This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 23

1959 IMO Shortlist, 2
Tags: square root , algebra , equation
For what real values of $x$ is \[ \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A \] given a) $A=\sqrt{2}$; b) $A=1$; c) $A=2$, where only non-negative real numbers are admitted for square roots?

2012 German National Olympiad, 4
Tags: real number , square root , inequalities , algebra
Let $a,b$ be positive real numbers and $n\geq 2$ a positive integer. Prove that if $x^n \leq ax+b$ holds for a positive real number $x$, then it also satisfies the inequality $x < \sqrt[n-1]{2a} + \sqrt[n]{2b}.$

1959 IMO, 2
Tags: square root , equation , algebra
For what real values of $x$ is \[ \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A \] given a) $A=\sqrt{2}$; b) $A=1$; c) $A=2$, where only non-negative real numbers are admitted for square roots?

2023 Brazil EGMO Team Selection Test, 3
Tags: square root , jensen , square root inequality , inequalities
Let $a_1, a_2, \ldots , a_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = 1$. Prove that $$\dfrac{a_1}{\sqrt{1-a_1}}+\cdots+\dfrac{a_n}{\sqrt{1-a_n}} \geq \dfrac{1}{\sqrt{n-1}}(\sqrt{a_1}+\cdots+\sqrt{a_n}).$$

2013 Bosnia And Herzegovina - Regional Olympiad, 3
Tags: square root , rational , number theory
Find all integers $a$ such that $\sqrt{\frac{9a+4}{a-6}}$ is rational number

1985 Traian Lălescu, 1.1
Tags: equation , square root
Solve the equation $ \frac{\sqrt{2+x} +\sqrt{2-x}}{\sqrt{2+x} -\sqrt{2-x}} =\sqrt 3. $

2020 HK IMO Preliminary Selection Contest, 17
Tags: algebra , square root , system of equations
How many positive integer solutions does the following system of equations have? $$\begin{cases}\sqrt{2020}(\sqrt{a}+\sqrt{b})=\sqrt{(c+2020)(d+2020)}\\\sqrt{2020}(\sqrt{b}+\sqrt{c})=\sqrt{(d+2020)(a+2020)}\\\sqrt{2020}(\sqrt{c}+\sqrt{d})=\sqrt{(a+2020)(b+2020)}\\\sqrt{2020}(\sqrt{d}+\sqrt{a})=\sqrt{(b+2020)(c+2020)}\\ \end{cases}$$

2020 Malaysia IMONST 1, 6
Tags: square root , sum
Find the sum of all integers between $-\sqrt {1442}$ and $\sqrt{2020}$.

2016 Japan MO Preliminary, 1
Tags: number theory , square root
Calculate the value of $\sqrt{\dfrac{11^4+100^4+111^4}{2}}$ and answer in the form of an integer.

2019 Azerbaijan Senior NMO, 1
Tags: square root , algebra
Solve the following equation $$\sqrt{\frac{x^2}3-ax+a^2}+\sqrt{\frac{x^2}3-bx+b^2}=\sqrt{a^2-ab+b^2}$$ where $a;b\in\mathbb{R^+}$

2006 Spain Mathematical Olympiad, 3
Tags: area , square root , diagonal , convex quadrilateral , geometry , geometric inequality
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$. Denotes by $S_1,S_2$ and $S$ the areas of the triangles $ABE$, $CDE$ and the quadrilateral $ABCD$ respectively. Prove that $\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S}$ . When equality is reached?

2014 BAMO, 3
Tags: square root , algebra , polynomial
Suppose that for two real numbers $x$ and $y$ the following equality is true: $$(x+ \sqrt{1+ x^2})(y+\sqrt{1+y^2})=1$$ Find (with proof) the value of $x+y$.

Russian TST 2017, P1
Tags: algebra , square root
Prove that $\sqrt{a_1}+\sqrt{a_2}+\cdots+\sqrt{a_{119}}$ is an integer, where \[a_n=2-\frac{1}{n^2+\sqrt{n^4+1/4}}.\]

1953 Putnam, A1
Tags: inequality , square root
Prove that for every positive integer $n$ $$ \frac{2}{3} n \sqrt{n} < \sqrt{1} + \sqrt{2} +\ldots +\sqrt{n} < \frac{4n+3}{6} \sqrt{n}.$$

2004 German National Olympiad, 4
Tags: integer , sum , square root , algebra
For a positive integer $n,$ let $a_n$ be the integer closest to $\sqrt{n}.$ Compute $$ \frac{1}{a_1 } + \frac{1}{a_2 }+ \cdots + \frac{1}{a_{2004}}.$$

1978 Austrian-Polish Competition, 8
Tags: limit , square root , algebra
For any positive integer $k$ consider the sequence $$a_n=\sqrt{k+\sqrt{k+\dots+\sqrt k}},$$ where there are $n$ square-root signs on the right-hand side. (a) Show that the sequence converges, for every fixed integer $k\ge 1$. (b) Find $k$ such that the limit is an integer. Furthermore, prove that if $k$ is odd, then the limit is irrational.

2016 Japan Mathematical Olympiad Preliminary, 1
Tags: square root , number theory
Calculate the value of $\sqrt{\dfrac{11^4+100^4+111^4}{2}}$ and answer in the form of an integer.

2020 OMMock - Mexico National Olympiad Mock Exam, 1
Tags: inequalities , algebra , inequality , square root
Let $a$, $b$, $c$ and $d$ positive real numbers with $a > c$ and $b < d$. Assume that \[a + \sqrt{b} \ge c + \sqrt{d} \qquad \text{and} \qquad \sqrt{a} + b \le \sqrt{c} + d\] Prove that $a + b + c + d > 1$. [i]Proposed by Victor Domínguez[/i]

2016 KOSOVO TST, 1
Tags: algebra , square root
Solve equation : $\sqrt{x+\sqrt{4x+\sqrt{16x}+..+\sqrt{4^nx+3}}}-\sqrt{x}=1$

2021 Polish Junior MO Finals, 1
Tags: algebra , positive integer , square root
Positive integers $a$, $b$ an $n$ satisfy \[ \frac{a}{b}=\frac{a^2+n^2}{b^2+n^2}. \] Prove that $\sqrt{ab}$ is an integer.

2017 Junior Regional Olympiad - FBH, 3
Tags: equation , square root
Find all real numbers $x$ such that: $$ \sqrt{\frac{x-7}{2015}}+\sqrt{\frac{x-6}{2016}}+\sqrt{\frac{x-5}{2017}}=\sqrt{\frac{x-2015}{7}}+\sqrt{\frac{x-2016}{6}}+\sqrt{\frac{x-2017}{5}}$$

2016 Nigerian Senior MO Round 2, Problem 4
Tags: algebra , square root
Find the real number satisfying $x=\sqrt{1+\sqrt{1+\sqrt{1+x}}}$.

2011 Hanoi Open Mathematics Competitions, 3
Tags: floor function , algebra , square root
What is the largest integer less than to $\sqrt[3]{(2011)^3 + 3 \times (2011)^2 + 4 \times 2011+ 5}$ ? (A) $2010$, (B) $2011$, (C) $2012$, (D) $2013$, (E) None of the above.