Found problems: 95
1948 Moscow Mathematical Olympiad, 153
* What is the radius of the largest possible circle inscribed into a cube with side $a$?
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 $n> 3$.
1997 Israel Grosman Mathematical Olympiad, 3
Find all real solutions of $\sqrt[4]{13+x}+ \sqrt[4]{14-x} = 3$.
2002 Junior Balkan Team Selection Tests - Romania, 1
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.
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 $a\sqrt{b}+ b\sqrt{a} = 126$.
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 $$f (1) + f (2) + f (3) + ... + f (2016).$$
1997 All-Russian Olympiad Regional Round, 10.8
Prove that if
$$\sqrt{x + a} +\sqrt{y+b}+\sqrt{z + c} =\sqrt{y + a} +\sqrt{z + b} +\sqrt{x + c} =\sqrt{z + a} +\sqrt{x+b}+\sqrt{y+c}$$
for some $a, b, c, x, y, z$, then $x = y = z$ or $a = b = c$.
III Soros Olympiad 1996 - 97 (Russia), 9.1
Is rational or irrational,the number
$$\left(\dfrac{2}{\sqrt[3]{25}+\sqrt[3]{15}+\sqrt[3]{9}}+\dfrac{1}{\sqrt[3]{9}+\sqrt[3]{6}+\sqrt[3]{4}}\right) \times \left(\sqrt[3]{25}+\sqrt[3]{10}+\sqrt[3]{4}\right)?$$
1939 Moscow Mathematical Olympiad, 046
Solve the equation $\sqrt{a-\sqrt{a+ x}} = x$ for $x$.
2016 India PRMO, 12
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 $n \ge 1$.
2001 Czech And Slovak Olympiad IIIA, 3
Find all triples of real numbers $(a,b,c)$ for which the set of solutions $x$ of $\sqrt{2x^2 +ax+b} > x-c$ is the set $(-\infty,0]\cup(1,\infty)$.
III Soros Olympiad 1996 - 97 (Russia), 9.9
What is the smallest value that the expression $$\sqrt{3x-2y-1}+\sqrt{2x+y+2}+\sqrt{3y-x}$$ can take?
III Soros Olympiad 1996 - 97 (Russia), 9.2
How many solutions, depending on the value of the parameter $a$, has the equation $$\sqrt{x^2-4}+\sqrt{2x^2-7x+5}=a ?$$
1987 Tournament Of Towns, (147) 4
For any natural $n$ prove the inequality
$$\sqrt{2\sqrt{2}{\sqrt{3}\sqrt{4 ...\sqrt{n-1\sqrt{n}}}}} <3$$
IV Soros Olympiad 1997 - 98 (Russia), 11.3
Solve the inequality
$$\sqrt{(x-2)^2(x-x^2)}<\sqrt{4x-1-(x^2-3x)^2}$$
2014 CHMMC (Fall), 5
Determine the value of
$$\prod^{\infty}_{n=1} 3^{n/3^n}= \sqrt[3]{3} \sqrt[3^2]{3^2} \sqrt[3^3]{3^3} ...$$
1949 Moscow Mathematical Olympiad, 161
Find the real roots of the equation $x^2 + 2ax + \frac{1}{16} = -a +\sqrt{ a^2 + x - \frac{1}{16} }$ , $\left(0 < a < \frac14 \right)$ .
2021 Auckland Mathematical Olympiad, 1
Solve the equation $\sqrt{x^2 - 4x + 13} + 1 = 2x$
2018 Ecuador NMO (OMEC), 6
Reduce $$\frac{2}{\sqrt{4-3\sqrt[4]{5} + 2\sqrt[4]{25}-\sqrt[4]{125}}}$$ to its lowest form.
Then generalize this result and show that it holds for any positive $n$.
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 .
2016 Irish Math Olympiad, 9
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
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 $a_i=b_i$ , $i=1,...,n$
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
$$\frac{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}>\frac14$$
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.
1979 Swedish Mathematical Competition, 2
Find rational $x$ in $(3,4)$ such that $\sqrt{x-3}$ and $\sqrt{x+1}$ are rational.