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: 15

2021 Polish Junior MO Finals, 1
Tags: positive integers , square roots , algebra
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.

2011 Hanoi Open Mathematics Competitions, 3
Tags: square roots , floor function , algebra
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.

1959 IMO Shortlist, 2
Tags: algebra , equation , square roots , IMO , IMO 1959
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?

2014 BAMO, 3
Tags: algebra , polynomial , square roots
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 $x+y$.

2006 Spain Mathematical Olympiad, 3
Tags: geometry , geometric inequality , convex quadrilateral , diagonals , square roots , areas
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?

2019 Azerbaijan Senior NMO, 1
Tags: algebra , square roots
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 $a;b\in\mathbb{R^+}$

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

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

1978 Austrian-Polish Competition, 8
Tags: square roots , algebra , limit
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.

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

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

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

1959 IMO, 2
Tags: algebra , equation , square roots , IMO , IMO 1959
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?

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

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