Olympiad problems

2024 CMI B.Sc. Entrance Exam, 4

Tags: inequalities , algebra

(a) For non negetive $a,b,c, r$ prove that \[a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r (c-a)(c-b) \geq 0 \] (b) Find an inequality for non negative $a,b,c$ with $a^4+b^4+c^4 + abc(a+b+c)$ on the greater side. (c) Prove that if $abc = 1$ for non negative $a,b,c$, $a^4+b^4+c^4+a^3+b^3+c^3+a+b+c \geq \frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}+3$

2002 China National Olympiad, 3

Tags: inequalities

Suppose that $c\in\left(\frac{1}{2},1\right)$. Find the least $M$ such that for every integer $n\ge 2$ and real numbers $0<a_1\le a_2\le\ldots \le a_n$, if $\frac{1}{n}\sum_{k=1}^{n}ka_{k}=c\sum_{k=1}^{n}a_{k}$, then we always have that $\sum_{k=1}^{n}a_{k}\le M\sum_{k=1}^{m}a_{k}$ where $m=[cn]$

1995 India Regional Mathematical Olympiad, 5

Tags: quadratic , inequalities

Show that for any triangle $ABC$, the following inequality is true: \[ a^2 + b^2 +c^2 > \sqrt{3} max \{ |a^2 - b^2|, |b^2 -c^2|, |c^2 -a^2| \} . \]

2004 Romania Team Selection Test, 16

Tags: parallelogram , geometry , geometric transformation , inequalities , circumcircle , homothety , complex number

Three circles $\mathcal{K}_1$, $\mathcal{K}_2$, $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$. The point $O$ lies inside the triangle $ABC$. Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$. Let $ \alpha = \frac {OA_1}{AA_1} $, $ \beta= \frac {OB_1}{BB_1} $ and $ \gamma = \frac {OC_1}{CC_1} $ and let $R$ be the circumradius of the triangle $ABC$. Prove that \[ \alpha R_1 + \beta R_2 + \gamma R_3 \geq R. \]

2012 JBMO TST - Turkey, 4

Tags: function , inequalities , 3-variable inequality

Find the greatest real number $M$ for which \[ a^2+b^2+c^2+3abc \geq M(ab+bc+ca) \] for all non-negative real numbers $a,b,c$ satisfying $a+b+c=4.$

2018 Stars of Mathematics, 4

Tags: inequalities

Let be a natural number $ n\ge 4 $ and $ n $ nonnegative numbers $ a,b,\ldots ,c. $ Prove that $$ \prod_{\text{cyc} } (a+b+c)^2 \ge 2^n\prod_{\text{cyc} } (a+b)^2, $$ and tell in which circumstances equality happens.

2002 Iran MO (3rd Round), 1

Tags: inequalities

Let $a,b,c\in\mathbb R^{n}, a+b+c=0$ and $\lambda>0$. Prove that \[\prod_{cycle}\frac{|a|+|b|+(2\lambda+1)|c|}{|a|+|b|+|c|}\geq(2\lambda+3)^{3}\]

2018 VJIMC, 3

Tags: inequalities , algebra

Let $n$ be a positive integer and let $x_1,\dotsc,x_n$ be positive real numbers satisfying $\vert x_i-x_j\vert \le 1$ for all pairs $(i,j)$ with $1 \le i<j \le n$. Prove that \[\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1} \ge \frac{x_2+1}{x_1+1}+\frac{x_3+1}{x_2+1}+\dots+\frac{x_n+1}{x_{n-1}+1}+\frac{x_1+1}{x_n+1}.\]

2019 Junior Balkan Team Selection Tests - Romania, 3

Tags: inequalities , inequality

Real numbers $a,b,c,d$ such that $|a|>1$ , $|b|>1$ , $|c|>1$ , $|d|>1$ and $ab(c+d)+dc(a+b)+a+b+c+d=0$ then prove that $\frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}+\frac{1}{d-1} >0$

1998 China Team Selection Test, 3

Tags: trigonometry , inequalities

For a fixed $\theta \in \lbrack 0, \frac{\pi}{2} \rbrack$, find the smallest $a \in \mathbb{R}^{+}$ which satisfies the following conditions: [b]I. [/b] $\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} > 1$. [b]II.[/b] There exists $x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta}, \frac{\sqrt a}{\cos \theta} \rbrack$ such that $\lbrack (1 - x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} + \lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta} \rbrack^{2} \leq a$.

2022 Assam Mathematical Olympiad, 6

Tags: inequalities

Prove that $n! \geq n^{\frac{n}{2}}$ for all natural numbers $n$. Also, show that the inequality is strict for $n > 2$.

2012 Mathcenter Contest + Longlist, 4

Tags: geometry , inequalities , geometric inequality , algebra

Let $a,b,c$ be the side lengths of any triangle. Prove that $$\frac{a}{\sqrt{2b^2+2c^2-a^2}}+\frac{b}{\sqrt{2c^2+2a^2-b^2 }}+\frac{c}{\sqrt{2a^2+2b^2-c^2}}\ge \sqrt{3}.$$ [i](Zhuge Liang)[/i]

2007 Ukraine Team Selection Test, 8

Tags: algebra , polynomial , inequalities

$ F(x)$ is polynomial with real coefficients. $ F(x) \equal{} x^{4}\plus{}a_{1}x^{3}\plus{}a_{2}x^{2}\plus{}a_{1}x^{1}\plus{}a_{0}$. $ M$ is local maximum and $ m$ is minimum. Prove that $ \frac{3}{10}(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}< M\minus{}m < 3(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}$

2003 VJIMC, Problem 4

Tags: function , calculus , integration , inequalities

Let $f,g:[0,1]\to(0,+\infty)$ be two continuous functions such that $f$ and $\frac gf$ are increasing. Prove that $$\int^1_0\frac{\int^x_0f(t)\text dt}{\int^x_0g(t)\text dt}\text dx\le2\int^1_0\frac{f(t)}{g(t)}\text dt.$$

2022 China Northern MO, 3

Tags: inequalities , algebra , recurrence relation

Let $\{a_n\}$ be a sequence of positive terms such that $a_{n+1}=a_n+ \frac{n^2}{a_n}$ . Let $b_n =a_n-n$ . (1) Are there infinitely many $n$ such that $b_n \ge 0$ ? (2) Prove that there is a positive number $M$ such that $\sum^{\infty}_{n=3} \frac{b_n}{n+1}<M$.

2020 Jozsef Wildt International Math Competition, W37

Tags: absolute value , inequalities , trigonometry

For all $x>0$ prove $$\frac{\sin^2x-x}{\ln\left(\frac{\sin^2x}x\right)^{\sqrt x}}+\frac{\cos^2x-x}{\ln\left(\frac{\cos^2x}x\right)^{\sqrt x}}>|\sin x|+|\cos x|$$ [i]Proposed by Pirkulyiev Rovsen[/i]

2019 Middle European Mathematical Olympiad, 7

Tags: inequalities , number theory

Let $a,b$ and $c$ be positive integers satisfying $a<b<c<a+b$. Prove that $c(a-1)+b$ does not divide $c(b-1)+a$. [i]Proposed by Dominik Burek, Poland[/i]

2014 ELMO Shortlist, 6

Tags: inequalities

Let $a,b,c$ be positive reals such that $a+b+c=ab+bc+ca$. Prove that \[ (a+b)^{ab-bc}(b+c)^{bc-ca}(c+a)^{ca-ab} \ge a^{ca}b^{ab}c^{bc}. \][i]Proposed by Sammy Luo[/i]

2010 Switzerland - Final Round, 4

Tags: inequalities , algebra , function , triangle inequality , three variable inequality

Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that \[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]

2017 Mathematical Talent Reward Programme, SAQ: P 2

Tags: inequalities

Let $a$, $b$, $c$ be positive reals such that $a+b+c=3$. Show that $$\sqrt{\frac{a}{b+c}} + \sqrt{\frac{b}{c+a}} + \sqrt{\frac{c}{a+b}} \leq \frac{6}{\sqrt(a+b)(b+c)(c+a)}$$

2013 Saudi Arabia Pre-TST, 1.1

Tags: algebra , inequalities

Let $-1 \le x, y \le 1$. Prove the inequality $$2\sqrt{(1- x^2)(1 - y^2) } \le 2(1 - x)(1 - y) + 1 $$

2000 France Team Selection Test, 3

Tags: inequalities

$a,b,c,d$ are positive reals with sum $1$. Show that $\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a} \ge \frac{1}{2}$ with equality iff $a=b=c=d=\frac{1}{4}$.

2012 Grigore Moisil Intercounty, 1

Tags: geometry , inequalities

For $ x\in\mathbb{R} , $ determine the minimum of $ \sqrt{(x-1)^2+\left( x^2-5\right)^2} +\sqrt{(x+2)^2+\left( x^2+1 \right)^2} $ and the maximum of $ \sqrt{(x-1)^2+\left( x^2-5\right)^2} -\sqrt{(x+2)^2+\left( x^2+1 \right)^2} . $ [i]Vasile Pop[/i]

1996 Romania Team Selection Test, 13

Tags: inequalities

Let $ x_1,x_2,\ldots,x_n $ be positive real numbers and $ x_{n+1} = x_1 + x_2 + \cdots + x_n $. Prove that \[ \sum_{k=1}^n \sqrt { x_k (x_{n+1} - x_k)} \leq \sqrt { \sum_{k=1}^n x_{n+1}(x_{n+1}-x_k)}. \] [i]Mircea Becheanu[/i]

PEN J Problems, 10

Tags: inequalities , divisor function

Show that [list=a] [*] if $n>49$, then there are positive integers $a>1$ and $b>1$ such that $a+b=n$ and $\frac{\phi(a)}{a}+\frac{\phi(b)}{b}<1$. [*] if $n>4$, then there are $a>1$ and $b>1$ such that $a+b=n$ and $\frac{\phi(a)}{a}+\frac{\phi(b)}{b}>1$.[/list]