Olympiad problems

1975 Chisinau City MO, 86

Tags: algebra , radical

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

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.

1998 ITAMO, 1

Tags: sum , algebra , 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.

1997 All-Russian Olympiad Regional Round, 10.8

Tags: algebra , radical

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 $a = b = c$.

1984 All Soviet Union Mathematical Olympiad, 372

Tags: inequalities , algebra , radical

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

2016 Irish Math Olympiad, 9

Tags: algebra , integer , radical

Show that the number $a^3$ where $a=\frac{251}{ \frac{1}{\sqrt[3]{252}-5\sqrt[3]{2}}-10\sqrt[3]{63}}+\frac{1}{\frac{251}{\sqrt[3]{252}+5\sqrt[3]{2}}+10\sqrt[3]{63}}$ is an integer and find its value

2016 India PRMO, 12

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

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 $n \ge 1$.

1998 Singapore Senior Math Olympiad, 3

Tags: radical , algebra , inequalities

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

III Soros Olympiad 1996 - 97 (Russia), 9.1

Tags: radical , algebra

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

2019 Saudi Arabia JBMO TST, 1

Tags: inequalities , sum , radical

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

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$

2017 India PRMO, 7

Tags: inequalities , radical , algebra

Find the number of positive integers $n$, such that $\sqrt{n} + \sqrt{n + 1} < 11$.

1990 Spain Mathematical Olympiad, 1

Tags: radical , algebra

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: algebra , rational root theorem , radical

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

2012 Hanoi Open Mathematics Competitions, 4

Tags: floor function , radical , 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.

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 $$\sqrt{x^2-4}+\sqrt{2x^2-7x+5}=a ?$$

1953 Moscow Mathematical Olympiad, 238

Tags: inequalities , radical , nested radical , algebra

Prove that if in the following fraction we have $n$ radicals in the numerator and $n - 1$ in the denominator, then $$\frac{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}>\frac14$$

2004 Thailand Mathematical Olympiad, 4

Tags: algebra , radical , equation

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

2005 iTest, 23

Tags: radical , algebra

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

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)