Olympiad problems

VI Soros Olympiad 1999 - 2000 (Russia), 11.1

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

2004 Thailand Mathematical Olympiad, 4

Tags: equation , radical , algebra

Find all real solutions $x$ to the equation $$x =\sqrt{x -\frac{1}{x}} +\sqrt{1 -\frac{1}{x}}$$

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.

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), 10.4

Tags: radical , algebra

Solve the equation $$ \sqrt{\sqrt{2x^2+x-3}+2x^2-3}=x.$$

IV Soros Olympiad 1997 - 98 (Russia), 9.1

Tags: radical , algebra

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

2006 Thailand Mathematical Olympiad, 8

Tags: radical , algebra , 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 $s^4 -18s^2 - 8s$ .

1996 Swedish Mathematical Competition, 3

Tags: radical , sequence , algebra

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

1998 ITAMO, 1

Tags: algebra , sum , radical

Calculate the sum $\sum_{n=1}^{1.000.000}[ \sqrt{n} ]$ . You may use the formula $\sum_{i=1}^{k} i^2=\frac{k(k +1)(2k +1)}{6}$ without a proof.

1982 Tournament Of Towns, (027) 1

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

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)

2005 Estonia National Olympiad, 4

Tags: algebra , radical

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

2005 iTest, 23

Tags: radical , algebra

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

2020 Canadian Mathematical Olympiad Qualification, 1

Tags: radical , algebra , floor function

Show that for all integers $a \ge 1$,$ \lfloor \sqrt{a}+\sqrt{a+1}+\sqrt{a+2}\rfloor = \lfloor \sqrt{9a+8}\rfloor$

2006 AIME Problems, 5

Tags: radical

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 $a\cdot b\cdot c$.

1996 Spain Mathematical Olympiad, 4

Tags: equation , radical , algebra

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

1949-56 Chisinau City MO, 53

Tags: equation , radical , algebra

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

2004 Thailand Mathematical Olympiad, 5

Tags: equation , algebra , sum , radical

Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =|2nx - n - 2n^2|$

2007 Greece JBMO TST, 4

Tags: radical , sum , algebra

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 Rioplatense Mathematical Olympiad, Level 3, 3

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

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

VI Soros Olympiad 1999 - 2000 (Russia), 11.1

2004 Thailand Mathematical Olympiad, 4

1964 Swedish Mathematical Competition, 3

1950 Moscow Mathematical Olympiad, 180

IV Soros Olympiad 1997 - 98 (Russia), 10.4

IV Soros Olympiad 1997 - 98 (Russia), 9.1

2006 Thailand Mathematical Olympiad, 8

1996 Swedish Mathematical Competition, 3

1998 ITAMO, 1

1982 Tournament Of Towns, (027) 1

2005 Estonia National Olympiad, 4

2005 iTest, 23

2020 Canadian Mathematical Olympiad Qualification, 1

2006 AIME Problems, 5

1996 Spain Mathematical Olympiad, 4

1949-56 Chisinau City MO, 53

2004 Thailand Mathematical Olympiad, 5

2007 Greece JBMO TST, 4

2001 Rioplatense Mathematical Olympiad, Level 3, 3

VI Soros Olympiad 1999 - 2000 (Russia), 9.1

1965 Czech and Slovak Olympiad III A, 3

1985 Poland - Second Round, 4

2009 German National Olympiad, 4

2020 Canadian Mathematical Olympiad Qualification, 8

IV Soros Olympiad 1997 - 98 (Russia), 10.2