Olympiad problems

2017 Denmark MO - Mohr Contest, 4

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 $$x =\sqrt{A +\sqrt{B +\sqrt{C +\sqrt{D + x}}}}.$$

V Soros Olympiad 1998 - 99 (Russia), 10.5

Tags: algebra , radical

Solve the equation $$\sqrt{2+\sqrt{2-\sqrt{2+x}}}=x.$$

1998 Singapore Senior Math Olympiad, 3

Tags: algebra , inequalities , radical

Prove that $\sqrt1+ \sqrt2+\sqrt3+...+ \sqrt{n^2-1}+\sqrt{n^2} \ge \frac{2n^3+n}{3}$ for any positive integer $n$.

1992 Tournament Of Towns, (326) 3

Tags: algebra , inequalities , compare , radical

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)

2016 Hanoi Open Mathematics Competitions, 14

Tags: algebra , radical , natural

Given natural numbers $a,b$ such that $2015a^2+a = 2016b^2+b$. Prove that $\sqrt{a-b}$ is a natural number.

2015 Finnish National High School Mathematics Comp, 1

Tags: equation , algebra , radical

Solve the equation $\sqrt{1+\sqrt {1+x}}=\sqrt[3]{x}$ for $x \ge 0$.

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. $(4,\infty)$

1990 Greece National Olympiad, 2

Tags: algebra , radical

Find all real solutions of $\sqrt{x-1}+\sqrt{x^2-1}=\sqrt{x^3}$

1990 Spain Mathematical Olympiad, 1

Tags: algebra , radical

Prove that $\sqrt{x}+\sqrt{y}+\sqrt{xy}$ is equal to$ \sqrt{x}+\sqrt{y+xy+2y\sqrt{x}}$ and compare the numbers $\sqrt{3}+\sqrt{10+2\sqrt{3}}$ and $\sqrt{5+\sqrt{22}}+\sqrt{8- \sqrt{22}+2\sqrt{15-3\sqrt{22}}}$.

2010 Bundeswettbewerb Mathematik, 2

Tags: floor function , algebra , recurrence relation , sequence , radical

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.

2001 Swedish Mathematical Competition, 2

Tags: algebra , rational , radical

Show that $\sqrt[3]{\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5}$ is rational.

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.

2017 India PRMO, 7

Tags: algebra , radical , inequalities

Find the number of positive integers $n$, such that $\sqrt{n} + \sqrt{n + 1} < 11$.

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 $$\sqrt{a+\sqrt{b + \sqrt{c}}} +\sqrt{b+\sqrt{c + \sqrt{a}}} +\sqrt{c +\sqrt{a + \sqrt{b}}}\le 3\sqrt{m+\sqrt{m + \sqrt{m}}}.$$

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.

1996 Romania National Olympiad, 2

Tags: algebra , radical

Find all real numbers $x$ for which the following equality holds : $$\sqrt{\frac{x-7}{1989}}+\sqrt{\frac{x-6}{1990}}+\sqrt{\frac{x-5}{1991}}=\sqrt{\frac{x-1989}{7}}+\sqrt{\frac{x-1990}{6}}+\sqrt{\frac{x-1991}{5}}$$

2002 Singapore Senior Math Olympiad, 3

Tags: number theory , diophantine , radical , 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 $25$.)

2013 Czech-Polish-Slovak Junior Match, 1

Tags: number theory , diophantine equation , diophantine , radical

Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$

2017 Latvia Baltic Way TST, 13

Tags: rational , radical , algebra

Prove that the number $$\sqrt{1 + \frac{1}{n^2} + \frac{1}{(n+1)^2}}$$ is rational for all natural $n$.

1978 Chisinau City MO, 154

Tags: algebra , radical , compare

What's more $\sqrt[4]{7}+\sqrt[4]{11}$ or $2\sqrt{\frac{\sqrt{7}+\sqrt{11}}{2}}$ ?