This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 6530

2024 Thailand Mathematical Olympiad, 8
Tags: geometry , inequalities , vector geometry
Let $ABCDEF$ be a convex hexagon and denote $U$,$V$,$W$,$X$,$Y$ and $Z$ be the midpoint of $AB$,$BC$,$CD$,$DE$,$EF$ and $FA$ respectively. Prove that the length of $UX$,$VY$,$WZ$ can be the length of each sides of some triangle.

1995 IMC, 6
Tags: inequalities , real analysis
Let $p>1$. Show that there exists a constant $K_{p} >0$ such that for every $x,y\in \mathbb{R}$ with $|x|^{p}+|y|^{p}=2$, we have $$(x-y)^{2} \leq K_{p}(4-(x+y)^{2}).$$

2021 Latvia TST, 2.5
Tags: algebra , inequality , 4-variable inequality , inequalities
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

1989 Romania Team Selection Test, 1
Tags: induction , inequalities , limit , algebra
Prove that $\sqrt {1+\sqrt {2+\ldots +\sqrt {n}}}<2$, $\forall n\ge 1$.

2002 China Team Selection Test, 2
Tags: geometry , incenter , circumcircle , geometric transformation , reflection , parallelogram , inequalities
Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively, such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively. Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$. Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$

2024 Auckland Mathematical Olympiad, 3
Tags: inequalities
Prove that for arbitrary real numbers $a$ and $b$ the following inequality is true $$a^2 +ab+b^2 \geq 3(a+b-1).$$

2023 Greece JBMO TST, 3
Tags: algebra , inequalities
Let $a, b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 $$ When equality holds?

2023 Brazil EGMO Team Selection Test, 3
Tags: inequalities , jensen , square root inequality , square root
Let $a_1, a_2, \ldots , a_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = 1$. Prove that $$\dfrac{a_1}{\sqrt{1-a_1}}+\cdots+\dfrac{a_n}{\sqrt{1-a_n}} \geq \dfrac{1}{\sqrt{n-1}}(\sqrt{a_1}+\cdots+\sqrt{a_n}).$$

2002 Bosnia Herzegovina Team Selection Test, 1
Tags: inequalities
Let $a,b,c$ be real numbers such that $a^2+b^2+c^2=1$. Prove that \[\frac{a^2}{1+2bc}+\frac{b^2}{1+2ca}+\frac{c^2}{1+2ab} \ge \frac35\]

2005 Tuymaada Olympiad, 4
Tags: inequalities , trigonometry , 3d geometry , tetrahedron , triangle inequality , geometry
In a triangle $ABC$, let $A_{1}$, $B_{1}$, $C_{1}$ be the points where the excircles touch the sides $BC$, $CA$ and $AB$ respectively. Prove that $A A_{1}$, $B B_{1}$ and $C C_{1}$ are the sidelenghts of a triangle. [i]Proposed by L. Emelyanov[/i]

1966 IMO Longlists, 50
Tags: inequalities , trigonometry , geometry
For any quadrilateral with the side lengths $a,$ $b,$ $c,$ $d$ and the area $S,$ prove the inequality$S\leq \frac{a+c}{2}\cdot \frac{b+d}{2}.$

2012 China Girls Math Olympiad, 1
Tags: induction , marvio , inequalities
Let $ a_1, a_2,\ldots, a_n$ be non-negative real numbers. Prove that $\frac{1}{1+ a_1}+\frac{ a_1}{(1+ a_1)(1+ a_2)}+\frac{ a_1 a_2}{(1+ a_1)(1+ a_2)(1+ a_3)}+$ $\cdots+\frac{ a_1 a_2\cdots a_{n-1}}{(1+ a_1)(1+ a_2)\cdots (1+ a_n)} \le 1.$

2017 Silk Road, 3
Tags: inequalities
Prove that among any $42$ numbers from the interval $[1,10^6]$, you can choose four numbers so that for any permutation $(a, b, c, d)$ of these numbers, the inequality $$25 (ab + cd) (ad + bc) \ge 16 (ac + bd)^ 2$$ holds.

1998 Vietnam National Olympiad, 2
Tags: analytic geometry , trigonometry , inequalities
Find minimum value of $F(x,y)=\sqrt{(x+1)^{2}+(y-1)^{2}}+\sqrt{(x-1)^{2}+(y+1)^{2}}+\sqrt{(x+2)^{2}+(y+2)^{2}}$, where $x,y\in\mathbb{R}$.

2009 Jozsef Wildt International Math Competition, W. 26
Tags: inequalities
If $a_i >0$ ($i=1, 2, \cdots , n$) and $\sum \limits_{i=1}^n a_i^k=1$, where $1\leq k\leq n+1$, then $$\sum \limits_{i=1}^n a_i + \frac{1}{\prod \limits_{i=1}^n a_i} \geq n^{1-\frac{1}{k}}+n^{\frac{n}{k}}$$

2003 China Girls Math Olympiad, 1
Tags: inequalities , cauchy inequality , geometry
Let $ ABC$ be a triangle. Points $ D$ and $ E$ are on sides $ AB$ and $ AC,$ respectively, and point $ F$ is on line segment $ DE.$ Let $ \frac {AD}{AB} \equal{} x,$ $ \frac {AE}{AC} \equal{} y,$ $ \frac {DF}{DE} \equal{} z.$ Prove that (1) $ S_{\triangle BDF} \equal{} (1 \minus{} x)y S_{\triangle ABC}$ and $ S_{\triangle CEF} \equal{} x(1 \minus{} y) (1 \minus{} z)S_{\triangle ABC};$ (2) $ \sqrt [3]{S_{\triangle BDF}} \plus{} \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.$

2010 Bundeswettbewerb Mathematik, 1
Tags: inequalities , algebra , sidelengths , geometric inequality
Let $a, b, c$ be the side lengths of an non-degenerate triangle with $a \le b \le c$. With $t (a, b, c)$ denote the minimum of the quotients $\frac{b}{a}$ and $\frac{c}{b}$ . Find all values that $t (a, b, c)$ can take.

2014 Thailand TSTST, 1
Tags: inequalities
Let $x, y, z$ be positive real numbers. Prove that $$4(x^2+y^2+z^2)\geq3(xy+yz+zx).$$

2019 Jozsef Wildt International Math Competition, W. 55
Tags: product , summation , inequalities
Let $a_1,a_2,\cdots ,a_n$ be $n$ positive numbers such that $\sum \limits_{i=1}^n\sqrt{a_i}=\sqrt{n}$. Then$$\prod \limits_{i=1}^{n-1}\left(1+\frac{1}{a_i}\right)^{a_{i+1}}\left(1+\frac{1}{a_n}\right)^{a_1}\geq 1+\frac{n}{\sum \limits_{i=1}^na_i}$$

2008 Romania Team Selection Test, 1
Tags: function , inequalities
Let $ n \geq 3$ be an odd integer. Determine the maximum value of \[ \sqrt{|x_{1}\minus{}x_{2}|}\plus{}\sqrt{|x_{2}\minus{}x_{3}|}\plus{}\ldots\plus{}\sqrt{|x_{n\minus{}1}\minus{}x_{n}|}\plus{}\sqrt{|x_{n}\minus{}x_{1}|},\] where $ x_{i}$ are positive real numbers from the interval $ [0,1]$.

2014 China Western Mathematical Olympiad, 6
Tags: ceiling function , inequalities
Let $n\ge 2$ is a given integer , $x_1,x_2,\ldots,x_n $ be real numbers such that $(1) x_1+x_2+\ldots+x_n=0 $, $(2) |x_i|\le 1$ $(i=1,2,\cdots,n)$. Find the maximum of Min$\{|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|\}$.

1992 China National Olympiad, 1
Tags: inequalities , triangle inequality , algebra
Let equation $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots +a_1x+a_0=0$ with real coefficients satisfy $0<a_0\le a_1\le a_2\le \dots \le a_{n-1}\le 1$. Suppose that $\lambda$ ($|\lambda|>1$) is a complex root of the equation, prove that $\lambda^{n+1}=1$.

2013 Greece JBMO TST, 1
Tags: algebra , inequalities
If x,y<0 prove that $\left(x+\frac{2}{y} \right) \left(\frac{y}{x}+2 \right)\geq 8$. When do we have equality?

2019 Silk Road, 2
Tags: algebra , inequalities
Let $ a_1, $ $ a_2, $ $ \ldots, $ $ a_ {99} $ be positive real numbers such that $ ia_j + ja_i \ge i + j $ for all $ 1 \le i <j \le 99. $ Prove , that $ (a_1 + 1) (a_2 + 2) \ldots (a_ {99} +99) \ge 100!$ .

2014 Balkan MO Shortlist, A4
Tags: inequalities
$\boxed{A4}$Let $m_1,m_2,m_3,n_1,n_2$ and $n_3$ be positive real numbers such that \[(m_1-n_1)(m_2-n_2)(m_3-n_3)=m_1m_2m_3-n_1n_2n_3\] Prove that \[(m_1+n_1)(m_2+n_2)(m_3+n_3)\geq8m_1m_2m_3\]