that $a$ and $b$ are even, it follows that $c^{2}$ is also even, because rearranging the first equation gives:

$$
c^{2}=2\left(\frac{a^{2}}{4}-2 \frac{b^{2}}{4}+s_{a}^{2}\right)
$$

and each term in the parentheses is an integer. We have seen that the square of an odd number is odd, so $c$ can only be an integer if it is even. Therefore, among the given measurements, only 16, 18, and 22 can be the sides.

Now, with $a=22, b=18, c=16$, we have $s_{a}=13, s_{b}=17$, and these values are among the given data, so $s_{c}$ is the one we are looking for: $s_{c}=\sqrt{340} \approx 18.44$ units.

Krassai Éva (Székesfehérvár, Teleki B. lg. II. o. t.)

Remarks. 1. It would be nice to conclude the solution as follows: "The sum of the expressions (1) can be transformed as follows:

$$
s_{a}^{2}+s_{b}^{2}+s_{c}^{2}=3\left[\left(\frac{a}{2}\right)^{2}+\left(\frac{b}{2}\right)^{2}+\left(\frac{c}{2}\right)^{2}\right]=3(64+81+121)=798
$$

The sum of the known medians squared is 458, so the square of the third median is 340."

However, this is not strictly correct. We had 5 pieces of data to determine the triangle, even if we did not precisely know their roles. We needed to verify that the selection of the 3 sides is consistent with the other 2 data points.

2. The side lengths can also be selected by a slightly different method. Each median is smaller than the larger of the two sides emanating from the same vertex, because in the smaller triangle formed by the median, the angle opposite the larger side is obtuse, and in an isosceles triangle, it is a right angle. Therefore, the largest given measurement: 22, can only be a side, let this be $a$. With this, the expressions (1) become:

$$
s_{a}^{2}=-121+\frac{b^{2}+c^{2}}{2}, \quad s_{b}^{2}=242+\frac{2 c^{2}-b^{2}}{4}, \quad s_{c}^{2}=242+\frac{2 b^{2}-c^{2}}{4}
$$

and at least two of these must be integers. If $b$ is odd, then $s_{b}^{2}$ is not an integer, so $s_{a}^{2}$ and $s_{c}^{2}$ must be integers. The first is only an integer with an odd $c$, but then $s_{c}^{2}$ is not an integer. Therefore, $b$ cannot be odd, and similarly, $c$ cannot be odd.

Mátrai Miklós (Hódmezővásárhely, Bethlen G. g. II. o. t.)[^0]

[^0]:    ${ }^{1}$ See, for example, Exercise 752, K. M. L. 25 (1962/11) p. 150.