Olympiad problems

2001 Balkan MO, 3

Let $a$, $b$, $c$ be positive real numbers with $abc \leq a+b+c$. Show that \[ a^2 + b^2 + c^2 \geq \sqrt 3 abc. \] [i]Cristinel Mortici, Romania[/i]

1990 IMO Longlists, 9

Tags: inequalities , inequalities proposed

Let $\{ a_1, a_2, \ldots, a_n\} = \{1, 2, \ldots, n\}$. Prove that \[\frac 12 +\frac 23 +\cdots+\frac{n-1}{n} \leq \frac{a_1}{a_2} + \frac{a_2}{a_3} +\cdots+\frac{a_{n-1}}{a_n}.\]

2005 Turkey MO (2nd round), 1

Tags: inequalities , inequalities proposed

For all positive real numbers $a,b,c,d$ prove the inequality \[\sqrt{a^4+c^4}+\sqrt{a^4+d^4}+\sqrt{b^4+c^4}+\sqrt{b^4+d^4} \ge 2\sqrt{2}(ad+bc)\]

2004 Moldova Team Selection Test, 9

Tags: inequalities , inequalities proposed

Let $a,b$ and $c$ be positive real numbers . Prove that\[\left | \frac{4(b^3-c^3)}{b+c}+ \frac{4(c^3-a^3)}{c+a}+ \frac{4(a^3-b^3)}{a+b} \right |\leq (b-c)^2+(c-a)^2+(a-b)^2.\]

2017 German National Olympiad, 5

Tags: inequalities , inequalities proposed , Symmetric inequality , Germany

Prove that for all non-negative numbers $x,y,z$ satisfying $x+y+z=1$, one has \[1 \le \frac{x}{1-yz}+\frac{y}{1-zx}+\frac{z}{1-xy} \le \frac{9}{8}.\]

2019 Kazakhstan National Olympiad, 1

Tags: inequalities proposed , inequalities

Prove for any positives $a,b,c$ the inequality $$ \sqrt[3]{\dfrac{a}{b}}+\sqrt[5]{\dfrac{b}{c}}+\sqrt[7]{\dfrac{c}{a}}>\dfrac{5}{2}$$

2006 Baltic Way, 2

Tags: inequalities , function , inequalities proposed

Suppose that the real numbers $a_i\in [-2,17],\ i=1,2,\ldots,59,$ satisfy $a_1+a_2+\ldots+a_{59}=0.$ Prove that \[a_1^2+a_2^2+\ldots+a_{59}^2\le 2006\]

1987 IMO Longlists, 14

Tags: inequalities , inequalities proposed

Given $n$ real numbers $0 < t_1 \leq t_2 \leq \cdots \leq t_n < 1$, prove that \[(1-t_n^2) \left( \frac{t_1}{(1-t_1^2)^2}+\frac{t_2}{(1-t_2^3)^2}+\cdots +\frac{t_n}{(1-t_n^{n+1})^2} \right) < 1.\]

2014 Iran Team Selection Test, 5

Tags: inequalities , inequalities proposed

if $x,y,z>0$ are postive real numbers such that $x^{2}+y^{2}+z^{2}=x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}$ prove that \[((x-y)(y-z)(z-x))^{2}\leq 2((x^{2}-y^{2})^{2}+(y^{2}-z^{2})^{2}+(z^{2}-x^{2})^{2})\]

2008 Romania National Olympiad, 3

Tags: inequalities proposed , inequalities

Let $ n$ be a positive integer and let $ a_i$ be real numbers, $ i \equal{} 1,2,\ldots,n$ such that $ |a_i|\leq 1$ and $ \sum_{i\equal{}1}^n a_i \equal{} 0$. Show that $ \sum_{i\equal{}1}^n |x \minus{} a_i|\leq n$, for every $ x\in \mathbb{R}$ with $ |x|\le 1$.

2007 Middle European Mathematical Olympiad, 1

Tags: inequalities , inequalities proposed

Let $ a,b,c,d$ be positive real numbers with $ a\plus{}b\plus{}c\plus{}d \equal{} 4$. Prove that \[ a^{2}bc\plus{}b^{2}cd\plus{}c^{2}da\plus{}d^{2}ab\leq 4.\]

2006 Taiwan National Olympiad, 3

Tags: inequalities , induction , function , inequalities proposed

$a_1, a_2, ..., a_{95}$ are positive reals. Show that $\displaystyle \sum_{k=1}^{95}{a_k} \le 94+ \prod_{k=1}^{95}{\max{\{1,a_k\}}}$

2010 Moldova Team Selection Test, 2

Tags: trigonometry , inequalities proposed , inequalities

Let $ x_1, x_2, \ldots, x_n$ be positive real numbers with sum $ 1$. Find the integer part of: $ E\equal{}x_1\plus{}\dfrac{x_2}{\sqrt{1\minus{}x_1^2}}\plus{}\dfrac{x_3}{\sqrt{1\minus{}(x_1\plus{}x_2)^2}}\plus{}\cdots\plus{}\dfrac{x_n}{\sqrt{1\minus{}(x_1\plus{}x_2\plus{}\cdots\plus{}x_{n\minus{}1})^2}}$

2013 Canadian Mathematical Olympiad Qualification Repechage, 6

Tags: inequalities proposed , inequalities

Let $x, y, z$ be real numbers that are greater than or equal to $0$ and less than or equal to $\frac{1}{2}$ [list] [*] (a) Determine the minimum possible value of \[x+y+z-xy-yz-zx\] and determine all triples $(x,y,z)$ for which this minimum is obtained. [*] (b) Determine the maximum possible value of \[x+y+z-xy-yz-zx\] and determine all triples $(x,y,z)$ for which this maximum is obtained.[/list]

2010 Stars Of Mathematics, 3

Tags: inequalities , inequalities proposed

Find the largest constant $K>0$ such that for any $0\le k\le K$ and non-negative reals $a,b,c$ satisfying $a^2+b^2+c^2+kabc=k+3$ we have $a+b+c\le 3$. (Dan Schwarz)

2023 China Northern MO, 2

Tags: inequalities , algebra , inequalities proposed , Inequality

Let $ a,b,c \in (0,1) $ and $ab+bc+ca=4abc .$ Prove that $$\sqrt{a+b+c}\geq \sqrt{1-a}+\sqrt{1-b}+\sqrt{1-c}$$

2021 German National Olympiad, 5

Tags: inequalities , inequalities proposed , Symmetric inequality , equality case

a) Determine the largest real number $A$ with the following property: For all non-negative real numbers $x,y,z$, one has \[\frac{1+yz}{1+x^2}+\frac{1+zx}{1+y^2}+\frac{1+xy}{1+z^2} \ge A.\] b) For this real number $A$, find all triples $(x,y,z)$ of non-negative real numbers for which equality holds in the above inequality.

2007 IberoAmerican Olympiad For University Students, 2

Tags: inequalities , induction , inequalities proposed

Prove that for all positive integers $n$ and for all real numbers $x$ such that $0\le x\le1$, the following inequality holds: $\left(1-x+\frac{x^2}{2}\right)^n-(1-x)^n\le\frac{x}{2}$.

2018 Pan-African Shortlist, A6

Tags: inequalities , inequalities proposed , algebra

Let $a, b, c$ be positive real numbers such that $a^3 + b^3 + c^3 = 5abc$. Show that \[ \left( \frac{a + b}{c} \right) \left( \frac{b + c}{a} \right) \left( \frac{c + a}{b} \right) \geq 9. \]

2006 Romania National Olympiad, 2

Tags: trigonometry , inequalities , inequalities proposed

Prove that for all $\displaystyle a,b \in \left( 0 ,\frac{\pi}{4} \right)$ and $\displaystyle n \in \mathbb N^\ast$ we have \[ \frac{\sin^n a + \sin^n b}{\left( \sin a + \sin b \right)^n} \geq \frac{\sin^n 2a + \sin^n 2b}{\left( \sin 2a + \sin 2b \right)^n} . \]

2003 China Team Selection Test, 3

Tags: inequalities , inequalities proposed

Let $a_{1},a_{2},...,a_{n}$ be positive real number $(n \geq 2)$,not all equal,such that $\sum_{k=1}^n a_{k}^{-2n}=1$,prove that: $\sum_{k=1}^n a_{k}^{2n}-n^2.\sum_{1 \leq i<j \leq n}(\frac{a_{i}}{a_{j}}-\frac{a_{j}}{a_{i}})^2 >n^2$

2006 Silk Road, 2

Tags: inequalities , inequalities proposed

For positive $a,b,c$, such that $abc=1$ prove the inequality: $4(\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{c}}+\sqrt[3]{\frac{c}{a}}) \leq 3(2+a+b+c+\frac{1}{a}+\frac{1}{b}+ \frac{1}{c})^{\frac{2}{3}}$.

2012 ELMO Shortlist, 1

Tags: inequalities , trigonometry , inequalities proposed

Let $x_1,x_2,x_3,y_1,y_2,y_3$ be nonzero real numbers satisfying $x_1+x_2+x_3=0, y_1+y_2+y_3=0$. Prove that \[\frac{x_1x_2+y_1y_2}{\sqrt{(x_1^2+y_1^2)(x_2^2+y_2^2)}}+\frac{x_2x_3+y_2y_3}{\sqrt{(x_2^2+y_2^2)(x_3^2+y_3^2)}}+\frac{x_3x_1+y_3y_1}{\sqrt{(x_3^2+y_3^2)(x_1^2+y_1^2)}} \ge -\frac32.\] [i]Ray Li, Max Schindler.[/i]

2023 China Northern MO, 4

Tags: inequalities , algebra , minimum , inequalities proposed

Given the sequence $(a_n) $ satisfies $1=a_1< a_2 < a_3< \cdots<a_n $ and there exist real number $m$ such that $$\displaystyle\sum_{i=1}^{n-1} \sqrt[3]{\frac{a_{i+1}-a_i}{(2+a_i)^4}}\leq m $$ for any positive integer $ n $ not less than 2 . Find the minimum of $m.$

2002 Romania National Olympiad, 1

Tags: inequalities , inequalities proposed

Let $ab+bc+ca=1$. Show that \[\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge\sqrt{3}+\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\]