Found problems: 95
1976 Chisinau City MO, 120
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 $n> 3$.
2016 Costa Rica - Final Round, F2
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 $$f (1) + f (2) + f (3) + ... + f (2016).$$
2010 Bundeswettbewerb Mathematik, 2
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.
1997 Israel Grosman Mathematical Olympiad, 3
Find all real solutions of $\sqrt[4]{13+x}+ \sqrt[4]{14-x} = 3$.
IV Soros Olympiad 1997 - 98 (Russia), 10.2
Solve the equation $$\sqrt[3]{x^3+6x^2-6x-1}=\sqrt{x^2+4x+1}$$
2011 Dutch Mathematical Olympiad, 4
Determine all pairs of positive real numbers $(a, b)$ with $a > b$ that satisfy the following equations:
$a\sqrt{a}+ b\sqrt{b} = 134$ and $a\sqrt{b}+ b\sqrt{a} = 126$.
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 $a\cdot b\cdot c$.
VI Soros Olympiad 1999 - 2000 (Russia), 9.1
Which of the two numbers is bigger :
$\sqrt{1997}+2\sqrt{1999} + 2\sqrt{2001} + \sqrt{2003}$ or $2\sqrt{1998} +2\sqrt{2000}+2\sqrt{2002}$ ?
2013 Czech-Polish-Slovak Junior Match, 1
Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$
2020 Canadian Mathematical Olympiad Qualification, 8
Find all pairs $(a, b)$ of positive rational numbers such that $\sqrt[b]{a}= ab$
2009 Greece JBMO TST, 3
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 .
1994 All-Russian Olympiad, 1
Prove that if $(x+\sqrt{x^2 +1}) (y+\sqrt{y^2 +1}) = 1$, then $x+y = 0$.
2005 Estonia National Olympiad, 4
Represent the number $\sqrt[3]{1342\sqrt{167}+2005}$ in the form where it contains only addition, subtraction, multiplication, division and square roots.
1996 Romania National Olympiad, 2
Find all real numbers $x$ for which the following equality holds :
$$\sqrt{\frac{x-7}{1989}}+\sqrt{\frac{x-6}{1990}}+\sqrt{\frac{x-5}{1991}}=\sqrt{\frac{x-1989}{7}}+\sqrt{\frac{x-1990}{6}}+\sqrt{\frac{x-1991}{5}}$$
1949-56 Chisinau City MO, 53
Solve the equation: $\sqrt[3]{a+\sqrt{x}}+\sqrt[3]{a-\sqrt{x}}=\sqrt[3]{b}$
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 $x, y, z$.
1990 Spain Mathematical Olympiad, 4
Prove that the sum $\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}} +\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}$
is independent of $a$ for $ a \ge - \frac{3}{4}$ and evaluate it.
2009 Grand Duchy of Lithuania, 3
Solve the equation $x^2+ 2 = 4\sqrt{x^3+1}$
IV Soros Olympiad 1997 - 98 (Russia), 11.3
Solve the inequality
$$\sqrt{(x-2)^2(x-x^2)}<\sqrt{4x-1-(x^2-3x)^2}$$
2001 Rioplatense Mathematical Olympiad, Level 3, 3
For every integer $n > 1$, the sequence $\left( {{S}_{n}} \right)$ is defined by ${{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor $
where $\left\lfloor x \right\rfloor$ denotes the floor function of $x$. Prove that ${{S}_{2001}}=2\,{{S}_{2000}}+1$.
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1998 Chile National Olympiad, 3
Evaluate $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}}$.
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 $a_i=b_i$ , $i=1,...,n$
1979 Swedish Mathematical Competition, 2
Find rational $x$ in $(3,4)$ such that $\sqrt{x-3}$ and $\sqrt{x+1}$ are rational.
II Soros Olympiad 1995 - 96 (Russia), 9.6
Without using a calculator (especially a computer), find out what is more:
$$\sqrt[3]{5\sqrt{13}+18}- \sqrt[3]{2\sqrt{13}+5} \,\,\, or \,\,\, 1 $$
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.