Found problems: 95
2006 AIME Problems, 5
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 $a\cdot b\cdot c$.
V Soros Olympiad 1998 - 99 (Russia), 10.5
Solve the equation $$\sqrt{2+\sqrt{2-\sqrt{2+x}}}=x.$$
2019 Saudi Arabia JBMO TST, 1
Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove that
$$\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c} \ge 2\sqrt2 \left( \sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}}\right)$$
2008 Dutch IMO TST, 4
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.
2016 Hanoi Open Mathematics Competitions, 14
Given natural numbers $a,b$ such that $2015a^2+a = 2016b^2+b$.
Prove that $\sqrt{a-b}$ is a natural number.
III Soros Olympiad 1996 - 97 (Russia), 10.3
Solve the equation
$$\sqrt{x(x+7)}+\sqrt{(x+7)(x+17)}+\sqrt{(x+17)(x+24)}=12+17\sqrt2$$
1977 Vietnam National Olympiad, 1
Find all real $x$ such that $ \sqrt{x - \frac{1}{x}} + \sqrt{1 - \frac{1}{x}}> \frac{x - 1}{x}$
1979 Swedish Mathematical Competition, 2
Find rational $x$ in $(3,4)$ such that $\sqrt{x-3}$ and $\sqrt{x+1}$ are rational.
2004 Thailand Mathematical Olympiad, 5
Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =| 2nx - n - 2n^2|$
1985 Poland - Second Round, 4
Prove that if for natural numbers $ a, b $ the number $ \sqrt[3]{a} + \sqrt[3]{b} $ is rational, then $ a, b $ are cubes of natural numbers.
2017 India PRMO, 7
Find the number of positive integers $n$, such that $\sqrt{n} + \sqrt{n + 1} < 11$.
1998 ITAMO, 1
Calculate the sum $\sum_{n=1}^{1.000.000}[ \sqrt{n} ]$ .
You may use the formula $\sum_{i=1}^{k} i^2=\frac{k(k +1)(2k +1)}{6}$ without a proof.
VI Soros Olympiad 1999 - 2000 (Russia), 11.1
Solve the system of equations
$$\begin{cases} x^2+arc siny =y^2+arcsin x \\ x^2+y^2-3x=2y\sqrt{x^2-2x-y}+1 \end{cases}$$
2006 Thailand Mathematical Olympiad, 8
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 $s^4 -18s^2 - 8s$ .
1953 Moscow Mathematical Olympiad, 238
Prove that if in the following fraction we have $n$ radicals in the numerator and $n - 1$ in the denominator, then
$$\frac{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}>\frac14$$
1974 Chisinau City MO, 73
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 $a_i=b_i$ , $i=1,...,n$
2001 Swedish Mathematical Competition, 2
Show that $\sqrt[3]{\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5}$ is rational.
2009 Grand Duchy of Lithuania, 3
Solve the equation $x^2+ 2 = 4\sqrt{x^3+1}$
1992 Tournament Of Towns, (326) 3
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)
1975 Chisinau City MO, 116
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 $x, y, z$.