Let's calculate the value of the expression $P$, where the number of nested radicals is $n$ (>1).

$$
\begin{aligned}
& P=\sqrt{a+\sqrt{a+\ldots+\sqrt{a+\sqrt{b}}}}, \text{ where } \\
& a=\sqrt{76+24 \sqrt{10}}-2 \sqrt{19-3 \sqrt{40}} \text{ and } \\
& b=\sqrt{165+4 \sqrt{1616}}-\sqrt{165-4 \sqrt{1616}}
\end{aligned}
$$

Create a similar expression with the same properties.
I. The numbers $a$ and $b$ can be transformed as follows:

$$
\begin{aligned}
a & =2 \sqrt{9+6 \sqrt{10}+10}-2 \sqrt{9-6 \sqrt{10}+10}=2(3+\sqrt{10})- \\
& -2 \cdot|3-\sqrt{10}|=12 \\
b & =\sqrt{64+16 \sqrt{101}+101}-\sqrt{101-16 \sqrt{101}+64}= \\
& =8+\sqrt{101}-(\sqrt{101}-8)=16
\end{aligned}
$$

Accordingly, $\sqrt{b}=4, a+\sqrt{b}=16=4^{2}$, and in $P$, when calculating the inner roots sequentially (i.e., from right to left), the result of each square root is 4, so $P=4$.

II. We primarily want to convey the property of $P$ that $b$ is a (positive) number such that adding its square root to $a$ gives back $b$:

$$
a+\sqrt{b}=b, \quad b=(b-a)^{2}
$$

In this case, by alternately adding and taking the square root, we can repeat the operations any number of times, and two results will alternate: $b$ and $\sqrt{b}$.

Accordingly, $b$ can be represented by pairs of numbers, such as 36 and 30, or 25 and 20, or $9/4$ and $3/4$, etc.

Furthermore, we want to obtain the rational numbers $a$ and $b$ as a difference, for example, in the form:

$$
a=\sqrt{x+\sqrt{y}}-\sqrt{x-\sqrt{y}}
$$

where $x, y$ are rational numbers, and $x \pm \sqrt{y}$ are the squares of some positive rational numbers $z, u$: $\sqrt{z} \pm u$, i.e., $z+u^{2} \pm 2 u \sqrt{z}$, or $x=z+u^{2}, y=4 u^{2} z$. This corresponds to $u=a / 2$ and any $z$ greater than $u^{2}$ (also, $z$ should not be a perfect square). For example:

For $a=30$, $u=15, z=513, y=259200$,

For $b=36$, $u=18, z=360, x=684, y=466560$, thus for the expression of $P$:

$$
\begin{gathered}
a=\sqrt{513+360 \sqrt{2}}-\sqrt{513-360 \sqrt{2}} \\
b=2 \sqrt{171+54 \sqrt{10}}-\sqrt{684-216 \sqrt{10}}
\end{gathered}
$$

Bérczi Szaniszló (Szeged, Ságvári E. Gym. III. grade) Várhegyi Éva (Budapest, Berzsenyi D. Gym. I. grade) Lörincz András (Budapest, Kölcsey F. Gym. II. grade)

Note. Instead of addition and square root in $P$, we can also alternate other two operations (or even more than two). Two such examples are:

$$
R=\sqrt[3]{a^{2}+a \sqrt[3]{a^{2}+\ldots+a \sqrt[3]{a^{2}+a \sqrt[3]{b}}}}
$$

where $a^{2}+a \sqrt[3]{b}=b$, for example, $b=8, a=2$ (although here 8 is a perfect cube);

$$
S=\frac{1}{a+\frac{1}{a+\frac{1}{a+}},} \quad \text { if } a+\frac{1}{b}=b,
$$

Hárs László (Budapest, Kölcsey F. Gym. II. grade)