Olympiad problems

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Found problems: 95

2021 Auckland Mathematical Olympiad, 3

Tags: nested radical , algebra , 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?

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)

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.

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. $(4,\infty)$

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.

2007 Greece JBMO TST, 4

Tags: algebra , sum , radical

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

2001 Swedish Mathematical Competition, 2

Tags: algebra , rational , radical

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

1987 Tournament Of Towns, (147) 4

Tags: inequalities , radical , algebra

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

2015 Finnish National High School Mathematics Comp, 1

Tags: algebra , radical , equation

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

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.

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

1978 Chisinau City MO, 154

Tags: radical , compare , algebra

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

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 $p_{mn}(x) = p_m(p_n(x))$.

2003 Junior Balkan Team Selection Tests - Romania, 3

Tags: diophantine , radical , inequalities , number theory

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} $

1950 Moscow Mathematical Olympiad, 180

Tags: equation , radical , algebra

Solve the equation $\sqrt {x + 3 - 4 \sqrt{x -1}} +\sqrt{x + 8 - 6 \sqrt{x - 1}}= 1$.

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) $10^{-200}$

1948 Moscow Mathematical Olympiad, 153

Tags: maximum , radical , cube , 3d geometry , combinatorial geometry , geometry , circles

* What is the radius of the largest possible circle inscribed into a cube with side $a$?

1985 Poland - Second Round, 4

Tags: algebra , rational , radical , perfect cube , number theory

Prove that if for natural numbers $ a, b $ the number $ \sqrt[3]{a} + \sqrt[3]{b} $ is rational, then $ a, b $ are cubes of natural numbers.

2011 Saudi Arabia Pre-TST, 3

Tags: radical , integer , number theory

Find all integers $n \ge 2$ for which $\sqrt[n]{3^n+ 4^n+5^n+8^n+10^n}$ is an integer.

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 $n$.

VI Soros Olympiad 1999 - 2000 (Russia), 11.1

Tags: algebra , system of equations , trigonometry , radical

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

2006 Thailand Mathematical Olympiad, 8

Tags: algebra , radical , cubic

Let $a, b, c$ be the roots of the equation $x^3-9x^2+11x-1 = 0$, and define $s =\sqrt{a}+\sqrt{b}+\sqrt{c}$. Compute $s^4 -18s^2 - 8s$ .

2017 Denmark MO - Mohr Contest, 4

Tags: algebra , radical , digit

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

IV Soros Olympiad 1997 - 98 (Russia), 9.1

Tags: algebra , radical

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

2002 Junior Balkan Team Selection Tests - Romania, 1

Tags: rational , algebra , radical , irrational

Let $a$ be an integer. Prove that for any real number $x, x^3 < 3$, both the numbers $\sqrt{3 -x^2}$ and $\sqrt{a - x^3}$ cannot be rational.