We select two points independently at random on the interval $(0,1)$ of the number line. (By saying that a point $P$ is chosen at random on the segment $(a, b)$, we mean that the probability that $P$ falls within any subinterval $(c, d)$ is $\frac{d-c}{b-a}$.)

We know about a number $h$ that the probability that the distance between the two chosen points is less than $h$ lies between $\frac{1}{4}$ and $\frac{3}{4}$.

What are the bounds within which $h$ lies?
We denote the chosen numbers by $x$ and $y$. According to the conditions of the problem, the point $P$ corresponding to the pair $(x, y)$ is selected from the square $N$ defined by $0 \leqq x \leqq 1, 0 \leqq y \leqq 1$ in the coordinate plane, according to a geometric distribution. This means that the probability that $P$ falls into some subset $A$ of $N$ is equal to the area of $A$.

The distance between the chosen numbers in the original $(0,1)$ interval is equal to the absolute difference $|x-y|$ of the $x$ and $y$ coordinates of the point $P$. This is less than $h$ if

$$
x-h < y < x+h
$$

If $0 < h < 1$, the points of $N$ that satisfy the inequality (1) form the set $A$, which is obtained from $N$ by cutting off the $(1-h)$-sided isosceles right triangles from the corners $(1; 0)$ and $(0; 1)$. Therefore, the area of $A$ is

$$
1 - (1-h)^2 = h(2-h)
$$

and this is also the probability of the event (1).

According to the problem, this probability lies between $1/4$ and $3/4$, so

$$
\frac{1}{4} < (1-h)^2 < \frac{3}{4}
$$

which means that the number $h$ lies between $\frac{1}{2}$ and $1 - \frac{\sqrt{3}}{2}$.

Páles Zsolt (Sátoraljaújhely, Kossuth L. Gymnasium, 4th grade)

Note. In our solution, we did not prove why the conditions of the problem imply that the point $P(x, y)$ is selected from $N$ according to a geometric probability. Generally, we prefer a formulation of a probability problem that contains the conditions in a more intuitive, practical way. However, before solving the problem, we need to translate the given conditions into the language of probability. In our case, the problem was already formulated in the language of probability, but the material temporarily included in the high school curriculum was not sufficient for the above reformulation.