Olympiad problems

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Found problems: 95

1988 All Soviet Union Mathematical Olympiad, 486

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

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.

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

1996 Tuymaada Olympiad, 5

Tags: algebra , radical , equation

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

1977 Vietnam National Olympiad, 1

Tags: inequalities , algebra , radical

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

V Soros Olympiad 1998 - 99 (Russia), 10.5

Tags: algebra , radical

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

2004 Thailand Mathematical Olympiad, 5

Tags: algebra , sum , equation , radical

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

1964 All Russian Mathematical Olympiad, 046

Tags: number theory , algebra , radical

Find integer solutions $(x,y)$ of the equation ($1964$ times "$\sqrt{}$"): $$\sqrt{x+\sqrt{x+\sqrt{....\sqrt{x+\sqrt{x}}}}}=y$$

2009 German National Olympiad, 4

Tags: inequalities , inequality , algebra , radical

Let $a$ and $b$ be two fixed positive real numbers. Find all real numbers $x$, such that inequality holds $$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$$

1985 Swedish Mathematical Competition, 1

Tags: inequalities , algebra , radical

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

1996 Spain Mathematical Olympiad, 4

Tags: algebra , equation , radical

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

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 $n \ge 1$.

2001 Czech And Slovak Olympiad IIIA, 3

Tags: radical , algebra , inequalities

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 $(-\infty,0]\cup(1,\infty)$.

1990 Greece National Olympiad, 2

Tags: algebra , radical

Find all real solutions of $\sqrt{x-1}+\sqrt{x^2-1}=\sqrt{x^3}$

III Soros Olympiad 1996 - 97 (Russia), 10.3

Tags: algebra , radical

Solve the equation $$\sqrt{x(x+7)}+\sqrt{(x+7)(x+17)}+\sqrt{(x+17)(x+24)}=12+17\sqrt2$$

III Soros Olympiad 1996 - 97 (Russia), 9.9

Tags: algebra , inequalities , radical

What is the smallest value that the expression $$\sqrt{3x-2y-1}+\sqrt{2x+y+2}+\sqrt{3y-x}$$ can take?

IV Soros Olympiad 1997 - 98 (Russia), 9.2

Tags: algebra , radical

Solve the equation $$2\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)(x+5)}}}}=x$$

1949 Moscow Mathematical Olympiad, 161

Tags: radical , equation , algebra , parameter

Find the real roots of the equation $x^2 + 2ax + \frac{1}{16} = -a +\sqrt{ a^2 + x - \frac{1}{16} }$ , $\left(0 < a < \frac14 \right)$ .

1998 Tuymaada Olympiad, 2

Tags: algebra , radical , equation

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

2021 Auckland Mathematical Olympiad, 1

Tags: algebra , radical

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

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 $\frac{D}{A}$

2017 Latvia Baltic Way TST, 13

Tags: algebra , rational , radical

Prove that the number $$\sqrt{1 + \frac{1}{n^2} + \frac{1}{(n+1)^2}}$$ is rational for all natural $n$.

IV Soros Olympiad 1997 - 98 (Russia), 10.4

Tags: algebra , radical

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

2002 Singapore Senior Math Olympiad, 3

Tags: diophantine , radical , number theory , diophantine equation

Prove that for natural numbers $p$ and $q$, there exists a natural number $x$ such that $$(\sqrt{p}+\sqrt{p-1})^q=\sqrt{x}+\sqrt{x-1}$$ (As an example, if $p = 3, q = 2$, then $x$ can be taken to be $25$.)

2014 CHMMC (Fall), 5

Tags: algebra , radical

Determine the value of $$\prod^{\infty}_{n=1} 3^{n/3^n}= \sqrt[3]{3} \sqrt[3^2]{3^2} \sqrt[3^3]{3^3} ...$$

1939 Moscow Mathematical Olympiad, 046

Tags: radical , parameter , equation , algebra

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