Olympiad problems

2021 Thailand TST, 2

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

2007 Pre-Preparation Course Examination, 4

Tags: inequalities

Prove that \[\sum_{i=-2007}^{2007}\frac{\sqrt{|i+1|}}{(\sqrt2)^{|i|}}>\sum_{i=-2007}^{2007}\frac{\sqrt{|i|}}{(\sqrt2)^{|i|}}\]

2013 District Olympiad, 1

Tags: limit , inequalities , calculus , calculus computations

Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate $\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).$

2010 Germany Team Selection Test, 3

Tags: function , inequalities , algebra , functional inequality

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] [i]Proposed by Igor Voronovich, Belarus[/i]

2008 Bundeswettbewerb Mathematik, 2

Tags: inequalities , greatest common divisor , least common multiple , relatively prime , number theory

Let the positive integers $ a,b,c$ chosen such that the quotients $ \frac{bc}{b\plus{}c},$ $ \frac{ca}{c\plus{}a}$ and $ \frac{ab}{a\plus{}b}$ are integers. Prove that $ a,b,c$ have a common divisor greater than 1.

2020 Jozsef Wildt International Math Competition, W7

Tags: inequalities

If $a,b>0$ then prove: $$\left(\frac{a+b}2-\frac{2ab}{a+b}\right)\operatorname{arctan}\left(\sqrt{ab}\right)+\left(\frac{2ab}{a+b}-\sqrt{ab}\right)\operatorname{arctan}\left(\frac{a+b}2\right)+\left(\sqrt{ab}-\frac{2ab}{a+b}\right)\operatorname{arctan}\left(\sqrt{\frac{a^2+b^2}2}\right)\ge0$$ [i]Proposed by Daniel Sitaru[/i]

2001 Saint Petersburg Mathematical Olympiad, 10.1

Tags: inequalities , quadratic , quadratic trinomial , algebra

Quadratic trinomials $f$ and $g$ with integer coefficients obtain only positive values and the inequality $\dfrac{f(x)}{g(x)}\geq \sqrt{2}$ is true $\forall x\in\mathbb{R}$. Prove that $\dfrac{f(x)}{g(x)}>\sqrt{2}$ is true $\forall x\in\mathbb{R}$ [I]Proposed by A. Khrabrov[/i]

2017 China Girls Math Olympiad, 5

Tags: inequalities , n-variable inequality

Let $0=x_0<x_1<\cdots<x_n=1$ .Find the largest real number$ C$ such that for any positive integer $ n $ , we have $$\sum_{k=1}^n x^2_k (x_k - x_{k-1})>C$$

2014 Cuba MO, 8

Tags: algebra , inequalities

Let $a$ and $b$ be real numbers. It is known that the graph of the parabola $y =ax^2 +b$ cuts the graph of the curve $y = x+1/x$ in exactly three points. Prove that $3ab < 1$.

2015 Saudi Arabia IMO TST, 3

Tags: inequalities , n-variable inequality , algebra

Let $a_1, a_2, ...,a_n$ be positive real numbers such that $$a_1 + a_2 + ... + a_n = a_1^2 + a_2^2 + ... + a_n^2$$ Prove that $$\sum_{1\le i<j\le n} a_ia_j(1 - a_ia_j) \ge 0$$ Võ Quốc Bá Cẩn.

2010 Macedonia National Olympiad, 2

Tags: rearrangement inequality , inequalities

Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality \[\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3\]

2006 Costa Rica - Final Round, 2

Tags: inequalities

If $ a$, $ b$, $ c$ are the sidelengths of a triangle, then prove that $ \frac {3\left(a^4 \plus{} b^4 \plus{} c^4\right)}{\left(a^2 \plus{} b^2 \plus{} c^2\right)^2} \plus{} \frac {bc \plus{} ca \plus{} ab}{a^2 \plus{} b^2 \plus{} c^2}\geq 2$.

2021 New Zealand MO, 7

Tags: algebra , inequalities , max

Let $a, b, c, d$ be integers such that $a > b > c > d \ge -2021$ and $$\frac{a + b}{b + c}=\frac{c + d}{d + a}$$ (and $b + c \ne 0 \ne d + a$). What is the maximum possible value of $ac$?

2009 China Team Selection Test, 3

Tags: induction , inequalities

Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y.$ Prove that $ 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|$

2024 Israel Olympic Revenge, P2

Tags: algebra , inequalities

Let $n\geq 2$ be an integer. For each natural $m$ and each integer sequence $0<k_1<k_2<\cdots <k_m$ for which $k_1+\cdots+k_m=n$, Michael wrote down the number $\frac{1}{k_1\cdot k_2\cdots k_m} $ on the board. Prove that the sum of the numbers on the board is less than $1$.

2012 IFYM, Sozopol, 4

Tags: geometry , geometric inequality , inequalities

In the right-angled $\Delta ABC$, with area $S$, a circle with area $S_1$ is inscribed and a circle with area $S_2$ is circumscribed. Prove the following inequality: $\pi \frac{S-S_1}{S_2} <\frac{1}{\pi-1}$.

2015 IFYM, Sozopol, 3

Tags: inequalities

Let $ a,b,c>0$ prove that:\[ \frac{a^{3}}{(a+b)^{3}}+\frac{b^{3}}{(b+c)^{3}}+\frac{c^{3}}{(c+a)^{3}}\geq \frac{3}{8} \] Good luck! :D

1972 USAMO, 4

Tags: geometry , 3d geometry , inequalities

Let $ R$ denote a non-negative rational number. Determine a fixed set of integers $ a,b,c,d,e,f$, such that for [i]every[/i] choice of $ R$, \[ \left| \frac{aR^2\plus{}bR\plus{}c}{dR^2\plus{}eR\plus{}f}\minus{}\sqrt[3]{2}\right| < \left|R\minus{}\sqrt[3]{2}\right|.\]

2010 Spain Mathematical Olympiad, 1

Tags: three variable inequality , inequalities

Let $a,b,c$ be three positive real numbers. Show that \[ \frac {a+b+3c}{3a+3b+2c}+\frac {a+3b+c}{3a+2b+3c}+\frac {3a+b+c}{2a+3b+3c} \ge \frac {15}{8}\]

2011 ISI B.Stat Entrance Exam, 4

Tags: function , integration , inequalities , calculus , calculus computations

Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0$ and $f''(x) \le f(x)$, for all $x\ge 0$. Show that $f'(0) \ge -\sqrt2$.

2003 China Girls Math Olympiad, 5

Tags: inequalities , function , induction , algebra

Let $ \{a_n\}^{\infty}_1$ be a sequence of real numbers such that $ a_1 \equal{} 2,$ and \[ a_{n\plus{}1} \equal{} a^2_n \minus{} a_n \plus{} 1, \forall n \in \mathbb{N}.\] Prove that \[ 1 \minus{} \frac{1}{2003^{2003}} < \sum^{2003}_{i\equal{}1} \frac{1}{a_i} < 1.\]

2013 Kosovo National Mathematical Olympiad, 3

Tags: inequalities

Find all numbers $x$ such that: $1+2\cdot2^x+3\cdot3^x<6^x$

1989 IMO Longlists, 73

Tags: symmetry , geometry , perimeter , calculus , trigonometry , integration , inequalities

We are given a finite collection of segments in the plane, of total length 1. Prove that there exists a line $ l$ such that the sum of the lengths of the projections of the given segments to the line $ l$ is less than $ \frac{2}{\pi}.$

1990 IMO Longlists, 4

Tags: function , trigonometry , inequalities

Find the minimal value of the function \[\begin{array}{c}\ f(x) =\sqrt{15 - 12 \cos x} + \sqrt{4 -2 \sqrt 3 \sin x}+\sqrt{7-4\sqrt 3 \sin x} +\sqrt{10-4 \sqrt 3 \sin x - 6 \cos x}\end{array}\]

1971 IMO Longlists, 18

Tags: inequalities

Let $a_1, a_2, \ldots, a_n$ be positive numbers, $m_g = \sqrt[n]{(a_1a_2 \cdots a_n)}$ their geometric mean, and $m_a = \frac{(a_1 + a_2 + \cdots + a_n)}{n}$ their arithmetic mean. Prove that \[(1 + m_g)^n \leq (1 + a_1) \cdots(1 + a_n) \leq (1 + m_a)^n.\]