Olympiad problems

2013 China Northern MO, 2

If $a_1,a_2,\cdots,a_{2013}\in[-2,2]$ and $a_1+a_2+\cdots+a_{2013}=0$ , find the maximum of $a^3_1+a^3_2+\cdots+a^3_{2013}$.

2010 Romania Team Selection Test, 1

Tags: induction , strong induction , inequalities proposed , inequalities

Given a positive integer number $n$, determine the minimum of \[\max \left\{\dfrac{x_1}{1 + x_1},\, \dfrac{x_2}{1 + x_1 + x_2},\, \cdots,\, \dfrac{x_n}{1 + x_1 + x_2 + \cdots + x_n}\right\},\] as $x_1, x_2, \ldots, x_n$ run through all non-negative real numbers which add up to $1$. [i]Kvant Magazine[/i]

2010 ISI B.Math Entrance Exam, 4

Tags: inequalities , inequalities proposed

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

2012 European Mathematical Cup, 3

Tags: inequalities , inequalities proposed

Prove that the following inequality holds for all positive real numbers $a$, $b$, $c$, $d$, $e$ and $f$ \[\sqrt[3]{\frac{abc}{a+b+d}}+\sqrt[3]{\frac{def}{c+e+f}} < \sqrt[3]{(a+b+d)(c+e+f)} \text{.}\] [i]Proposed by Dimitar Trenevski.[/i]

2008 Junior Balkan Team Selection Tests - Moldova, 2

Tags: inequalities , inequalities proposed , algebra

[b]BJ2. [/b] Positive real numbers $ a,b,c$ satisfy inequality $ \frac {3}{2}\geq a \plus{} b \plus{} c$. Find the smallest possible value for $ S \equal{} abc \plus{} \frac {1}{abc}$

2011 Morocco National Olympiad, 2

Tags: inequalities , geometry , perimeter , trigonometry , inequalities proposed

Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle. $(a)$ Prove that \[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\] $(b)$ When do we have equality?

2002 Turkey Team Selection Test, 3

Tags: inequalities proposed , inequalities

A positive integer $n$ and real numbers $a_1,\dots, a_n$ are given. Show that there exists integers $m$ and $k$ such that \[|\sum\limits_{i=1}^m a_i -\sum\limits_{i=m+1}^n a_i | \leq |a_k|.\]

2006 France Team Selection Test, 2

Tags: inequalities , inequalities proposed , algebra

Let $a,b,c$ be three positive real numbers such that $abc=1$. Show that: \[ \displaystyle \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+ \frac{c}{(c+1)(a+1)} \geq \frac{3}{4}. \] When is there equality?

2025 Kosovo EGMO Team Selection Test, P4

Tags: inequalities , inequalities proposed

Let $a,b$ be positive real numbers such that $a^3+b^3=2(a^2+b^2)$. Prove the following inequality: $$ \sqrt{a^3+1} + \sqrt{b^3+1} \leq a+b+2. $$ When is equality achieved?

2004 Regional Olympiad - Republic of Srpska, 2

Tags: inequalities , trigonometry , inequalities proposed

Let $0<x<\pi/2$. Prove the inequality \[\sin x>\frac{4x}{x^2+4} .\]

2012 China Team Selection Test, 2

Tags: function , inequalities , vector , inequalities proposed

Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression \[f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^{\frac{s}{r}})^{\frac{1}{s}}}.\]

1994 Baltic Way, 2

Tags: inequalities , trigonometry , algebra , polynomial , inequalities proposed

Let $a_1,a_2,\ldots ,a_9$ be any non-negative numbers such that $a_1=a_9=0$ and at least one of the numbers is non-zero. Prove that for some $i$, $2\le i\le 8$, the inequality $a_{i-1}+a_{i+1}<2a_i$ holds. Will the statement remain true if we change the number $2$ in the last inequality to $1.9$?

2006 India IMO Training Camp, 1

Tags: geometry , inradius , circumcircle , inequalities , inequalities proposed

Let $ABC$ be a triangle with inradius $r$, circumradius $R$, and with sides $a=BC,b=CA,c=AB$. Prove that \[\frac{R}{2r} \ge \left(\frac{64a^2b^2c^2}{(4a^2-(b-c)^2)(4b^2-(c-a)^2)(4c^2-(a-b)^2)}\right)^2.\]

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]

1997 Turkey Team Selection Test, 3

Tags: inequalities , inequalities proposed

If $x_{1}, x_{2},\ldots ,x_{n}$ are positive real numbers with $x_{1}^2+x_2^{2}+\ldots +x_{n}^{2}=1$, ﬁnd the minimum value of $\sum_{i=1}^{n}\frac{x_{i}^{5}}{x_{1}+x_{2}+\ldots +x_{n}-x_{i}}$.

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.

1997 Iran MO (2nd round), 1

Tags: inequalities proposed , inequalities

Let $x_1,x_2,x_3,x_4$ be positive reals such that $x_1x_2x_3x_4=1$. Prove that: \[ \sum_{i=1}^{4}{x_i^3}\geq\max\{ \sum_{i=1}^{4}{x_i},\sum_{i=1}^{4}{\frac{1}{x_i}} \}. \]

2009 China Team Selection Test, 2

Tags: induction , ratio , LaTeX , inequalities , blogs , inequalities proposed

Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1,$ then $ (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.$

2006 JBMO ShortLists, 2

Tags: inequalities , inequalities proposed

Let $ x,y,z$ be positive real numbers such that $ x\plus{}2y\plus{}3z\equal{}\frac{11}{12}$. Prove the inequality $ 6(3xy\plus{}4xz\plus{}2yz)\plus{}6x\plus{}3y\plus{}4z\plus{}72xyz\le \frac{107}{18}$.

2007 Serbia National Math Olympiad, 3

Tags: inequalities , inequalities proposed

Let $k$ be a given natural number. Prove that for any positive numbers $x; y; z$ with the sum $1$ the following inequality holds: \[\frac{x^{k+2}}{x^{k+1}+y^{k}+z^{k}}+\frac{y^{k+2}}{y^{k+1}+z^{k}+x^{k}}+\frac{z^{k+2}}{z^{k+1}+x^{k}+y^{k}}\geq \frac{1}{7}.\] When does equality occur?

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 $a_{1}+a_{2}+...+a_{n}\leq\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}}$.

2011 Kosovo National Mathematical Olympiad, 3

Tags: inequalities , logarithms , inequalities proposed

Prove that the following inequality holds: \[ \left( \log_{24}48 \right)^2+ \left( \log_{12}54 \right)^2>4\]

2025 Bulgarian Spring Mathematical Competition, 12.1

Tags: calculus , Derivatives , algebra , inequalities proposed

In terms of the real numbers $a$ and $b$ determine the minimum value of $$ \sqrt{(x+a)^2+1}+\sqrt{(x+1-a)^2+1}+\sqrt{(x+b)^2+1}+\sqrt{(x+1-b)^2+1}$$ as well as all values of $x$ which attain it.

2006 International Zhautykov Olympiad, 2

Tags: inequalities , calculus , inequalities proposed

Let $ a,b,c,d$ be real numbers with sum 0. Prove the inequality: \[ (ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd)^2 \plus{} 12\geq 6(abc \plus{} abd \plus{} acd \plus{} bcd). \]

1994 Hungary-Israel Binational, 2

Tags: inequalities proposed , inequalities

Let $ a_1$, $ \ldots$, $ a_k$, $ a_{k\plus{}1}$, $ \ldots$, $ a_n$ be $ n$ positive numbers ($ k<n$). Suppose that the values of $ a_{k\plus{}1}$, $ a_{k\plus{}2}$, $ \ldots$, $ a_n$ are fixed. Choose the values of $ a_1$, $ a_2$, $ \ldots$, $ a_k$ that minimize the sum $ \sum_{i, j, i\neq j}\frac{a_i}{a_j}$