Olympiad problems

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 $n$.

1996 Romania National Olympiad, 2

Tags: algebra , Radicals

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

2006 AIME Problems, 5

Tags: USAMTS , AMC , Radicals

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 $a\cdot b\cdot c$.

III Soros Olympiad 1996 - 97 (Russia), 9.1

Tags: algebra , Radicals

Without using a calculator, find out which number is greater: $$|\sqrt[3]{5}-\sqrt3|-\sqrt3| \,\,\,\, \text{or} \,\,\,\, 0.01$$

VI Soros Olympiad 1999 - 2000 (Russia), 11.1

Tags: algebra , system of equations , Radicals , trigonometry

Solve the system of equations $$\begin{cases} x^2+arc siny =y^2+arcsin x \\ x^2+y^2-3x=2y\sqrt{x^2-2x-y}+1 \end{cases}$$

III Soros Olympiad 1996 - 97 (Russia), 9.2

Tags: algebra , Radicals , parameter

How many solutions, depending on the value of the parameter $a$, has the equation $$\sqrt{x^2-4}+\sqrt{2x^2-7x+5}=a ?$$

2014 CHMMC (Fall), 5

Tags: Radicals , algebra , CHMMC

Determine the value of $$\prod^{\infty}_{n=1} 3^{n/3^n}= \sqrt[3]{3} \sqrt[3^2]{3^2} \sqrt[3^3]{3^3} ...$$

2012 Hanoi Open Mathematics Competitions, 4

Tags: algebra , floor function , Radicals

What is the largest integer less than or equal to $4x^3 - 3x$, where $x=\frac{\sqrt[3]{2+\sqrt3}+\sqrt[3]{2-\sqrt3}}{2}$ ? (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) None of the above.

V Soros Olympiad 1998 - 99 (Russia), 10.5

Tags: algebra , Radicals

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

III Soros Olympiad 1996 - 97 (Russia), 9.1

Tags: algebra , Rational Root Theorem , Radicals

Is rational or irrational,the number $$\left(\dfrac{2}{\sqrt[3]{25}+\sqrt[3]{15}+\sqrt[3]{9}}+\dfrac{1}{\sqrt[3]{9}+\sqrt[3]{6}+\sqrt[3]{4}}\right) \times \left(\sqrt[3]{25}+\sqrt[3]{10}+\sqrt[3]{4}\right)?$$

IV Soros Olympiad 1997 - 98 (Russia), 11.3

Tags: algebra , inequalities , Radicals

Solve the inequality $$\sqrt{(x-2)^2(x-x^2)}<\sqrt{4x-1-(x^2-3x)^2}$$

IV Soros Olympiad 1997 - 98 (Russia), 9.1

Tags: algebra , Radicals

Solve the equation $$2(x-6)=\dfrac{x^2}{(1+\sqrt{x+1})^2}$$

IV Soros Olympiad 1997 - 98 (Russia), 11.8

Tags: algebra , Radicals

Calculate $\sqrt{5,44...4}$ (the decimal point is followed by $100$ fours) with approximation to: a) $10^{-100}$, b) $10^{-200}$

IV Soros Olympiad 1997 - 98 (Russia), 9.2

Tags: algebra , Radicals

Solve the equation $$2\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)(x+5)}}}}=x$$

VI Soros Olympiad 1999 - 2000 (Russia), 9.1

Tags: algebra , Radicals

Which of the two numbers is bigger : $\sqrt{1997}+2\sqrt{1999} + 2\sqrt{2001} + \sqrt{2003}$ or $2\sqrt{1998} +2\sqrt{2000}+2\sqrt{2002}$ ?

II Soros Olympiad 1995 - 96 (Russia), 9.6

Tags: algebra , Radicals

Without using a calculator (especially a computer), find out what is more: $$\sqrt[3]{5\sqrt{13}+18}- \sqrt[3]{2\sqrt{13}+5} \,\,\, or \,\,\, 1 $$

IV Soros Olympiad 1997 - 98 (Russia), 10.7

Tags: algebra , Radicals

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 $\frac{D}{A}$