Olympiad problems

1972 IMO Shortlist, 3

The least number is $m$ and the greatest number is $M$ among $ a_1 ,a_2 ,\ldots,a_n$ satisfying $ a_1 \plus{}a_2 \plus{}...\plus{}a_n \equal{}0$. Prove that \[ a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM\]

2019 JBMO Shortlist, A2

Tags: algebra , Inequality , TST , Balkan MO Shortlist , AZE JBMO TST

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

2017 Azerbaijan JBMO TST, 1

Tags: algebra , Inequality

a,b,c>0 and $abc\ge 1$.Prove that: $\dfrac{1}{a^3+2b^3+6}+\dfrac{1}{b^3+2c^3+6}+\dfrac{1}{c^3+2a^3+6} \le \dfrac{1}{3}$

1975 IMO, 1

Tags: rearrangement inequality , Inequality , permutation , algebra , IMO , IMO 1975

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

2010 Junior Balkan Team Selection Tests - Romania, 3

Tags: algebra , Inequality , TST , AZE IzHO TST

Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that: $$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.

2020 Stars of Mathematics, 1

Tags: algebra , Inequality , romania

Let $a_1,a_2,a_3,a_4$ be positive real numbers satisfying \[\sum_{i<j}a_ia_j=1.\]Prove that \[\sum_{\text{sym}}\frac{a_1a_2}{1+a_3a_4}\geq\frac{6}{7}.\][i]* * *[/i]

2014 Contests, 1

Tags: algebra , Inequality , inequalities

Prove that for $\forall$ $a,b,c\in [\frac{1}{3},3]$ the following inequality is true: $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\geq \frac{7}{5}$.

2019 ELMO Shortlist, A1

Tags: algebra , Inequality , inequalities

Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$ [i]Proposed by Milan Haiman[/i]

1988 IMO Shortlist, 24

Tags: inequality system , Inequality , Sequence , Linear Recurrences , IMO Shortlist

Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that: \[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0 \] and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that: \[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2} \] for all $ k \equal{} 1,2, \ldots$.

2018 Junior Balkan Team Selection Tests - Moldova, 3

Tags: Inequality , inequalities

Let $a,b,c \in\mathbb{R^*_+}$.Prove the inequality $\frac{a^2+4}{b+c}+\frac{b^2+9}{c+a}+\frac{c^2+16}{a+b}\ge9$.

2021 Brazil Team Selection Test, 4

Tags: algebra , IMO Shortlist , Inequality , IMO Shortlist 2020 , inequalities

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

2021 Science ON grade X, 1

Tags: algebra , complex numbers , Inequality

Consider the complex numbers $x,y,z$ such that $|x|=|y|=|z|=1$. Define the number $$a=\left (1+\frac xy\right )\left (1+\frac yz\right )\left (1+\frac zx\right ).$$ $\textbf{(a)}$ Prove that $a$ is a real number. $\textbf{(b)}$ Find the minimal and maximal value $a$ can achieve, when $x,y,z$ vary subject to $|x|=|y|=|z|=1$. [i] (Stefan Bălăucă & Vlad Robu)[/i]

2010 Saint Petersburg Mathematical Olympiad, 6

Tags: algebra , Inequality

For positive numbers is true that $$ab+ac+bc=a+b+c$$ Prove $$a+b+c+1 \geq 4abc$$

2020 Centroamerican and Caribbean Math Olympiad, 5

Tags: algebra , polynomial , Inequality , inequalities , n-variable inequality

Let $P(x)$ be a polynomial with real non-negative coefficients. Let $k$ be a positive integer and $x_1, x_2, \dots, x_k$ positive real numbers such that $x_1x_2\cdots x_k=1$. Prove that $$P(x_1)+P(x_2)+\cdots+P(x_k)\geq kP(1).$$

2021 Thailand TST, 2

Tags: algebra , Inequality , IMO Shortlist , IMO Shortlist 2020 , 4-variable inequality , inequalities

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2018 JBMO Shortlist, A6

Tags: Inequality , algebra , inequalities

For $a,b,c$ positive real numbers such that $ab+bc+ca=3$, prove: $ \frac{a}{\sqrt{a^3+5}}+\frac{b}{\sqrt{b^3+5}}+\frac{c}{\sqrt{c^3+5}} \leq \frac{\sqrt{6}}{2}$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

2000 Moldova National Olympiad, Problem 6

Tags: Inequality , inequalities

Assuming that real numbers $x$ and $y$ satisfy $y\left(1+x^2\right)=x\left(\sqrt{1-4y^2}-1\right)$, find the maximum value of $xy$.

2012 Junior Balkan MO, 1

Tags: JBMO , algebra , Inequality , JBMO Shortlist

Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that \[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\] When does equality hold?

2016 PAMO, 4

Tags: Inequality , algebra

Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that $\frac{1}{(x+1)^2+y^2+1}$ $+$ $\frac{1}{(y+1)^2+z^2+1}$ $+$ $\frac{1}{(z+1)^2+x^2+1}$ $\leq$ ${\frac{1}{2}}$.

2014 Contests, 1

Tags: algebra , Inequality , BMO , BMO Shortlist

Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds. [i]UK - David Monk[/i]

2021 Macedonian Mathematical Olympiad, Problem 1

Tags: Sequence , recursive , Inequality , factorial , AM-GM , inequalities

Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$ For every $n > 1$, prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$.

2009 JBMO Shortlist, 5

Tags: Inequality , algebra

$\boxed{\text{A5}}$ Let $x,y,z$ be positive reals. Prove that $(x^2+y+1)(x^2+z+1)(y^2+x+1)(y^2+z+1)(z^2+x+1)(z^2+y+1)\geq (x+y+z)^6$

1983 IMO Shortlist, 2

Tags: number theory , Divisors , sum of divisors , Inequality , IMO Shortlist

Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$ Prove that there exists an infinity of [i]superabundant[/i] numbers.

2020 IMO Shortlist, A1

Tags: algebra , IMO Shortlist , Inequality , IMO Shortlist 2020 , inequalities

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

2025 Kyiv City MO Round 1, Problem 5

Tags: algebra , Inequality

Determine the largest possible constant $ C $ such that for any positive real numbers $ x, y, z $, which are the sides of a triangle, the following inequality holds: \[ \frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C. \] [i]Proposed by Vadym Solomka[/i]