Olympiad problems

2007 APMO, 4

Tags: inequalities

Let $x; y$ and $z$ be positive real numbers such that $\sqrt{x}+\sqrt{y}+\sqrt{z}= 1$. Prove that $\frac{x^{2}+yz}{\sqrt{2x^{2}(y+z)}}+\frac{y^{2}+zx}{\sqrt{2y^{2}(z+x)}}+\frac{z^{2}+xy}{\sqrt{2z^{2}(x+y)}}\geq 1.$

2010 India IMO Training Camp, 9

Tags: rectangle , geometry , algebra , inequalities

Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$. Show that there exists $j<k$ and $l<m$ such that \[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]

2008 China Team Selection Test, 3

Tags: inequalities , rearrangement inequality , induction

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.$

2005 China Team Selection Test, 1

Tags: rearrangement inequality , combinatorics , inequalities

Let $a_{1}$, $a_{2}$, …, $a_{6}$; $b_{1}$, $b_{2}$, …, $b_{6}$ and $c_{1}$, $c_{2}$, …, $c_{6}$ are all permutations of $1$, $2$, …, $6$, respectively. Find the minimum value of $\sum_{i=1}^{6}a_{i}b_{i}c_{i}$.

2008 China Girls Math Olympiad, 4

Tags: geometry , inequalities

Equilateral triangles $ ABQ$, $ BCR$, $ CDS$, $ DAP$ are erected outside of the convex quadrilateral $ ABCD$. Let $ X$, $ Y$, $ Z$, $ W$ be the midpoints of the segments $ PQ$, $ QR$, $ RS$, $ SP$, respectively. Determine the maximum value of \[ \frac {XZ\plus{}YW}{AC \plus{} BD}. \]

2019 LIMIT Category C, Problem 2

Tags: absolute value , inequalities

Let $x,y\in[0,\infty)$. Which of the following is true? $\textbf{(A)}~\left|\log\left(1+x^2\right)-\log\left(1+y^2\right)\right|\le|x-y|$ $\textbf{(B)}~\left|\sin^2x-\sin^2y\right|\le|x-y|$ $\textbf{(C)}~\left|\tan^{-1}x-\tan^{-1}y\right|\le|x-y|$ $\textbf{(D)}~\text{None of the above}$

2020 IMO Shortlist, A1

Tags: inequality , algebra , inequalities

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

1978 Austrian-Polish Competition, 3

Tags: trigonometry , inequalities

Prove that $$\sqrt[44]{\tan 1^\circ\cdot \tan 2^\circ\cdot \dots\cdot \tan 44^\circ}<\sqrt 2-1<\frac{\tan 1^\circ+ \tan 2^\circ+\dots+\tan 44^\circ}{44}.$$

2018 Junior Regional Olympiad - FBH, 4

Tags: positive real , inequalities

Let $a$, $b$ and $c$ be positive real numbers such that $a \geq b \geq c$. Prove the inequality: $\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \leq \frac{b}{a}+\frac{c}{b}+\frac{a}{c}$

1981 USAMO, 3

Tags: geometry , trigonometry , inequalities

If $A,B,C$ are the angles of a triangle, prove that \[-2 \le \sin{3A}+\sin{3B}+\sin{3C} \le \frac{3\sqrt{3}}{2}\] and determine when equality holds.

2017 Korea USCM, 7

Tags: trigonometry , real analysis , series , trigonometric series , inequalities

Prove the following inequality holds if $\{a_n\}$ is a deceasing sequence of positive reals, and $0<\theta<\frac{\pi}{2}$. $$\left|\sum_{n=1}^{2017} a_n \cos n\theta \right| \leq \frac{\pi a_1}{\theta}$$

1997 Poland - Second Round, 4

Tags: number theory , modular arithmetic , greatest common divisor , inequalities

There is a set with three elements: (2,3,5). It has got an interesting property: (2*3) mod 5=(2*5) mod 3=(3*5) mod 2. Prove that it is the only one set with such property.

2011 USAJMO, 2

Tags: inequalities

Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2+(a+b+c)^2\leq4$. Prove that \[\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\geq 3.\]

2021 Indonesia MO, 4

Tags: algebra , inequalities

Let $x,y$ and $z$ be positive reals such that $x + y + z = 3$. Prove that \[ 2 \sqrt{x + \sqrt{y}} + 2 \sqrt{y + \sqrt{z}} + 2 \sqrt{z + \sqrt{x}} \le \sqrt{8 + x - y} + \sqrt{8 + y - z} + \sqrt{8 + z - x} \]

VI Soros Olympiad 1999 - 2000 (Russia), 10.8

Tags: algebra , inequalities

There are $100$ positive numbers $a_1$, $a_2$, $...$, $a_{100}$ such that $$\frac{1}{a_1+1}+\frac{1}{a_2+1}+...+\frac{1}{a_{100}+1} \le 1.$$ Prove that $$a_1 \cdot a_2\cdot ... \cdot a_{100} \ge 99^{100}.$$

2015 District Olympiad, 1

Tags: algebra , inequalities

If $ a,b,c $ represent the lengths of the sides of a triangle, prove the inequality: $$ 3\le\sum_{\text{cyc}}\sqrt{\frac{a}{-a+b+c}} . $$

2003 Abels Math Contest (Norwegian MO), 2b

Tags: number theory , sum , inequalities

Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$

2012 USAJMO, 3

Tags: algebra , inequalities , polynomial , ratio

Let $a,b,c$ be positive real numbers. Prove that $\frac{a^3+3b^3}{5a+b}+\frac{b^3+3c^3}{5b+c}+\frac{c^3+3a^3}{5c+a} \geq \frac{2}{3}(a^2+b^2+c^2)$.

1986 IMO Longlists, 45

Tags: inequalities

Given $n$ real numbers $a_1 \leq a_2 \leq \cdots \leq a_n$, define \[M_1=\frac 1n \sum_{i=1}^{n} a_i , \quad M_2=\frac{2}{n(n-1)} \sum_{1 \leq i<j \leq n} a_ia_j, \quad Q=\sqrt{M_1^2-M_2}\] Prove that \[a_1 \leq M_1 - Q \leq M_1 + Q \leq a_n\] and that equality holds if and only if $a_1 = a_2 = \cdots = a_n.$

2015 Saudi Arabia JBMO TST, 1

Tags: inequalities , 3-variable inequality , three variable inequality , algebraic inequality , inequality

Let $a,b,c$ be positive real numbers. Prove that: $\left (a+b+c \right )\left ( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right ) \geq 9+3\sqrt[3]{\frac{(a-b)^2(b-c)^2(c-a)^2}{a^2b^2c^2}}$

2003 CentroAmerican, 3

Tags: logarithm , inequalities , induction , calculus

Let $a$ and $b$ be positive integers with $a>1$ and $b>2$. Prove that $a^b+1\ge b(a+1)$ and determine when there is inequality.

2014 JBMO Shortlist, 4

Tags: algebra , inequalities

With the conditions $a,b,c\in\mathbb{R^+}$ and $a+b+c=1$, prove that \[\frac{7+2b}{1+a}+\frac{7+2c}{1+b}+\frac{7+2a}{1+c}\geq\frac{69}{4}\]

2010 Iran MO (3rd Round), 2

Tags: inequalities , search

$a,b,c$ are positive real numbers. prove the following inequality: $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{(a+b+c)^2}\ge \frac{7}{25}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c})^2$ (20 points)

MathLinks Contest 7th, 1.3

Tags: induction , probability , vector , linear algebra , function , matrix , inequalities

We are given the finite sets $ X$, $ A_1$, $ A_2$, $ \dots$, $ A_{n \minus{} 1}$ and the functions $ f_i: \ X\rightarrow A_i$. A vector $ (x_1,x_2,\dots,x_n)\in X^n$ is called [i]nice[/i], if $ f_i(x_i) \equal{} f_i(x_{i \plus{} 1})$, for each $ i \equal{} 1,2,\dots,n \minus{} 1$. Prove that the number of nice vectors is at least \[ \frac {|X|^n}{\prod\limits_{i \equal{} 1}^{n \minus{} 1} |A_i|}. \]

2007 German National Olympiad, 1

Tags: real number , algebra , inequalities

Determine all real numbers $x$ such that for all positive integers $n$ the inequality $(1+x)^n \leq 1+(2^n -1)x$ is true.