Found problems: 8
Evaluate $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}}$.
The numbers $a_1, a_2, ..., a_n$ are equal to $1$ or $-1$. Prove that
$$2 \sin \left(a_1+\frac{a_1a_2}{2}+\frac{a_1a_2a_3}{4}+...+\frac{a_1a_2...a_n}{2^{n-1}}\right)\frac{\pi}{4}=a_1\sqrt{2+a_2\sqrt{2+a_3\sqrt{2+...+a_n\sqrt2}}}$$
In particular, for $a_1 = a_2 = ... = a_n = 1$ we have
$$2 \sin \left(1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^{n-1}}\right)\frac{\pi}{4}=2\cos \frac{\pi}{2^{n+1}}= \sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt2}}}$$
Show that if $ \left( s_n \right)_{n\ge 0} $ is a sequence that tends to $ 6, $ then, the sequence
$$ \left( \sqrt[3]{s_n+\sqrt[3]{s_{n-1}+\sqrt[3]{s_{n-2}+\sqrt[3]{\cdots +\sqrt[3]{s_0}}}}} \right)_{n\ge 0} $$
tends to $ 2. $
[i]Mihai Bălună[/i]
For how many integers $n$ between $ 1$ and $2021$ does the infinite nested expression $$\sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}}$$ give a rational number?
The vertices of a regular $45$-gon are painted into three colors so that the number of vertices of each color is the same. Prove that three vertices of each color can be selected so that three triangles formed by the chosen vertices of the same color are all equal.
Find the limit
$$\lim_{n\to\infty}\sqrt{1+2\sqrt{1+3\sqrt{\ldots+(n-1)\sqrt{1+n}}}}.$$
Prove that if in the following fraction we have $n$ radicals in the numerator and $n - 1$ in the denominator, then
$$\frac{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}>\frac14$$
Find all real numbers of the form $\sqrt[p]{2021+\sqrt[q]{a}}$ that can be expressed as a linear combination of roots of unity with rational coefficients, where $p$ and $q$ are (possible the same) prime numbers, and $a>1$ is an integer, which is not a $q$-th power.