Olympiad problems

1964 IMO, 2

Suppose $a,b,c$ are the sides of a triangle. Prove that \[ a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc \]

2021 China Girls Math Olympiad, 1

Tags: algebra , inequalities

Let $n \in \mathbb{N}^+,$ $x_1,x_2,...,x_{n+1},p,q\in \mathbb{R}^+ $ , $p<q$ and $x^p_{n+1}>\sum_{i=1}^{n}x^p_{i}.$ Prove that $(1)x^q_{n+1}>\sum_{i=1}^{n}x^q_{i};$ $(2)\left(x^p_{n+1}-\sum_{i=1}^{n}x^p_{i}\right)^{\frac{1}{p}}<\left(x^q_{n+1}-\sum_{i=1}^{n}x^q_{i}\right)^{\frac{1}{q}}.$

2002 All-Russian Olympiad, 3

Tags: geometry , circumcircle , inequalities

Let O be the circumcenter of a triangle ABC. Points M and N are choosen on the sides AB and BC respectively so that the angle AOC is two times greater than angle MON. Prove that the perimeter of triangle MBN is not less than the lenght of side AC

2017 Caucasus Mathematical Olympiad, 7

Tags: geometric inequality , 3d geometry , geometry , inequalities

$8$ ants are placed on the edges of the unit cube. Prove that there exists a pair of ants at a distance not exceeding $1$.

2000 Singapore Team Selection Test, 3

Tags: combinatorial geometry , combinatorics , geometric inequalities , inequalities

There are $n$ blue points and $n$ red points on a straight line. Prove that the sum of all distances between pairs of points of the same colour is less than or equal to the sum of all distances between pairs of points of different colours

2009 Romania Team Selection Test, 3

Tags: trigonometry , floor function , algebra , complex analysis , complex number , function , inequalities

Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.

1995 Singapore MO Open, 5

Tags: algebra , sum of powers , inequalities

Let $a, b, c, d$ be four positive real numbers. Prove that $$a^{10} + b^{10}+c^{10} + d^{10} \ge (0.1a + 0.2b + 0.3c + 0.4d)^{10} + (0.4a + 0.3b + 0.2c + 0.ld)^{10} + (0.2a + 0.4b + 0.1c + 0.3d)^{10} + (0.3a + 0.1b + 0.4c + 0.2d)^{10}$$

2002 APMO, 4

Tags: inequalities

Let $x,y,z$ be positive numbers such that \[ {1\over x}+{1\over y}+{1\over z}=1. \] Show that \[ \sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\ge\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z} \]

2021 Taiwan TST Round 2, A

Tags: inequalities , algebra , inequality

[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 $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

2012 Chile National Olympiad, 2

Tags: sum , number theory , inequalities

Let $a_1,a_2,...,a_n$ be all positive integers with $2012$ digits or less, none of which is a $9$. Prove that $$ \frac{1}{a_1}+\frac{1}{a_2}+ ... +\frac{1}{a_{n}}\le 80.$$

2002 IMC, 12

Tags: inequalities , gradient , real analysis

Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a convex function whose gradient $\nabla f$ exists at every point of $\mathbb{R}^{n}$ and satisfies the condition $$\exists L>0\; \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||\leq L||x_{1}-x_{2}||.$$ Prove that $$ \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||^{2}\leq L\langle\nabla f(x_{1})-\nabla f(x_{2}), x_{1}-x_{2}\rangle. $$

2009 Jozsef Wildt International Math Competition, W. 15

Tags: trigonometry , inequalities

Let a triangle $\triangle ABC$ and the real numbers $x$, $y$, $z>0$. Prove that $$x^n\cos\frac{A}{2}+y^n\cos\frac{B}{2}+z^n\cos\frac{C}{2}\geq (yz)^{\frac{n}{2}}\sin A +(zx)^{\frac{n}{2}}\sin B +(xy)^{\frac{n}{2}}\sin C$$

2000 Turkey MO (2nd round), 3

Tags: inequalities , algebra , function

Find all continuous functions $f:[0,1]\to [0,1]$ for which there exists a positive integer $n$ such that $f^{n}(x)=x$ for $x \in [0,1]$ where $f^{0} (x)=x$ and $f^{k+1}=f(f^{k}(x))$ for every positive integer $k$.

2002 Austrian-Polish Competition, 8

Tags: algebra , limit , trigonometry , inequalities

Determine the number of real solutions of the system \[\left\{ \begin{aligned}\cos x_{1}&= x_{2}\\ &\cdots \\ \cos x_{n-1}&= x_{n}\\ \cos x_{n}&= x_{1}\\ \end{aligned}\right.\]

1996 IMO Shortlist, 8

Tags: inequalities , trigonometry , circumcircle , geometry

Let $ ABCD$ be a convex quadrilateral, and let $ R_A, R_B, R_C, R_D$ denote the circumradii of the triangles $ DAB, ABC, BCD, CDA,$ respectively. Prove that $ R_A \plus{} R_C > R_B \plus{} R_D$ if and only if $ \angle A \plus{} \angle C > \angle B \plus{} \angle D.$

2019 Saudi Arabia JBMO TST, 4

Tags: inequalities

Prove that if $x, y, z$ are reals, then $x^2(3y^2+3z^2-2yz)=>yz(2xy+2xz-yz)$

1980 Vietnam National Olympiad, 3

Tags: 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 $p$.

1986 India National Olympiad, 5

Tags: inequalities , algebra , polynomial

If $ P(x)$ is a polynomial with integer coefficients and $ a$, $ b$, $ c$, three distinct integers, then show that it is impossible to have $ P(a)\equal{}b$, $ P(b)\equal{}c$, $ P(c)\equal{}a$.

1977 IMO Longlists, 54

Tags: function , inequalities

If $0 \leq a \leq b \leq c \leq d,$ prove that \[a^bb^cc^dd^a \geq b^ac^bd^ca^d.\]

1989 China National Olympiad, 2

Tags: inequalities

Let $x_1, x_2, \dots ,x_n$ ($n\ge 2$) be positive real numbers satisfying $\sum^{n}_{i=1}x_i=1$. Prove that:\[\sum^{n}_{i=1}\dfrac{x_i}{\sqrt{1-x_i}}\ge \dfrac{\sum_{i=1}^{n}\sqrt{x_i}}{\sqrt{n-1}}.\]

2011 Mediterranean Mathematics Olympiad, 2

Tags: inequalities , geometric sequence , function , ratio , combinatorics

Let $A$ be a finite set of positive reals, let $B = \{x/y\mid x,y\in A\}$ and let $C = \{xy\mid x,y\in A\}$. Show that $|A|\cdot|B|\le|C|^2$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

2008 Germany Team Selection Test, 2

Tags: geometry , trigonometry , inequalities , circumcircle

For three points $ X,Y,Z$ let $ R_{XYZ}$ be the circumcircle radius of the triangle $ XYZ.$ If $ ABC$ is a triangle with incircle centre $ I$ then we have: \[ \frac{1}{R_{ABI}} \plus{} \frac{1}{R_{BCI}} \plus{} \frac{1}{R_{CAI}} \leq \frac{1}{\bar{AI}} \plus{} \frac{1}{\bar{BI}} \plus{} \frac{1}{\bar{CI}}.\]

1998 Czech and Slovak Match, 3

Tags: hexagon , triangle inequality , convex polygon , inequalities

Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE, EF = FA$. Prove that $\frac{BC}{BE} +\frac{DE}{DA} +\frac{FA}{FC} \ge \frac{3}{2}$ . When does equality occur?

1958 AMC 12/AHSME, 17

Tags: inequalities , logarithm

If $ x$ is positive and $ \log{x} \ge \log{2} \plus{} \frac{1}{2}\log{x}$, then: $ \textbf{(A)}\ {x}\text{ has no minimum or maximum value}\qquad \\ \textbf{(B)}\ \text{the maximum value of }{x}\text{ is }{1}\qquad \\ \textbf{(C)}\ \text{the minimum value of }{x}\text{ is }{1}\qquad \\ \textbf{(D)}\ \text{the maximum value of }{x}\text{ is }{4}\qquad \\ \textbf{(E)}\ \text{the minimum value of }{x}\text{ is }{4}$

2019 Saint Petersburg Mathematical Olympiad, 3

Tags: inequalities

Let $a, b$ and $c$ be non-zero natural numbers such that $c \geq b$ . Show that $$a^b\left(a+b\right)^c>c^b a^c.$$