Found problems: 95
Find integer solutions $(x,y)$ of the equation ($1964$ times "$\sqrt{}$"):
$$\sqrt{x+\sqrt{x+\sqrt{....\sqrt{x+\sqrt{x}}}}}=y$$
How many solutions, depending on the value of the parameter $a$, has the equation $$\sqrt{x^2-4}+\sqrt{2x^2-7x+5}=a ?$$
What is the largest integer less than or equal to $4x^3 - 3x$, where $x=\frac{\sqrt[3]{2+\sqrt3}+\sqrt[3]{2-\sqrt3}}{2}$ ?
(A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) None of the above.
Let $A, B, C$ and $D$ denote the digits in a four-digit number $n = ABCD$. Determine the least $n$ greater than $2017$ satisfying that there exists an integer $x$ such that $$x =\sqrt{A +\sqrt{B +\sqrt{C +\sqrt{D + x}}}}.$$
Let $a, b, c$ be the roots of the equation $x^3-9x^2+11x-1 = 0$, and define $s =\sqrt{a}+\sqrt{b}+\sqrt{c}$.
Compute $s^4 -18s^2 - 8s$ .
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 all pairs $(a, b)$ of positive rational numbers such that $\sqrt[b]{a}= ab$
What's more $\sqrt[4]{7}+\sqrt[4]{11}$ or $2\sqrt{\frac{\sqrt{7}+\sqrt{11}}{2}}$ ?
Let $a = (\sqrt2 +\sqrt3 +\sqrt6)(\sqrt2 +\sqrt3 -\sqrt6)(\sqrt3 +\sqrt6 -\sqrt2)(\sqrt6 +\sqrt2 -\sqrt3)$
$b = (\sqrt2 +\sqrt3 +\sqrt5)(\sqrt2 +\sqrt3 -\sqrt5)(\sqrt3 +\sqrt5 -\sqrt2)(\sqrt5 +\sqrt2 -\sqrt3)$
The difference $a - b$ belongs to the set:
A. $(-\infty,-4)$ B. $[-4,0)$ C.$\{0\}$ D. $(0,4]$ E. $(4,\infty)$
Prove that if in the following fraction we have $n$ radicals in the numerator and $n - 1$ in the denominator, then
$$\frac{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}>\frac14$$
For the real numbers $a_1,...,a_n, b_1,...,b_m$ , the following relations hold:
1) $|a_i|= |b_j|=1$, $i=1,...,n$ ,$j=1,...,m$
2) $a_1\sqrt{2+a_2\sqrt{2+...+a_n\sqrt2}}=b_1\sqrt{2+b_2\sqrt{2+...+b_m\sqrt2}}$
Prove that $n = m$ and $a_i=b_i$ , $i=1,...,n$
The number \[ \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006} \] can be written as $a\sqrt{2}+b\sqrt{3}+c\sqrt{5},$ where $a, b,$ and $c$ are positive integers. Find $a\cdot b\cdot c$.
Given natural numbers $a,b$ such that $2015a^2+a = 2016b^2+b$.
Prove that $\sqrt{a-b}$ is a natural number.
Prove that for any tetrahedron the radius of the inscribed sphere $r <\frac{ ab}{ 2(a + b)}$, where $a$ and $b$ are the lengths of any pair of opposite edges.
Find all real solutions of $\sqrt{x-1}+\sqrt{x^2-1}=\sqrt{x^3}$
The sequence of numbers $a_1, a_2, a_3, ...$ is defined recursively by $a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor $ for $n \ge 1$. Find all numbers that appear more than twice at this sequence.
For every positive integer $n$, we define the function $p_n$ for $x\ge 1$ by
$$p_n(x) = \frac12 \left(\left(x+\sqrt{x^2-1}\right)^n+\left(x-\sqrt{x^2-1}\right)^n\right).$$
Prove that $p_n(x) \ge 1$ and that $p_{mn}(x) = p_m(p_n(x))$.
Calculate $\sqrt{5,44...4}$ (the decimal point is followed by $100$ fours) with approximation to:
a) $10^{-100}$,
b) $10^{-200}$
Let $a$ be an integer. Prove that for any real number $x, x^3 < 3$, both the numbers $\sqrt{3 -x^2}$ and $\sqrt{a - x^3}$ cannot be rational.
Solve the equation $$2(x-6)=\dfrac{x^2}{(1+\sqrt{x+1})^2}$$