Found problems: 95
Show that for all integers $a \ge 1$,$ \lfloor \sqrt{a}+\sqrt{a+1}+\sqrt{a+2}\rfloor = \lfloor \sqrt{9a+8}\rfloor$
Solve the equation $\sqrt{1+\sqrt {1+x}}=\sqrt[3]{x}$ for $x \ge 0$.
Without using a calculator, find out which number is greater:
$$|\sqrt[3]{5}-\sqrt3|-\sqrt3| \,\,\,\, \text{or} \,\,\,\, 0.01$$
Solve the equation $\sqrt {x + 3 - 4 \sqrt{x -1}} +\sqrt{x + 8 - 6 \sqrt{x - 1}}= 1$.
Prove that $\sqrt1+ \sqrt2+\sqrt3+...+ \sqrt{n^2-1}+\sqrt{n^2} \ge \frac{2n^3+n}{3}$ for any positive integer $n$.
Prove that every positive $a$ and $b$ satisfy inequality $$\frac{(a+b)^2}{2} + \frac{a+b}{4} \ge a\sqrt b + b\sqrt a$$
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?
Let $S = 1 + \frac{1}{\sqrt2}+ \frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+ \frac{1}{\sqrt{99}}+ \frac{1}{\sqrt{100}}$ . Find $[S]$.
You may use the fact that $\sqrt{n} < \frac12 (\sqrt{n} +\sqrt{n+1}) <\sqrt{n+1}$ for all integers $n \ge 1$.
The sides of a triangle are equal to $\sqrt2, \sqrt3, \sqrt4$ and its angles are $\alpha, \beta, \gamma$, respectively. Prove that the equation $x\sin \alpha + y\sin \beta + z\sin \gamma = 0$ has exactly one solution in integers $x, y, z$.
Solve the equation $$\sqrt{2+\sqrt{2-\sqrt{2+x}}}=x.$$
Find all real solutions of $\sqrt[4]{13+x}+ \sqrt[4]{14-x} = 3$.
Find the number of positive integers $n$, such that $\sqrt{n} + \sqrt{n + 1} < 11$.
Solve the equation $\sqrt{1981-\sqrt{1996+x}}=x+15$
Solve the system of equations
$$\begin{cases} x^2+arc siny =y^2+arcsin x \\ x^2+y^2-3x=2y\sqrt{x^2-2x-y}+1 \end{cases}$$
Prove that if $(x+\sqrt{x^2 +1}) (y+\sqrt{y^2 +1}) = 1$, then $x+y = 0$.
If $a > b > 0$, prove the inequality $$\frac{(a-b)^2}{8a}< \frac{a+b}{2}- \sqrt{ab} < \frac{(a-b)^2}{8b}.$$
$\sqrt[3]{x+\sqrt[3]{x+\sqrt[3]{x+ \sqrt[3]{x ...}}}}= 8$. Find $x$.
What is the smallest value that the expression $$\sqrt{3x-2y-1}+\sqrt{2x+y+2}+\sqrt{3y-x}$$ can take?
Show that the number $a^3$ where $a=\frac{251}{ \frac{1}{\sqrt[3]{252}-5\sqrt[3]{2}}-10\sqrt[3]{63}}+\frac{1}{\frac{251}{\sqrt[3]{252}+5\sqrt[3]{2}}+10\sqrt[3]{63}}$
is an integer and find its value
Prove that $\sqrt{x}+\sqrt{y}+\sqrt{xy}$ is equal to$ \sqrt{x}+\sqrt{y+xy+2y\sqrt{x}}$
and compare the numbers $\sqrt{3}+\sqrt{10+2\sqrt{3}}$ and $\sqrt{5+\sqrt{22}}+\sqrt{8-
\sqrt{22}+2\sqrt{15-3\sqrt{22}}}$.
Determine the value of
$$\prod^{\infty}_{n=1} 3^{n/3^n}= \sqrt[3]{3} \sqrt[3^2]{3^2} \sqrt[3^3]{3^3} ...$$
Show that $\sqrt[3]{\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5}$ is rational.
Solve the equation
$$2\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)(x+5)}}}}=x$$
Calculate the sum $$S=\sqrt{1+\frac{8\cdot 1^2-1}{1^2\cdot 3^2}}+\sqrt{1+\frac{8\cdot 2^2-1}{3^2\cdot 5^2}}+...+ \sqrt{1+\frac{8\cdot 1003^2-1}{2005^2\cdot 2007^2}}$$
Is rational or irrational,the number
$$\left(\dfrac{2}{\sqrt[3]{25}+\sqrt[3]{15}+\sqrt[3]{9}}+\dfrac{1}{\sqrt[3]{9}+\sqrt[3]{6}+\sqrt[3]{4}}\right) \times \left(\sqrt[3]{25}+\sqrt[3]{10}+\sqrt[3]{4}\right)?$$