Olympiad problems

2010 Contests, 2

Find all non-decreasing functions $f:\mathbb R^+\cup\{0\}\rightarrow\mathbb R^+\cup\{0\}$ such that for each $x,y\in \mathbb R^+\cup\{0\}$ \[f\left(\frac{x+f(x)}2+y\right)=2x-f(x)+f(f(y)).\]

2014 Mediterranean Mathematics Olympiad, 1

Tags: induction , triangle inequality , n-variable inequality , inequalities , algebra

Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that $ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$ (Proposed by Gerhard Woeginger, Austria)

1996 Moldova Team Selection Test, 4

Tags: inequalities , function , induction

Let $f: (0,+\infty)\rightarrow (0,+\infty)$ be a function satisfying the following condition: for arbitrary positive real numbers $x$ and $y$, we have $f(xy)\le f(x)f(y)$. Show that for arbitrary positive real number $x$ and natural number $n$, inequality $f(x^n)\le f(x)f(x^2)^{\dfrac{1}{2}}\dots f(x^n)^{\dfrac{1}{n}}$ holds.

2019 Mediterranean Mathematics Olympiad, 2

Tags: algebra , sequence , inequalities

Let $m_1<m_2<\cdots<m_s$ be a sequence of $s\ge2$ positive integers, none of which can be written as the sum of (two or more) distinct other numbers in the sequence. For every integer $r$ with $1\le r<s$, prove that \[ r\cdot m_r+m_s ~\ge~ (r+1)(s-1). \] (Proposed by Gerhard Woeginger, Austria)

2022 Iran MO (3rd Round), 6

Tags: inequalities , algebra , inequality

Prove that among any $9$ distinct real numbers, there exist $4$ distinct numbers $a,b,c,d$ such that $$(ac+bd)^2\ge\frac{9}{10}(a^2+b^2)(c^2+d^2)$$

2007 Romania Team Selection Test, 1

Tags: three variable inequality , trigonometry , fourier , inequalities

For $n\in\mathbb{N}$, $n\geq 2$, $a_{i}, b_{i}\in\mathbb{R}$, $1\leq i\leq n$, such that \[\sum_{i=1}^{n}a_{i}^{2}=\sum_{i=1}^{n}b_{i}^{2}=1, \sum_{i=1}^{n}a_{i}b_{i}=0. \] Prove that \[\left(\sum_{i=1}^{n}a_{i}\right)^{2}+\left(\sum_{i=1}^{n}b_{i}\right)^{2}\leq n. \] [i]Cezar Lupu & Tudorel Lupu[/i]

1991 IberoAmerican, 2

Tags: rotation , geometry , parallelogram , geometric transformation , inequalities

A square is divided in four parts by two perpendicular lines, in such a way that three of these parts have areas equal to 1. Show that the square has area equal to 4.

1996 Estonia Team Selection Test, 2

Tags: geometric inequality , inequalities , inradius , geometry

Let $a,b,c$ be the sides of a triangle, $\alpha ,\beta ,\gamma$ the corresponding angles and $r$ the inradius. Prove that $$a\cdot sin\alpha+b\cdot sin\beta+c\cdot sin\gamma\geq 9r$$

2017 China Northern MO, 1

Tags: sequence , inequalities

A sequence $\{a_n\}$ is defined as follows: $a_1 = 1$, $a_2 = \frac{1}{3}$, and for all $n \geq 1,$ $\frac{(1+a_n)(1+a_{n+2})}{(1+a_n+1)^2} = \frac{a_na_{n+2}}{a_{n+1}^2}$. Prove that, for all $n \geq 1$, $a_1 + a_2 + ... + a_n < \frac{34}{21}$.

2009 Singapore MO Open, 4

Tags: inequalities

find largest constant C st $\sum_{i=1}^{4} (x_i+1/x_i)^3 \geq C$ for all positive real numbers $x_1,..,x_4$ st $x_1^3+x_3^3+3x_1x_3=x_2+x_4=1$

2022 Latvia Baltic Way TST, P2

Tags: inequalities

Prove that for positive real numbers $a,b,c$ satisfying $abc=1$ the following inequality holds: $$ \frac{a}{b}+\frac{b}{c}+\frac{c}{a} \ge \frac{a^2+1}{2a}+\frac{b^2+1}{2b}+\frac{c^2+1}{2c}.$$

1994 Romania TST for IMO, 4:

Tags: inequalities , triangle inequality , geometry

Let be given two concentric circles of radii $R$ and $R_1 > R$. Let quadrilateral $ABCD$ is inscribed in the smaller circle and let the rays $CD, DA, AB, BC$ meet the larger circle at $A_1, B_1, C_1, D_1$ respectively. Prove that $$ \frac{\sigma(A_1B_1C_1D_1)}{\sigma(ABCD)} \geq \frac{R_1^2}{R^2}$$ where $\sigma(P)$ denotes the area of a polygon $P.$

1986 Flanders Math Olympiad, 2

Tags: induction , inequalities

Prove that for integer $n$ we have: \[n! \le \left( \frac{n+1}{2} \right)^n\] [size=75][i](please note that the pupils in the competition never heard of AM-GM or alikes, it is intended to be solved without any knowledge on inequalities)[/i][/size]

2006 Tournament of Towns, 3

Tags: inequalities

Let $a$ be some positive number. Find the number of integer solutions $x$ of inequality $100 < xa < 1000$ given that inequality $10 < xa < 100$ has exactly $5$ integer solutions. Consider all possible cases. [i](4 points)[/i]

2009 Indonesia TST, 1

Tags: inequalities

Let $ x_1,x_2,\dots,x_n$ be positive real numbers. Let $ m\equal{}\min\{x_1,x_2,\dots,x_n\}$, $ M\equal{}\max\{x_1,x_2,\dots,x_n\}$, $ A\equal{}\frac{1}{n}(x_1\plus{}x_2\plus{}\dots\plus{}x_n)$, and $ G\equal{}\sqrt[n]{x_1x_2 \dots x_n}$. Prove that \[ A\minus{}G \ge \frac{1}{n}(\sqrt{M}\minus{}\sqrt{m})^2.\]

2012 China Team Selection Test, 2

Tags: floor function , function , pigeonhole principle , difference of squares , inequalities , algebra , ceiling function

Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that \[0<|xy-zw|<C\alpha ^{-4}\] where $\alpha =\frac{|X|}{n}$.

1984 IMO Longlists, 10

Tags: geometry , tetrahedron , ratio , 3d geometry , vector , inequalities

Assume that the bisecting plane of the dihedral angle at edge $AB$ of the tetrahedron $ABCD$ meets the edge $CD$ at point $E$. Denote by $S_1, S_2, S_3$, respectively the areas of the triangles $ABC, ABE$, and $ABD$. Prove that no tetrahedron exists for which $S_1, S_2, S_3$ (in this order) form an arithmetic or geometric progression.

2013 Bosnia and Herzegovina Junior BMO TST, 2

Tags: inequalities , inequality , positive real , algebra

Let $a$, $b$ and $c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove the following inequality: $\frac{a}{3c(a^2-ab+b^2)} + \frac{b}{3a(b^2-bc+c^2)} + \frac{c}{3b(c^2-ca+a^2)} \leq \frac{1}{abc}$

MathLinks Contest 4th, 6.3

Tags: algebra , inequalities

If $n>2$ is an integer and $x_1, \ldots ,x_n$ are positive reals such that \[ \frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n} = n \] then the following inequality takes place \[ \frac{x_2^2+\cdots+x_n^2}{n-1}\cdot \frac {x_1^2+x_3^2+\cdots +x_n^2} {n-1} \cdots \frac{x_1^2+\cdots+x_{n-1}^2}{n-1}\geq \left(\frac{x_1^2+...+x_n^2}{n}\right)^{n-1}. \]

2013 South East Mathematical Olympiad, 1

Tags: algebra , cubic function , root , inequalities

Let $a,b$ be real numbers such that the equation $x^3-ax^2+bx-a=0$ has three positive real roots . Find the minimum of $\frac{2a^3-3ab+3a}{b+1}$.

2011 N.N. Mihăileanu Individual, 2

Tags: equation , algebra , inequalities

Determine the real numbers $ x,y,z $ from the interval $ (0,1) $ that satisfies $ x+y+z=1, $ and $$ \sqrt{\frac{x(1-y^2)}{2}} +\sqrt{\frac{y(1-z^2)}{2}} +\sqrt{\frac{z(1-x^2)}{2}} =\sqrt{1+xy+yz+zx} . $$ [i]Gabriela Constantinescu[/i]

2008 Romania National Olympiad, 2

Tags: algebra , circumcircle , complex number , geometry , inequalities

Let $ a,b,c$ be 3 complex numbers such that \[ a|bc| \plus{} b|ca| \plus{} c|ab| \equal{} 0.\] Prove that \[ |(a\minus{}b)(b\minus{}c)(c\minus{}a)| \geq 3\sqrt 3 |abc|.\]

2018 Czech-Polish-Slovak Match, 5

Tags: combinatorics , inequalities

In a $2 \times 3$ rectangle there is a polyline of length $36$, which can have self-intersections. Show that there exists a line parallel to two sides of the rectangle, which intersects the other two sides in their interior points and intersects the polyline in fewer than $10$ points. [i]Proposed by Josef Tkadlec, Czechia and Vojtech Bálint, Slovakia[/i]

2005 Iran MO (3rd Round), 5

Tags: geometry , trigonometry , inequalities

Suppose $a,b,c \in \mathbb R^+$and \[\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}=2\] Prove that $ab+ac+bc\leq \frac32$

1961 IMO Shortlist, 4

Tags: ratio , geometry , triangle , intersection , inequalities

Consider triangle $P_1P_2P_3$ and a point $p$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers \[ \dfrac{P_1P}{PQ_1}, \dfrac{P_2P}{PQ_2}, \dfrac{P_3P}{PQ_3} \] at least one is $\leq 2$ and at least one is $\geq 2$