Found problems: 95
Prove that the number $\left(\sqrt2+\sqrt3+\sqrt5\right)^{1997}$ can be represented as $$A\sqrt2+B\sqrt3+C\sqrt5+D\sqrt{30}$$ where $A$, $B$, $C$, $D$ are integers. Find with approximation to $10^{-10}$ the ratio $\frac{D}{A}$
Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =| 2nx - n - 2n^2|$
Find all integers $n \ge 2$ for which $\sqrt[n]{3^n+ 4^n+5^n+8^n+10^n}$ is an integer.
Solve the equation: $\sqrt[3]{a+\sqrt{x}}+\sqrt[3]{a-\sqrt{x}}=\sqrt[3]{b}$
Solve the inequality
$$\sqrt{(x-2)^2(x-x^2)}<\sqrt{4x-1-(x^2-3x)^2}$$
For each real value of $p$, find all real solutions of the equation $\sqrt{x^2 - p}+2\sqrt{x^2-1} = x$.
Solve the equation $(x^3-1000)^{1/2}=(x^2+100)^{1/3}$
Let $n$ be a positive integer. Prove that there are no positive integers $x$ and $y$ such as $\sqrt{n}+\sqrt{n+1} < \sqrt{x}+\sqrt{y} <\sqrt{4n+2} $
Let $n, m, k$ be natural numbers, with $m > n$. Which of the numbers is greater:
$$\sqrt{n+\sqrt{m+\sqrt{n+...}}}\,\,\, or \,\,\,\, \sqrt{m+\sqrt{n+\sqrt{m+...}}}\,\, ?$$
Note: Each of the expressions contains $k$ square root signs; $n, m$ alternate within each expression.
(N. Kurlandchik)
Find rational $x$ in $(3,4)$ such that $\sqrt{x-3}$ and $\sqrt{x+1}$ are rational.
Find the real roots of the equation $x^2 + 2ax + \frac{1}{16} = -a +\sqrt{ a^2 + x - \frac{1}{16} }$ , $\left(0 < a < \frac14 \right)$ .
Solve the equation $$\sqrt[3]{x^3+6x^2-6x-1}=\sqrt{x^2+4x+1}$$
Solve the equation $x^2+ 2 = 4\sqrt{x^3+1}$
Find all real numbers $x$ for which the following equality holds :
$$\sqrt{\frac{x-7}{1989}}+\sqrt{\frac{x-6}{1990}}+\sqrt{\frac{x-5}{1991}}=\sqrt{\frac{x-1989}{7}}+\sqrt{\frac{x-1990}{6}}+\sqrt{\frac{x-1991}{5}}$$
Prove that the number
$$\sqrt{1 + \frac{1}{n^2} + \frac{1}{(n+1)^2}}$$
is rational for all natural $n$.
Sea $f: R^+ \to R$ defined as $$f (x) = \frac{1}{\sqrt[3]{x^2 + 6x + 9} + \sqrt[3]{x^2 + 4x + 3} + \sqrt[3]{x^2 + 2x + 1}}$$
Calculate $$f (1) + f (2) + f (3) + ... + f (2016).$$
Given are the non zero natural numbers $a,b,c$ such that the number $\frac{a\sqrt2+b\sqrt3}{b\sqrt2+c\sqrt3}$ is rational.
Prove that the number $\frac{a^2+b^2+c^2}{a+b+c}$ is an integer .
The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}$ for $n \ge 1$.
Prove that there exists $a$ such that $\frac{1}{2} \le \frac{a_n}{n^a} \le 2$ for $n \ge 1$.
Which of the two numbers is bigger :
$\sqrt{1997}+2\sqrt{1999} + 2\sqrt{2001} + \sqrt{2003}$ or $2\sqrt{1998} +2\sqrt{2000}+2\sqrt{2002}$ ?
Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer.
Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.
Find the product of all numbers of the form $\sqrt[m]{m}-\sqrt[k]{k}$ , $m ,k$ are natural numbers satisfying the inequalities $1 \le k < m \le n$, where $n> 3$.
Solve the equation $\sqrt{a-\sqrt{a+ x}} = x$ for $x$.
Solve the equation
$$\sqrt{x(x+7)}+\sqrt{(x+7)(x+17)}+\sqrt{(x+17)(x+24)}=12+17\sqrt2$$
What is the number $x =\sqrt{4+\sqrt7}-\sqrt{4-\sqrt7}-\sqrt2$, positive, negative or zero?
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
$$2 \sin \left(1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^{n-1}}\right)\frac{\pi}{4}=2\cos \frac{\pi}{2^{n+1}}= \sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt2}}}$$