Olympiad problems

III Soros Olympiad 1996 - 97 (Russia), 9.1

Tags: radical , algebra

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

1998 Chile National Olympiad, 3

Tags: radical , nested radical , algebra

Evaluate $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}}$.

1988 All Soviet Union Mathematical Olympiad, 486

Tags: 3d geometry , geometric inequality , geometry , sphere , radical , tetrahedron

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.

2019 Saudi Arabia JBMO TST, 1

Tags: sum , radical , inequalities

Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove that $$\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c} \ge 2\sqrt2 \left( \sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}}\right)$$

1996 Tuymaada Olympiad, 5

Tags: radical , equation , algebra

Solve the equation $\sqrt{1981-\sqrt{1996+x}}=x+15$

1994 All-Russian Olympiad, 1

Tags: equation , radical , algebra

Prove that if $(x+\sqrt{x^2 +1}) (y+\sqrt{y^2 +1}) = 1$, then $x+y = 0$.

2008 Dutch IMO TST, 4

Tags: perfect square , radical , number theory

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.

2009 Grand Duchy of Lithuania, 3

Tags: algebra , radical

Solve the equation $x^2+ 2 = 4\sqrt{x^3+1}$

1998 Tuymaada Olympiad, 2

Tags: radical , equation , algebra

Solve the equation $(x^3-1000)^{1/2}=(x^2+100)^{1/3}$

2012 Hanoi Open Mathematics Competitions, 4

Tags: radical , floor function , algebra

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.

1985 Swedish Mathematical Competition, 1

Tags: radical , inequalities , algebra

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

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) $10^{-200}$

IV Soros Olympiad 1997 - 98 (Russia), 10.7

Tags: algebra , radical

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 $\frac{D}{A}$

1988 Swedish Mathematical Competition, 6

Tags: sequence , recurrence relation , radical , algebra , inequalities

The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}$ for $n \ge 1$. Prove that there exists $a$ such that $\frac{1}{2} \le \frac{a_n}{n^a} \le 2$ for $n \ge 1$.

1984 All Soviet Union Mathematical Olympiad, 372

Tags: radical , inequalities , algebra

Prove that every positive $a$ and $b$ satisfy inequality $$\frac{(a+b)^2}{2} + \frac{a+b}{4} \ge a\sqrt b + b\sqrt a$$

1977 Vietnam National Olympiad, 1

Tags: radical , algebra , inequalities

Find all real $x$ such that $ \sqrt{x - \frac{1}{x}} + \sqrt{1 - \frac{1}{x}}> \frac{x - 1}{x}$

2021 Auckland Mathematical Olympiad, 3

Tags: nested radical , radical , algebra

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?

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 $$2 \sin \left(1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^{n-1}}\right)\frac{\pi}{4}=2\cos \frac{\pi}{2^{n+1}}= \sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt2}}}$$