Olympiad problems

2013 China Team Selection Test, 1

Let $n$ and $k$ be two integers which are greater than $1$. Let $a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m$ be non-negative real numbers such that i) $a_1\ge a_2\ge\ldots\ge a_n$ and $a_1+a_2+\ldots+a_n=1$; ii) For any integer $m\in\{1,2,\ldots,n\}$, we have that $c_1+c_2+\ldots+c_m\le m^k$. Find the maximum of $c_1a_1^k+c_2a_2^k+\ldots+c_na_n^k$.

2006 Taiwan TST Round 1, 1

Tags: trigonometry , inequalities proposed , inequalities

Let the three sides of $\triangle ABC$ be $a,b,c$. Prove that $\displaystyle \frac{\sin^2A}{a}+\frac{\sin^2B}{b}+\frac{\sin^2C}{c} \le \frac{S^2}{abc}$ where $\displaystyle S=\frac{a+b+c}{2}$. Find the case where equality holds.

2013 JBMO TST - Turkey, 4

Tags: inequalities , quadratics , algebra , polynomial , inequalities proposed

For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that \[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]

2012 ELMO Shortlist, 9

Tags: inequalities , algebra , polynomial , function , induction , inequalities proposed

Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that \[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\] and \[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\] [i]Calvin Deng.[/i]

2008 China Team Selection Test, 3

Tags: inequalities , induction , rearrangement inequality , inequalities proposed

Let $ 0 < x_{1}\leq\frac {x_{2}}{2}\leq\cdots\leq\frac {x_{n}}{n}, 0 < y_{n}\leq y_{n \minus{} 1}\leq\cdots\leq y_{1},$ Prove that $ (\sum_{k \equal{} 1}^{n}x_{k}y_{k})^2\leq(\sum_{k \equal{} 1}^{n}y_{k})(\sum_{k \equal{} 1}^{n}(x_{k}^2 \minus{} \frac {1}{4}x_{k}x_{k \minus{} 1})y_{k}).$ where $ x_{0} \equal{} 0.$

2012 Greece Team Selection Test, 3

Tags: inequalities proposed , algebra , Inequality

Let $a,b,c$ be positive real numbers satisfying $a+b+c=3$.Prove that $\sum_{sym} \frac{a^{2}}{(b+c)^{3}}\geq \frac{3}{8}$

1998 Baltic Way, 9

Tags: trigonometry , inequalities proposed , inequalities

Let the numbers $\alpha ,\beta $ satisfy $0<\alpha <\beta <\frac{\pi}{2}$ and let $\gamma $ and $\delta $ be the numbers defined by the conditions: $(\text{i})\ 0<\gamma<\frac{\pi}{2}$, and $\tan\gamma$ is the arithmetic mean of $\tan\alpha$ and $\tan\beta$; $(\text{ii})\ 0<\delta<\frac{\pi}{2}$, and $\frac{1}{\cos\delta}$ is the arithmetic mean of $\frac{1}{\cos\alpha}$ and $\frac{1}{\cos\beta}$. Prove that $\gamma <\delta $.

2013 JBMO TST - Macedonia, 3

Tags: inequalities , inequalities proposed

$ a,b,c>0 $ and $ abc=1 $. Prove that $\frac{1}{2}\ (\sqrt{a}\ +\sqrt{b}\ + \sqrt{c}\ ) +\frac{1}{1+a}\ + \frac{1}{1+b}\ + \frac{1}{1+c}\ge\ 3 $. ( The official problem is with $ abc = 1 $ but it can be proved without using it. )

2012 Romania Team Selection Test, 4

Tags: algebra , function , domain , inequalities , inequalities proposed , Balkan

Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.

2017, SRMC, 3

Tags: inequalities proposed , inequalities , srmc

Prove that among any $42$ numbers from the interval $[1,10^6]$, you can choose four numbers so that for any permutation $(a, b, c, d)$ of these numbers, the inequality $$25 (ab + cd) (ad + bc) \ge 16 (ac + bd)^ 2$$ holds.

2005 Georgia Team Selection Test, 3

Tags: inequalities , calculus , derivative , Cauchy Inequality , inequalities proposed

Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.

1998 Baltic Way, 11

Tags: trigonometry , inequalities , geometry , inequalities proposed

If $a,b,c$ be the lengths of the sides of a triangle. Let $R$ denote its circumradius. Prove that \[ R\ge \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}\] When does equality hold?

2005 Moldova Team Selection Test, 2

Tags: inequalities , function , inequalities proposed

Let $ a$, $ b$, $ c$ be positive reals such that $ a^4 \plus{} b^4 \plus{} c^4 \equal{} 3$. Prove that $ \sum\frac1{4 \minus{} ab}\leq1$, where the $ \sum$ sign stands for cyclic summation. [i]Alternative formulation:[/i] For any positive reals $ a$, $ b$, $ c$ satisfying $ a^4 \plus{} b^4 \plus{} c^4 \equal{} 3$, prove the inequality $ \frac{1}{4\minus{}bc}\plus{}\frac{1}{4\minus{}ca}\plus{}\frac{1}{4\minus{}ab}\leq 1$.

2012 China Northern MO, 6

Tags: function , logarithms , inequalities proposed , inequalities

Prove that\[(1+\frac{1}{3})(1+\frac{1}{3^2})\cdots(1+\frac{1}{3^n})< 2.\]

2001 Irish Math Olympiad, 5

Tags: inequalities proposed , inequalities

Prove that for all real numbers $ a,b$ with $ ab>0$ we have: $ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}10ab\plus{}b^2}{12}$ and find the cases of equality. Hence, or otherwise, prove that for all real numbers $ a,b$ $ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}ab\plus{}b^2}{3}$ and find the cases of equality.

2015 Vietnam National Olympiad, 2

Tags: inequalities , induction , calculus , derivative , inequalities proposed

If $a,b,c$ are nonnegative real numbers, then \[{ 3(a^2+b^2+c^2) \geq (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2 \geq (a+b+c)^2.}\]

1980 Vietnam National Olympiad, 3

Tags: inequalities proposed , inequalities

Let be given an integer $n\ge 2$ and a positive real number $p$. Find the maximum of \[\displaystyle\sum_{i=1}^{n-1} x_ix_{i+1},\] where $x_i$ are non-negative real numbers with sum $p$.

2012 JBMO ShortLists, 5

Tags: inequalities , inequalities proposed

Find the largest positive integer $n$ for which the inequality \[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\] holds true for all $a, b, c \in [0,1]$. Here we make the convention $\sqrt[1]{abc}=abc$.

1980 Vietnam National Olympiad, 2

Tags: function , inequalities , inequalities proposed

Let $m_1, m_2, \cdots ,m_k$ be positive numbers with the sum $S$. Prove that \[\displaystyle\sum_{i=1}^k\left(m_i +\frac{1}{m_i}\right)^2 \ge k\left(\frac{k}{S}+\frac{S}{k}\right)^2\]

2009 Bulgaria National Olympiad, 6

Tags: integration , calculus , derivative , inequalities proposed , inequalities

Prove that if $ a_{1},a_{2},\ldots,a_{n}$, $ b_{1},b_{2},\ldots,b_{n}$ are arbitrary taken real numbers and $ c_{1},c_{2},\ldots,c_{n}$ are positive real numbers, than $ \left(\sum_{i,j \equal{} 1}^{n}\frac {a_{i}a_{j}}{c_{i} \plus{} c_{j}}\right)\left(\sum_{i,j \equal{} 1}^{n}\frac {b_{i}b_{j}}{c_{i} \plus{} c_{j}}\right)\ge \left(\sum_{i,j \equal{} 1}^{n}\frac {a_{i}b_{j}}{c_{i} \plus{} c_{j}}\right)^{2}$.

2012 Indonesia TST, 1

Tags: inequalities , inequalities proposed

Let $a,b,c \in \mathbb{C}$ such that $a|bc| + b|ca| + c|ab| = 0$. Prove that $|(a-b)(b-c)(c-a)| \ge 3\sqrt{3}|abc|$.

2009 Kyrgyzstan National Olympiad, 7

Tags: inequalities , inequalities proposed

Does $ a^2 \plus{} b^2 \plus{} c^2 \leqslant 2(ab \plus{} bc \plus{} ca)$ hold for every $ a,b,c$ if it is known that $ a^4 \plus{} b^4 \plus{} c^4 \leqslant 2(a^2 b^2 \plus{} b^2 c^2 \plus{} c^2 a^2 )$.

2008 Canada National Olympiad, 3

Tags: inequalities , trigonometry , inequalities proposed , algebra

Let $ a$, $ b$, $ c$ be positive real numbers for which $ a \plus{} b \plus{} c \equal{} 1$. Prove that \[ {{a\minus{}bc}\over{a\plus{}bc}} \plus{} {{b\minus{}ca}\over{b\plus{}ca}} \plus{} {{c\minus{}ab}\over{c\plus{}ab}} \leq {3 \over 2}.\]

2015 India Regional MathematicaI Olympiad, 1

Tags: inequalities , inequalities proposed , rearrangement inequality , AM-GM , geometry , geometry proposed

Let $ABCD$ be a convex quadrilateral with $AB=a$, $BC=b$, $CD=c$ and $DA=d$. Suppose \[a^2+b^2+c^2+d^2=ab+bc+cd+da,\] and the area of $ABCD$ is $60$ sq. units. If the length of one of the diagonals is $30$ units, determine the length of the other diagonal.

1974 IMO Longlists, 14

Tags: inequalities , inequalities proposed

Let $n$ and $k$ be natural numbers and $a_1,a_2,\ldots ,a_n$ be positive real numbers satisfying $a_1+a_2+\cdots +a_n=1$. Prove that \[\dfrac {1} {a_1^{k}}+\dfrac {1} {a_2^{k}}+\cdots +\dfrac {1} {a_n^{k}} \ge n^{k+1}.\]