This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 95

1994 All-Russian Olympiad, 1
Tags: algebra , radical , equation
Prove that if $(x+\sqrt{x^2 +1}) (y+\sqrt{y^2 +1}) = 1$, then $x+y = 0$.

1948 Moscow Mathematical Olympiad, 153
Tags: maximum , radical , cube , 3d geometry , combinatorial geometry , circles , geometry
* What is the radius of the largest possible circle inscribed into a cube with side $a$?

2011 Dutch Mathematical Olympiad, 4
Tags: algebra , system of equations , radical
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 $a\sqrt{b}+ b\sqrt{a} = 126$.

2003 Junior Balkan Team Selection Tests - Romania, 3
Tags: diophantine , number theory , radical , inequalities
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} $

1990 Greece National Olympiad, 2
Tags: radical , algebra
Find all real solutions of $\sqrt{x-1}+\sqrt{x^2-1}=\sqrt{x^3}$

1945 Moscow Mathematical Olympiad, 104
Tags: nested radical , radical , trigonometry , algebra
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 $$2 \sin \left(1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^{n-1}}\right)\frac{\pi}{4}=2\cos \frac{\pi}{2^{n+1}}= \sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt2}}}$$

2016 Irish Math Olympiad, 9
Tags: algebra , integer , radical
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

2011 Saudi Arabia Pre-TST, 3
Tags: number theory , radical , integer
Find all integers $n \ge 2$ for which $\sqrt[n]{3^n+ 4^n+5^n+8^n+10^n}$ is an integer.

2015 Finnish National High School Mathematics Comp, 1
Tags: algebra , radical , equation
Solve the equation $\sqrt{1+\sqrt {1+x}}=\sqrt[3]{x}$ for $x \ge 0$.

1988 All Soviet Union Mathematical Olympiad, 486
Tags: sphere , tetrahedron , radical , geometric inequality , 3d geometry , geometry
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.

2002 Singapore Senior Math Olympiad, 3
Tags: diophantine , radical , number theory , diophantine equation
Prove that for natural numbers $p$ and $q$, there exists a natural number $x$ such that $$(\sqrt{p}+\sqrt{p-1})^q=\sqrt{x}+\sqrt{x-1}$$ (As an example, if $p = 3, q = 2$, then $x$ can be taken to be $25$.)

IV Soros Olympiad 1997 - 98 (Russia), 10.2
Tags: radical , algebra
Solve the equation $$\sqrt[3]{x^3+6x^2-6x-1}=\sqrt{x^2+4x+1}$$

1996 Tuymaada Olympiad, 5
Tags: algebra , radical , equation
Solve the equation $\sqrt{1981-\sqrt{1996+x}}=x+15$

IV Soros Olympiad 1997 - 98 (Russia), 10.7
Tags: radical , algebra
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 $\frac{D}{A}$

2021 Auckland Mathematical Olympiad, 3
Tags: algebra , radical , nested radical
For how many integers $n$ between $ 1$ and $2021$ does the infinite nested expression $$\sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}}$$ give a rational number?

1976 Chisinau City MO, 120
Tags: algebra , product , radical
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 $n> 3$.

2018 Hanoi Open Mathematics Competitions, 4
Tags: algebra , radical expressions , radical
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. $(4,\infty)$

2013 Czech-Polish-Slovak Junior Match, 1
Tags: diophantine equation , diophantine , number theory , radical
Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$

III Soros Olympiad 1996 - 97 (Russia), 9.9
Tags: radical , algebra , inequalities
What is the smallest value that the expression $$\sqrt{3x-2y-1}+\sqrt{2x+y+2}+\sqrt{3y-x}$$ can take?

2016 Costa Rica - Final Round, F2
Tags: algebra , sum , radical
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 $$f (1) + f (2) + f (3) + ... + f (2016).$$

2008 Dutch IMO TST, 4
Tags: perfect square , number theory , radical
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.

1949 Moscow Mathematical Olympiad, 161
Tags: radical , equation , algebra , parameter
Find the real roots of the equation $x^2 + 2ax + \frac{1}{16} = -a +\sqrt{ a^2 + x - \frac{1}{16} }$ , $\left(0 < a < \frac14 \right)$ .

IV Soros Olympiad 1997 - 98 (Russia), 9.2
Tags: algebra , radical
Solve the equation $$2\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)(x+5)}}}}=x$$

2007 Switzerland - Final Round, 7
Tags: inequalities , algebra , radical
Let $a, b, c$ be nonnegative real numbers with arithmetic mean $m =\frac{a+b+c}{3}$ . Provethat $$\sqrt{a+\sqrt{b + \sqrt{c}}} +\sqrt{b+\sqrt{c + \sqrt{a}}} +\sqrt{c +\sqrt{a + \sqrt{b}}}\le 3\sqrt{m+\sqrt{m + \sqrt{m}}}.$$

1990 Spain Mathematical Olympiad, 4
Tags: algebra , radical
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.