Olympiad problems

2014 District Olympiad, 2

1994 Irish Math Olympiad, 5

Tags: inequalities , function , inequalities proposed

Let $ f(n)$ be defined for $ n \in \mathbb{N}$ by $ f(1)\equal{}2$ and $ f(n\plus{}1)\equal{}f(n)^2\minus{}f(n)\plus{}1$ for $ n \ge 1$. Prove that for all $ n >1:$ $ 1\minus{}\frac{1}{2^{2^{n\minus{}1}}}<\frac{1}{f(1)}\plus{}\frac{1}{f(2)}\plus{}...\plus{}\frac{1}{f(n)}<1\minus{}\frac{1}{2^{2^n}}$

2009 China Team Selection Test, 3

Tags: inequalities , inequalities proposed

Let nonnegative real numbers $ a_{1},a_{2},a_{3},a_{4}$ satisfy $ a_{1} \plus{} a_{2} \plus{} a_{3} \plus{} a_{4} \equal{} 1.$ Prove that $ max\{\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \minus{} 1} \plus{} a_{i \minus{} 1}^2 \plus{} a_{i \minus{} 1}a_{i \minus{} 2}}},\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \plus{} 1} \plus{} a_{i \plus{} 1}^2 \plus{} a_{i \plus{} 1}a_{i \plus{} 2}}}\}\ge 2.$ Where for all integers $ i, a_{i \plus{} 4} \equal{} a_{i}$ holds.

2002 Tournament Of Towns, 1

Tags: inequalities , inequalities proposed , BPSQ

Let $a,b,c$ be sides of a triangle. Show that $a^3+b^3+3abc>c^3$.

2009 Vietnam Team Selection Test, 1

Tags: inequalities , calculus , quadratics , function , logarithms , inequalities proposed

Let $ a,b,c$ be positive numbers.Find $ k$ such that: $ (k \plus{} \frac {a}{b \plus{} c})(k \plus{} \frac {b}{c \plus{} a})(k \plus{} \frac {c}{a \plus{} b}) \ge (k \plus{} \frac {1}{2})^3$

2014 Contests, 3

Tags: inequalities , parameterization , inequalities proposed

Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]

2014 China Girls Math Olympiad, 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 Moldova Team Selection Test, 2

Tags: inequalities , function , inequalities proposed

Let $ a_1,\ldots,a_n$ be positive reals so that $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le\frac n2$. Find the minimal value of $ \sqrt{a_1^2\plus{}\frac1{a_2^2}}\plus{}\sqrt{a_2^2\plus{}\frac1{a_3^2}}\plus{}\ldots\plus{}\sqrt{a_n^2\plus{}\frac1{a_1^2}}$.

2006 Germany Team Selection Test, 3

Tags: inequalities , induction , inequalities proposed

Let $n$ be a positive integer, and let $b_{1}$, $b_{2}$, ..., $b_{n}$ be $n$ positive reals. Set $a_{1}=\frac{b_{1}}{b_{1}+b_{2}+...+b_{n}}$ and $a_{k}=\frac{b_{1}+b_{2}+...+b_{k}}{b_{1}+b_{2}+...+b_{k-1}}$ for every $k>1$. Prove the inequality $a_{1}+a_{2}+...+a_{n}\leq\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}}$.

2007 France Team Selection Test, 2

Tags: inequalities , search , symmetry , inequalities proposed

Let $a,b,c,d$ be positive reals such taht $a+b+c+d=1$. Prove that: \[6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.\]

1996 Iran MO (3rd Round), 1

Tags: inequalities , inequalities proposed

Let $a,b,c,d$ be positive real numbers. Prove that \[\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c} \geq \frac{2}{3}.\]

1967 IMO Longlists, 37

Tags: Inequality , three variable inequality , Muirhead , IMO Shortlist , IMO Longlist , algebra , inequalities proposed

Prove that for arbitrary positive numbers the following inequality holds \[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]

2013 District Olympiad, 2

Tags: inequalities , trigonometry , function , inequalities proposed

Let $a,b\in \mathbb{C}$. Prove that $\left| az+b\bar{z} \right|\le 1$, for every $z\in \mathbb{C}$, with $\left| z \right|=1$, if and only if $\left| a \right|+\left| b \right|\le 1$.

2012 Romania National Olympiad, 3

Tags: inequalities , inequalities proposed

[color=darkred]Prove that if $n\ge 2$ is a natural number and $x_1,x_2,\ldots,x_n$ are positive real numbers, then: \[4\left(\frac {x_1^3-x_2^3}{x_1+x_2}+\frac {x_2^3-x_3^3}{x_2+x_3}+\ldots+\frac {x_{n-1}^3-x_n^3}{x_{n-1}+x_n}+\frac {x_n^3-x_1^3}{x_n+x_1}\right)\le \\ \\ \le(x_1-x_2)^2+(x_2-x_3)^2+\ldots+(x_{n-1}-x_n)^2+(x_n-x_1)^2\, .\][/color]

1966 IMO Longlists, 2

Tags: inequalities proposed , inequalities

Given $n$ positive numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ such that $a_{1}\cdot a_{2}\cdot ...\cdot a_{n}=1.$ Prove \[ \left( 1+a_{1}\right) \left( 1+a_{2}\right) ...\left(1+a_{n}\right) \geq 2^{n}.\]

2009 Macedonia National Olympiad, 4

Tags: inequalities , inequalities proposed

Let $a,b,c$ be positive real numbers for which $ab+bc+ca=\frac{1}{3}$. Prove the inequality \[ \frac{a}{a^2-bc+1}+\frac{b}{b^2-ca+1}+\frac{c}{c^2-ab+1}\ge\frac{1}{a+b+c}\]

2001 Italy TST, 2

Tags: inequalities , inequalities proposed , algebra

Let $0\le a\le b\le c$ be real numbers. Prove that \[(a+3b)(b+4c)(c+2a)\ge 60abc \]

2014 ELMO Shortlist, 3

Tags: inequalities , parameterization , inequalities proposed

Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]

1971 IMO Longlists, 18

Tags: inequalities , inequalities proposed

Let $a_1, a_2, \ldots, a_n$ be positive numbers, $m_g = \sqrt[n]{(a_1a_2 \cdots a_n)}$ their geometric mean, and $m_a = \frac{(a_1 + a_2 + \cdots + a_n)}{n}$ their arithmetic mean. Prove that \[(1 + m_g)^n \leq (1 + a_1) \cdots(1 + a_n) \leq (1 + m_a)^n.\]

2010 Contests, 4

Tags: inequalities , inequalities proposed

If $a,b,c\in (0,1)$ satisfy $a+b+c=2$ , prove that $\frac{abc}{(1-a)(1-b)(1-c)}\ge 8$

2014 Silk Road, 3

Tags: inequalities , inequalities proposed

$ a,b,c\ge 0,\ \ \ a^3+b^3+c^3+abc=4 $ Prove that $a^3b+b^3c+c^3b \le 3$

2024 Ukraine National Mathematical Olympiad, Problem 5

Tags: algebra , Inequality , inequalities proposed , inequalities

For real numbers $a, b, c, d \in [0, 1]$, find the largest possible value of the following expression: $$a^2+b^2+c^2+d^2-ab-bc-cd-da$$ [i]Proposed by Mykhailo Shtandenko[/i]

2010 ELMO Shortlist, 2

Tags: inequalities , inequalities proposed

Let $a,b,c$ be positive reals. Prove that \[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \] [i]Calvin Deng.[/i]

2003 Tuymaada Olympiad, 1

Tags: trigonometry , inequalities proposed , inequalities

Prove that for every $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ in the interval $(0,\pi/2)$ \[\left({1\over \sin \alpha_{1}}+{1\over \sin \alpha_{2}}+\ldots+{1\over \sin \alpha_{n}}\right) \left({1\over \cos \alpha_{1}}+{1\over \cos \alpha_{2}}+\ldots+{1\over \cos \alpha_{n}}\right) \leq\] \[\leq 2 \left({1\over \sin 2\alpha_{1}}+{1\over \sin 2\alpha_{2}}+\ldots+{1\over \sin 2\alpha_{n}}\right)^{2}.\] [i]Proposed by A. Khrabrov[/i]

1990 Turkey Team Selection Test, 2

Tags: algebra , polynomial , calculus , derivative , inequalities proposed , inequalities

For real numbers $x_i$, the statement \[ x_1 + x_2 + x_3 = 0 \Rightarrow x_1x_2 + x_2x_3 + x_3x_1 \leq 0\] is always true. (Prove!) For which $n\geq 4$ integers, the statement \[x_1 + x_2 + \dots + x_n = 0 \Rightarrow x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1 \leq 0\] is always true. Justify your answer.