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

2018 Kazakhstan National Olympiad, 4
Tags: inequalities , algebra , analysis
Prove that for all reas $a,b,c,d\in(0,1)$ we have $$\left(ab-cd\right)\left(ac+bd\right)\left(ad-bc\right)+\min{\left(a,b,c,d\right)} < 1.$$

1967 IMO Longlists, 4
Tags: inequalities , trigonometry , geometry
Suppose, medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that: (a) The medians of the triangle correspond to the sides of a right-angled triangle. (b) If $a,b,c$ are the side-lengths of the triangle, then, the following inequality holds:\[5(a^2+b^2-c^2)\geq 8ab\]

2018 Azerbaijan BMO TST, 2
Tags: three variable inequality , inequalities , algebra
Let $M = \{(a,b,c)\in R^3 :0 <a,b,c<\frac12$ with $a+b+c=1 \}$ and $f: M\to R$ given as $$f(a,b,c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{abc}$$ Find the best (real) bounds $\alpha$ and $\beta$ such that $f(M) = \{f(a,b,c): (a,b,c)\in M\}\subseteq [\alpha,\beta]$ and determine whether any of them is achievable.

2009 Abels Math Contest (Norwegian MO) Final, 4b
Tags: algebra , inequalities , sum
Let $x = 1 - 2^{-2009}$. Show that $x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010$ for all positive integers $m$.

2006 Estonia Math Open Junior Contests, 10
Tags: inequalities
Let a, b, c be positive integers. Prove that the inequality \[ (x\minus{}y)^a(x\minus{}z)^b(y\minus{}z)^c \ge 0 \] holds for all reals x, y, z if and only if a, b, c are even.

2022 Greece Junior Math Olympiad, 1
Tags: algebra , polynomial , inequalities
(a) Find the value of the real number $k$, for which the polynomial $P(x)=x^3-kx+2$ has the number $2$ as a root. In addition, for the value of $k$ you will find, write this polynomial as the product of two polynomials with integer coefficients. (b) If the positive real numbers $a,b$ satisfy the equation $$2a+b+\frac{4}{ab}=10,$$ find the maximum possible value of $a$.

2023 Regional Competition For Advanced Students, 1
Tags: inequalities , algebra
Let $a$, $b$ and $c$ be real numbers with $0 \le a, b, c \le 2$. Prove that $$(a - b)(b - c)(a- c) \le 2.$$ When does equality hold? [i](Karl Czakler)[/i]

1983 All Soviet Union Mathematical Olympiad, 355
Tags: geometry , geometric inequality , inequalities , area
The point $D$ is the midpoint of the side $[AB]$ of the triangle $ABC$ . The points $E$ and $F$ belong to $[AC]$ and $[BC]$ respectively. Prove that the area of triangle $DEF$ area does not exceed the sum of the areas of triangles $ADE$ and $BDF$.

2006 Iran Team Selection Test, 5
Tags: circumcircle , trigonometry , inequalities , function , incenter , geometry
Let $ABC$ be an acute angle triangle. Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$. Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$. Prove that \[ 2(PQ+QR+RP)\geq DE+EF+FD \]

2008 International Zhautykov Olympiad, 1
Tags: inequalities , modular arithmetic , number theory
For each positive integer $ n$,denote by $ S(n)$ the sum of all digits in decimal representation of $ n$. Find all positive integers $ n$,such that $ n\equal{}2S(n)^3\plus{}8$.

2003 Abels Math Contest (Norwegian MO), 2b
Tags: sum , inequalities , number theory
Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$

2008 Germany Team Selection Test, 1
Tags: inequalities
Let $ a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $ a^2_1 \plus{} a^2_2 \plus{} \ldots \plus{} a^2_{100} \equal{} 1.$ Prove that \[ a^2_1 \cdot a_2 \plus{} a^2_2 \cdot a_3 \plus{} \ldots \plus{} a^2_{100} \cdot a_1 < \frac {12}{25}. \] [i]Author: Marcin Kuzma, Poland[/i]

1983 AIME Problems, 9
Tags: trigonometry , inequalities , calculus
Find the minimum value of \[\frac{9x^2 \sin^2 x + 4}{x \sin x}\] for $0 < x < \pi$.

2009 Romania National Olympiad, 2
Tags: minimal , complex number , inequalities , complex disc
Let be a real number $ a\in \left[ 2+\sqrt 2,4 \right] . $ Find $ \inf_{\stackrel{z\in\mathbb{C}}{|z|\le 1}} \left| z^2-az+a \right| . $

2023 Mexican Girls' Contest, 7
Tags: inequalities , algebra
Suppose $a$ and $b$ are real numbers such that $0 < a < b < 1$. Let $$x= \frac{1}{\sqrt{b}} - \frac{1}{\sqrt{b+a}},\hspace{1cm} y= \frac{1}{b-a} - \frac{1}{b}\hspace{0.5cm}\textrm{and}\hspace{0.5cm} z= \frac{1}{\sqrt{b-a}} - \frac{1}{\sqrt{b}}.$$ Show that $x$, $y$, $z$ are always ordered from smallest to largest in the same way, regardless of the choice of $a$ and $b$. Find this order among $x$, $y$, $z$.

1966 IMO Longlists, 38
Tags: inequalities , circles , geometry , geometric inequality
Two concentric circles have radii $R$ and $r$ respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between $\frac 32 \cdot \frac{\sqrt R +\sqrt r }{\sqrt R -\sqrt r } -1$ and $\frac{63}{20} \cdot \frac{R+r}{R-r}.$

2009 Postal Coaching, 3
Tags: inequalities , polygon , cyclic , geometric inequalities
Let $\Omega$ be an $n$-gon inscribed in the unit circle, with vertices $P_1, P_2, ..., P_n$. (a) Show that there exists a point $P$ on the unit circle such that $PP_1 \cdot PP_2\cdot ... \cdot PP_n \ge 2$. (b) Suppose for each $P$ on the unit circle, the inequality $PP_1 \cdot PP_2\cdot ... \cdot PP_n \le 2$ holds. Prove that $\Omega$ is regular.

2005 Thailand Mathematical Olympiad, 21
Tags: trigonometry , algebra , min , inequalities
Compute the minimum value of $cos(a-b) + cos(b-c) + cos(c-a)$ as $a,b,c$ ranges over the real numbers.

2013 Junior Balkan Team Selection Tests - Romania, 3
Tags: min , max , inequalities , algebra
Find the minimum and the maximum value of the expression $\sqrt{4 -a^2} +\sqrt{4 -b^2} +\sqrt{4 -c^2}$ where $a,b, c$ are positive real numbers satisfying the condition $a^2 + b^2 + c^2=6$

2000 Korea - Final Round, 3
Tags: search , inequalities
The real numbers $a,b,c,x,y,$ and $z$ are such that $a>b>c>0$ and $x>y>z>0$. Prove that \[\frac {a^2x^2}{(by+cz)(bz+cy)}+\frac{b^2y^2}{(cz+ax)(cx+az)}+\frac{c^2z^2}{(ax+by)(ay+bx)}\ge \frac{3}{4}\]

2005 Belarusian National Olympiad, 1
Tags: algebra , inequalities
Prove for positive numbers: $$(a^2+b+\frac{3}{4})(b^2+a+\frac{3}{4}) \geq (2a+\frac{1}{2})(2b+\frac{1}{2})$$

2006 International Zhautykov Olympiad, 1
Tags: inequalities , function , number theory
Solve in positive integers the equation \[ n \equal{} \varphi(n) \plus{} 402 , \] where $ \varphi(n)$ is the number of positive integers less than $ n$ having no common prime factors with $ n$.

2014 China Western Mathematical Olympiad, 7
Tags: inequalities
In the plane, Point $ O$ is the center of the equilateral triangle $ABC$ , Points $P,Q$ such that $\overrightarrow{OQ}=2\overrightarrow{PO}$. Prove that\[|PA|+|PB|+|PC|\le |QA|+|QB|+|QC|.\]

1998 Akdeniz University MO, 3
Tags: inequalities , easy inequality
Let $x,y,z$ be non-negative numbers such that $x+y+z \leq 3$. Prove that $$\frac{2}{1+x}+\frac{2}{1+y}+\frac{2}{1+z} \geq 3$$

2019 Azerbaijan Senior NMO, 5
Tags: inequalities , algebra
Prove that for any $a;b;c\in\mathbb{R^+}$, we have $$(a+b)^2+(a+b+4c)^2\geq \frac{100abc}{a+b+c}$$ When does the equality hold?