Olympiad problems

2008 Junior Balkan Team Selection Tests - Romania, 3

Let $ n$ be a positive integer and let $ a_1,a_2,\ldots,a_n$ be positive real numbers such that: \[ \sum^n_{i \equal{} 1} a_i \equal{} \sum^n_{i \equal{} 1} \frac {1}{a_i^2}. \] Prove that for every $ i \equal{} 1,2,\ldots,n$ we can find $ i$ numbers with sum at least $ i$.

2015 Saint Petersburg Mathematical Olympiad, 4

Tags: inequalities , inequalities proposed , algebra

Positive numbers $x, y, z$ satisfy the condition $$xy + yz + zx + 2xyz = 1.$$ Prove that $4x + y + z \ge 2.$ [i]A. Khrabrov[/i]

2009 South East Mathematical Olympiad, 7

Tags: inequalities , inequalities proposed

Let $x,y,z\geq0$ be real numbers such that $x+y+z=1$ Define $f(x,y,z)$ in this way : \[f(x,y,z)=\frac{x(2y-z)}{1+x+3y}+\frac{y(2z-x)}{1+y+3z}+\frac{z(2x-y)}{1+z+3x}\] Find the minimum value and maximum value of $f(x,y,z)$ .

2003 Iran MO (2nd round), 1

Tags: inequalities , inequalities proposed

Let $x,y,z\in\mathbb{R}$ and $xyz=-1$. Prove that: \[ x^4+y^4+z^4+3(x+y+z)\geq\frac{x^2}{y}+\frac{x^2}{z}+\frac{y^2}{x}+\frac{y^2}{z}+\frac{z^2}{x}+\frac{z^2}{y}. \]

2012 Canada National Olympiad, 1

Tags: inequalities , induction , inequalities proposed

Let $x,y$ and $z$ be positive real numbers. Show that $x^2+xy^2+xyz^2\ge 4xyz-4$.

2015 Peru IMO TST, 14

Tags: symmetry , inequalities proposed , inequalities

Let $ n$ be a positive integer and let $ a_1,a_2,\ldots,a_n$ be positive real numbers such that: \[ \sum^n_{i \equal{} 1} a_i \equal{} \sum^n_{i \equal{} 1} \frac {1}{a_i^2}. \] Prove that for every $ i \equal{} 1,2,\ldots,n$ we can find $ i$ numbers with sum at least $ i$.

2011 China National Olympiad, 2

Tags: function , calculus , derivative , inequalities , rearrangement inequality , inequalities proposed

Let $a_i,b_i,i=1,\cdots,n$ are nonnegitive numbers,and $n\ge 4$,such that $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0$. Find the maximum of $\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}$

2020 Turkey MO (2nd round), 3

Tags: inequalities , inequalities proposed , Turkey

If $x, y, z$ are positive real numbers find the minimum value of $$2\sqrt{(x+y+z) \left( \frac{1}{x}+ \frac{1}{y} + \frac{1}{z} \right)} - \sqrt{ \left( 1+ \frac{x}{y} \right) \left( 1+ \frac{y}{z} \right)}$$

2014 Contests, 2

Tags: floor function , ceiling function , inequalities proposed , inequalities

Let $x_1,x_2,\ldots,x_n $ be real numbers, where $n\ge 2$ is a given integer, and let $\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor $ be a permutation of $1,2,\ldots,n$. Find the maximum and minimum of $\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor$ (here $\lfloor x\rfloor $ is the largest integer not greater than $x$).

2008 China Team Selection Test, 2

Tags: inequalities , algebra , polynomial , inequalities proposed

Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.$

2019 Turkey Junior National Olympiad, 2

Tags: inequalities , inequalities proposed

$x,y,z \in \mathbb{R}^+$ and $x^5+y^5+z^5=xy+yz+zx$. Prove that $$3 \ge x^2y+y^2z+z^2x$$

2009 Moldova Team Selection Test, 2

Tags: inequalities , function , calculus , derivative , logarithms , rearrangement inequality , inequalities proposed

[color=darkred]Let $ m,n\in \mathbb{N}$, $ n\ge 2$ and numbers $ a_i > 0$, $ i \equal{} \overline{1,n}$, such that $ \sum a_i \equal{} 1$. Prove that $ \small{\dfrac{a_1^{2 \minus{} m} \plus{} a_2 \plus{} ... \plus{} a_{n \minus{} 1}}{1 \minus{} a_1} \plus{} \dfrac{a_2^{2 \minus{} m} \plus{} a_3 \plus{} ... \plus{} a_n}{1 \minus{} a_1} \plus{} ... \plus{} \dfrac{a_n^{2 \minus{} m} \plus{} a_1 \plus{} ... \plus{} a_{n \minus{} 2}}{1 \minus{} a_1}\ge n \plus{} \dfrac{n^m \minus{} n}{n \minus{} 1}}$[/color]

2011 Iran MO (3rd Round), 4

Tags: inequalities , function , LaTeX , Cauchy Inequality , inequalities proposed

For positive real numbers $a,b$ and $c$ we have $a+b+c=3$. Prove $\frac{a}{1+(b+c)^2}+\frac{b}{1+(a+c)^2}+\frac{c}{1+(a+b)^2}\le \frac{3(a^2+b^2+c^2)}{a^2+b^2+c^2+12abc}$. [i]proposed by Mohammad Ahmadi[/i]

1991 Dutch Mathematical Olympiad, 1

Tags: inequalities , function , inequalities proposed

Prove that for any three positive real numbers $ a,b,c, \frac{1}{a\plus{}b}\plus{}\frac{1}{b\plus{}c}\plus{}\frac{1}{c\plus{}a} \ge \frac{9}{2} \cdot \frac{1}{a\plus{}b\plus{}c}$.

2023 ITAMO, 5

Tags: algebra , inequalities proposed

Let $a, b, c$ be reals satisfying $a^2+b^2+c^2=6$. Find the maximal values of the expressions a) $(a-b)^2+(b-c)^2+(c-a)^2$; b) $(a-b)^2 \cdot (b-c)^2 \cdot (c-a)^2$. In both cases, describe all triples for which equality holds.

1996 Turkey Team Selection Test, 3

Tags: inequalities proposed , inequalities

If $0=x_{1}<x_{2}<...<x_{2n+1}=1$ are real numbers with $x_{i+1}-x_{i} \leq h$ for $1 \leq i \leq 2n$, show that $\frac{1-h}{2}<\sum_{i=1}^{n}{x_{2i}(x_{2i+1}-x_{2i-1})}\leq \frac{1+h}{2}$

1982 IMO Longlists, 51

Tags: inequalities , inequalities proposed

Let n numbers $x_1, x_2, \ldots, x_n$ be chosen in such a way that $1 \geq x_1 \geq x_2 \geq \cdots \geq x_n \geq 0$. Prove that \[(1 + x_1 + x_2 + \cdots + x_n)^\alpha \leq 1 + x_1^\alpha+ 2^{\alpha-1}x_2^\alpha+ \cdots+ n^{\alpha-1}x_n^\alpha\] if $0 \leq \alpha \leq 1$.

2001 Junior Balkan Team Selection Tests - Romania, 3

Tags: inequalities , induction , limit , inequalities proposed

Let $n\ge 2$ be a positive integer. Find the positive integers $x$ \[\sqrt{x+\sqrt{x+\ldots +\sqrt{x}}}<n \] for any number of radicals.

1992 Balkan MO, 2

Tags: inequalities , inequalities proposed

Prove that for all positive integers $n$ the following inequality takes place \[ (2n^2+3n+1)^n \geq 6^n (n!)^2 . \] [i]Cyprus[/i]

2009 JBMO Shortlist, 4

Tags: symmetry , inequalities proposed , inequalities

Let $ x$, $ y$, $ z$ be real numbers such that $ 0 < x,y,z < 1$ and $ xyz \equal{} (1 \minus{} x)(1 \minus{} y)(1 \minus{} z)$. Show that at least one of the numbers $ (1 \minus{} x)y,(1 \minus{} y)z,(1 \minus{} z)x$ is greater than or equal to $ \frac {1}{4}$

2023 Austrian MO National Competition, 1

Tags: algebra , inequalities proposed

Let $a, b, c, d$ be positive reals strictly smaller than $1$, such that $a+b+c+d=2$. Prove that $$\sqrt{(1-a)(1-b)(1-c)(1-d)} \leq \frac{ac+bd}{2}. $$

2010 Balkan MO Shortlist, A1

Tags: inequalities , rearrangement inequality , inequalities proposed , Balkan

Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]

2024 Al-Khwarizmi IJMO, 3

Tags: inequalities , Al-Khwarizmi IJMO , inequalities proposed

Find all $x, y, z \in \left (0, \frac{1}{2}\right )$ such that $$ \begin{cases} (3 x^{2}+y^{2}) \sqrt{1-4 z^{2}} \geq z; \\ (3 y^{2}+z^{2}) \sqrt{1-4 x^{2}} \geq x; \\ (3 z^{2}+x^{2}) \sqrt{1-4 y^{2}} \geq y. \end{cases} $$ [i]Proposed by Ngo Van Trang, Vietnam[/i]

2012 JBMO ShortLists, 2

Tags: inequalities , inequalities proposed

Let $a$ , $b$ , $c$ be positive real numbers such that $abc=1$ . Show that : \[\frac{1}{a^3+bc}+\frac{1}{b^3+ca}+\frac{1}{c^3+ab} \leq \frac{ \left (ab+bc+ca \right )^2 }{6}\]

2000 Tuymaada Olympiad, 4

Tags: inequalities , logarithms , inequalities proposed

Prove for real $x_1$, $x_2$, ....., $x_n$, $0 < x_k \leq {1\over 2}$, the inequality \[ \left( {n \over x_1 + \dots + x_n} - 1 \right)^n \leq \left( {1 \over x_1} - 1 \right) \dots \left( {1 \over x_n} - 1 \right). \]