Olympiad problems

1996 Romania National Olympiad, 2

Tags: radical , algebra

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

IV Soros Olympiad 1997 - 98 (Russia), 10.4

Tags: radical , algebra

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

1950 Moscow Mathematical Olympiad, 180

Tags: equation , radical , algebra

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

2020 Canadian Mathematical Olympiad Qualification, 1

Tags: floor function , radical , algebra

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

1990 Spain Mathematical Olympiad, 1

Tags: algebra , radical

Prove that $\sqrt{x}+\sqrt{y}+\sqrt{xy}$ is equal to$ \sqrt{x}+\sqrt{y+xy+2y\sqrt{x}}$ and compare the numbers $\sqrt{3}+\sqrt{10+2\sqrt{3}}$ and $\sqrt{5+\sqrt{22}}+\sqrt{8- \sqrt{22}+2\sqrt{15-3\sqrt{22}}}$.

III Soros Olympiad 1996 - 97 (Russia), 9.1

Tags: radical , algebra , rational root theorem

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

1998 Tuymaada Olympiad, 2

Tags: algebra , radical , equation

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

2016 India PRMO, 12

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

Let $S = 1 + \frac{1}{\sqrt2}+ \frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+ \frac{1}{\sqrt{99}}+ \frac{1}{\sqrt{100}}$ . Find $[S]$. You may use the fact that $\sqrt{n} < \frac12 (\sqrt{n} +\sqrt{n+1}) <\sqrt{n+1}$ for all integers $n \ge 1$.

2001 Czech And Slovak Olympiad IIIA, 3

Tags: inequalities , radical , algebra

Find all triples of real numbers $(a,b,c)$ for which the set of solutions $x$ of $\sqrt{2x^2 +ax+b} > x-c$ is the set $(-\infty,0]\cup(1,\infty)$.

IV Soros Olympiad 1997 - 98 (Russia), 9.1

Tags: algebra , radical

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

VI Soros Olympiad 1999 - 2000 (Russia), 9.1

Tags: radical , algebra

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

1939 Moscow Mathematical Olympiad, 046

Tags: radical , parameter , equation , algebra

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

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

2004 Thailand Mathematical Olympiad, 4

Tags: algebra , radical , equation

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

2020 Canadian Mathematical Olympiad Qualification, 8

Tags: radical , rational , equation , algebra

Find all pairs $(a, b)$ of positive rational numbers such that $\sqrt[b]{a}= ab$

1975 Chisinau City MO, 86

Tags: algebra , radical

What is the number $x =\sqrt{4+\sqrt7}-\sqrt{4-\sqrt7}-\sqrt2$, positive, negative or zero?

1997 All-Russian Olympiad Regional Round, 10.8

Tags: radical , algebra

Prove that if $$\sqrt{x + a} +\sqrt{y+b}+\sqrt{z + c} =\sqrt{y + a} +\sqrt{z + b} +\sqrt{x + c} =\sqrt{z + a} +\sqrt{x+b}+\sqrt{y+c}$$ for some $a, b, c, x, y, z$, then $x = y = z$ or $a = b = c$.

1988 Swedish Mathematical Competition, 6

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

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 $n \ge 1$.

2021 Auckland Mathematical Olympiad, 1

Tags: algebra , radical

Solve the equation $\sqrt{x^2 - 4x + 13} + 1 = 2x$

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.

2010 Bundeswettbewerb Mathematik, 2

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

The sequence of numbers $a_1, a_2, a_3, ...$ is defined recursively by $a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor $ for $n \ge 1$. Find all numbers that appear more than twice at this sequence.

1997 Israel Grosman Mathematical Olympiad, 3

Tags: radical , equation , algebra

Find all real solutions of $\sqrt[4]{13+x}+ \sqrt[4]{14-x} = 3$.

1984 All Soviet Union Mathematical Olympiad, 372

Tags: inequalities , algebra , radical

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

1998 Chile National Olympiad, 3

Tags: radical , nested radical , algebra

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

1998 Singapore Senior Math Olympiad, 3

Tags: inequalities , radical , algebra

Prove that $\sqrt1+ \sqrt2+\sqrt3+...+ \sqrt{n^2-1}+\sqrt{n^2} \ge \frac{2n^3+n}{3}$ for any positive integer $n$.