Olympiad problems

2018 Ecuador NMO (OMEC), 6

Tags: algebra , radical

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 $n$.

II Soros Olympiad 1995 - 96 (Russia), 9.6

Tags: radical , algebra

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

1982 Tournament Of Towns, (027) 1

Tags: floor function , radical , logarithm , sum , algebra

Prove that for all natural numbers $n$ greater than $1$ : $$[\sqrt{n}] + [\sqrt[3]{n}] +...+[ \sqrt[n]{n}] = [\log_2 n] + [\log_3 n] + ... + [\log_n n]$$ (VV Kisil)

2009 Greece JBMO TST, 3

Tags: algebra , rational , integration , radical

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 .

IV Soros Olympiad 1997 - 98 (Russia), 11.3

Tags: algebra , inequalities , radical

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

III Soros Olympiad 1996 - 97 (Russia), 9.1

Tags: radical , algebra

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

2005 iTest, 23

Tags: algebra , radical

$\sqrt[3]{x+\sqrt[3]{x+\sqrt[3]{x+ \sqrt[3]{x ...}}}}= 8$. Find $x$.

2001 Rioplatense Mathematical Olympiad, Level 3, 3

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

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 ${{S}_{2001}}=2\,{{S}_{2000}}+1$. .

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 $n$.

2014 CHMMC (Fall), 5

Tags: algebra , radical

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

1978 Chisinau City MO, 154

Tags: radical , compare , algebra

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

2007 Greece JBMO TST, 4

Tags: algebra , sum , radical

Calculate the sum $$S=\sqrt{1+\frac{8\cdot 1^2-1}{1^2\cdot 3^2}}+\sqrt{1+\frac{8\cdot 2^2-1}{3^2\cdot 5^2}}+...+ \sqrt{1+\frac{8\cdot 1003^2-1}{2005^2\cdot 2007^2}}$$

1987 Tournament Of Towns, (147) 4

Tags: inequalities , radical , algebra

For any natural $n$ prove the inequality $$\sqrt{2\sqrt{2}{\sqrt{3}\sqrt{4 ...\sqrt{n-1\sqrt{n}}}}} <3$$

1949-56 Chisinau City MO, 53

Tags: radical , algebra , equation

Solve the equation: $\sqrt[3]{a+\sqrt{x}}+\sqrt[3]{a-\sqrt{x}}=\sqrt[3]{b}$

1964 Swedish Mathematical Competition, 3

Tags: radical , integer polynomial , polynomial , algebra

Find a polynomial with integer coefficients which has $\sqrt2 + \sqrt3$ and $\sqrt2 + \sqrt[3]{3}$ as roots.

2009 German National Olympiad, 4

Tags: inequalities , inequality , algebra , radical

Let $a$ and $b$ be two fixed positive real numbers. Find all real numbers $x$, such that inequality holds $$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$$

1964 All Russian Mathematical Olympiad, 046

Tags: number theory , algebra , radical

Find integer solutions $(x,y)$ of the equation ($1964$ times "$\sqrt{}$"): $$\sqrt{x+\sqrt{x+\sqrt{....\sqrt{x+\sqrt{x}}}}}=y$$

1996 Swedish Mathematical Competition, 3

Tags: algebra , radical , sequence

For every positive integer $n$, we define the function $p_n$ for $x\ge 1$ by $$p_n(x) = \frac12 \left(\left(x+\sqrt{x^2-1}\right)^n+\left(x-\sqrt{x^2-1}\right)^n\right).$$ Prove that $p_n(x) \ge 1$ and that $p_{mn}(x) = p_m(p_n(x))$.

III Soros Olympiad 1996 - 97 (Russia), 9.2

Tags: algebra , parameter , radical

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

1965 Czech and Slovak Olympiad III A, 3

Tags: algebra , parameter , radical , equation , real root

Find all real roots $x$ of the equation $$\sqrt{x^2-2x-1}+\sqrt{x^2+2x-1}=p,$$ where $p$ is a real parameter.

IV Soros Olympiad 1997 - 98 (Russia), 11.8

Tags: radical , algebra

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

1985 Swedish Mathematical Competition, 1

Tags: inequalities , algebra , radical

If $a > b > 0$, prove the inequality $$\frac{(a-b)^2}{8a}< \frac{a+b}{2}- \sqrt{ab} < \frac{(a-b)^2}{8b}.$$

1996 Spain Mathematical Olympiad, 4

Tags: algebra , equation , radical

For each real value of $p$, find all real solutions of the equation $\sqrt{x^2 - p}+2\sqrt{x^2-1} = x$.

2012 Hanoi Open Mathematics Competitions, 4

Tags: radical , algebra , floor function

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.

2005 Estonia National Olympiad, 4

Tags: radical , algebra

Represent the number $\sqrt[3]{1342\sqrt{167}+2005}$ in the form where it contains only addition, subtraction, multiplication, division and square roots.