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

2019 Ramnicean Hope, 1
Tags: inequalities
Show that $$ \frac{a^4}{(a+b)\left( a^2+b^2 \right)} +\frac{b^4}{(b+c)\left( b^2+c^2 \right)} +\frac{c^4}{(c+a)\left( c^2+a^2 \right)}\ge \frac{a+b+c}{4} , $$ for any positive real numbers $ a,b,c. $ [i]Costică Ambrinoc[/i]

2019 Taiwan TST Round 1, 1
Tags: inequalities
Assume $ a_{1} \ge a_{2} \ge \dots \ge a_{107} > 0 $ satisfy $ \sum\limits_{k=1}^{107}{a_{k}} \ge M $ and $ b_{107} \ge b_{106} \ge \dots \ge b_{1} > 0 $ satisfy $ \sum\limits_{k=1}^{107}{b_{k}} \le M $. Prove that for any $ m \in \{1,2, \dots, 107\} $, the arithmetic mean of the following numbers $$ \frac{a_{1}}{b_{1}}, \frac{a_{2}}{b_{2}}, \dots, \frac{a_{m}}{b_{m}} $$ is greater than or equal to $ \frac{M}{N} $

2019 Thailand TST, 3
Tags: inequalities
Find the maximal value of \[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\] where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$. [i]Proposed by Evan Chen, Taiwan[/i]

1998 Austrian-Polish Competition, 4
Tags: inequalities , sum , algebra
For positive integers $m, n$, denote $$S_m(n)=\sum_{1\le k \le n} \left[ \sqrt[k^2]{k^m}\right]$$ Prove that $S_m(n) \le n + m (\sqrt[4]{2^m}-1)$

2006 Romania Team Selection Test, 4
Tags: inequalities
Let $x_i$, $1\leq i\leq n$ be real numbers. Prove that \[ \sum_{1\leq i<j\leq n}|x_i+x_j|\geq\frac{n-2}{2}\sum_{i=1}^n|x_i|. \] [i]Discrete version by Dan Schwarz of a Putnam problem[/i]

1956 AMC 12/AHSME, 43
Tags: inequalities , integration , geometry , perimeter , calculus
The number of scalene triangles having all sides of integral lengths, and perimeter less than $ 13$ is: $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 18$

2015 Junior Balkan Team Selection Tests - Romania, 3
Tags: inequalities
Let $x$,$y$,$z>0$ . Show that : $$\frac{x^3}{z^3+x^2y}+\frac{y^3}{x^3+y^2z}+\frac{z^3}{y^3+z^2x} \geq \frac{3}{2}$$

2011 Saudi Arabia Pre-TST, 1.4
Tags: number theory , algebra , inequalities , fermat number
Let $f_n = 2^{2^n}+ 1$, $n = 1,2,3,...$, be the Fermat’s numbers. Find the least real number $C$ such that $$\frac{1}{f_1}+\frac{2}{f_2}+\frac{2^2}{f_3}+...+\frac{2^{n-1}}{f_n} <C$$ for all positive integers $n$

2005 Sharygin Geometry Olympiad, 14
Tags: inequalities , area , area of a triangle , geometric inequality , triangle inequality , geometry
Let $P$ be an arbitrary point inside the triangle $ABC$. Let $A_1, B_1$ and $C_1$ denote the intersection points of the straight lines $AP, BP$ and $CP$, respectively, with the sides $BC, CA$ and $AB$. We order the areas of the triangles $AB_1C_1,A_1BC_1,A_1B_1C$. Denote the smaller by $S_1$, the middle by $S_2$, and the larger by $S_3$. Prove that $\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3}$ ,where $S$ is the area of the triangle $A_1B_1S_1$.

2019 Peru MO (ONEM), 2
Tags: inequalities , algebra
Find all the real numbers $k$ that have the following property: For any non-zero real numbers $a$ and $b$, it is true that at least one of the following numbers: $$a, b,\frac{5}{a^2}+\frac{6}{b^3}$$is less than or equal to $k$.

1980 Austrian-Polish Competition, 6
Tags: inequalities , sequence , algebra , absolute value
Let $a_1,a_2,a_3,\dots$ be a sequence of real numbers satisfying the inequality \[ |a_{k+m}-a_k-a_m| \leq 1 \quad \text{for all} \ k,m \in \mathbb{Z}_{>0}. \] Show that the following inequality holds for all positive integers $k,m$ \[ \left| \frac{a_k}{k}-\frac{a_m}{m} \right| < \frac{1}{k}+\frac{1}{m}. \]

2021 Iran MO (3rd Round), 1
Tags: algebra , inequalities
Positive real numbers $a, b, c$ and $d$ are given such that $a+b+c+d = 4$ prove that $$\frac{ab}{a^2-\frac{4}{3}a+\frac{4}{3}} + \frac{bc}{b^2-\frac{4}{3}b+ \frac{4}{3}} + \frac{cd}{c^2-\frac{4}{3}c+ \frac{4}{3}} + \frac{da}{d^2-\frac{4}{3}d+ \frac{4}{3}}\leq 4.$$

2012 Flanders Math Olympiad, 3
Tags: algebra , trigonometry , inequalities
(a) Show that for any angle $\theta$ and for any natural number $m$: $$| \sin m\theta| \le m| \sin \theta|$$ (b) Show that for all angles $\theta_1$ and $\theta_2$ and for all even natural numbers $m$: $$| \sin m \theta_2 - \sin m \theta_1| \le m| \sin (\theta_2 - \theta_1)|$$ (c) Show that for every odd natural number $m$ there are two angles, resp. $\theta_1$ and $\theta_2$, exist for which the inequality in (b) is not valid.

2018 SIMO, Q2
Tags: algebra , inequalities
Let $x_1, x_2, x_3, y_1, y_2, y_3$ be real numbers in $[-1, 1]$. Find the maximum value of \[(x_1y_2-x_2y_1)(x_2y_3-x_3y_2)(x_3y_1-x_1y_3).\]

2013 Regional Competition For Advanced Students, 3
Tags: inequalities , arithmetic mean-geometric mean , sequence
For non-negative real numbers $a,$ $b$ let $A(a, b)$ be their arithmetic mean and $G(a, b)$ their geometric mean. We consider the sequence $\langle a_n \rangle$ with $a_0 = 0,$ $a_1 = 1$ and $a_{n+1} = A(A(a_{n-1}, a_n), G(a_{n-1}, a_n))$ for $n > 0.$ (a) Show that each $a_n = b^2_n$ is the square of a rational number (with $b_n \geq 0$). (b) Show that the inequality $\left|b_n - \frac{2}{3}\right| < \frac{1}{2^n}$ holds for all $n > 0.$

1982 Swedish Mathematical Competition, 2
Tags: inequalities , algebra
Show that \[ abc \geq (a+b-c)(b+c-a)(c+a-b) \] for positive reals $a$, $b$, $c$.

2007 ITest, 35
Tags: inequalities
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits.

2005 Germany Team Selection Test, 2
Tags: inequalities , function , 3-variable inequality , nesbitt
If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that \[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]

2016 Saudi Arabia Pre-TST, 1.1
Tags: inequalities , algebra
Let $x, y, z$ be positive real numbers satisfy the condition $x^2 +y^2 + z^2 = 2(x y + yz + z x)$. Prove that $x + y + z + \frac{1}{2x yz} \ge 4$

2015 Estonia Team Selection Test, 10
Tags: inequalities , algebra
Let $n$ be an integer and $a, b$ real numbers such that $n > 1$ and $a > b > 0$. Prove that $$(a^n - b^n) \left ( \frac{1}{b^{n- 1}} - \frac{1}{a^{n -1}}\right) > 4n(n -1)(\sqrt{a} - \sqrt{b})^2$$

2000 Canada National Olympiad, 5
Tags: inequalities
Suppose that the real numbers $a_1, a_2, \ldots, a_{100}$ satisfy \begin{eqnarray*} 0 \leq a_{100} \leq a_{99} \leq \cdots \leq a_2 &\leq& a_1 , \\ a_1+a_2 & \leq & 100 \\ a_3+a_4+\cdots+a_{100} &\leq & 100. \end{eqnarray*} Determine the maximum possible value of $a_1^2 + a_2^2 + \cdots + a_{100}^2$, and find all possible sequences $a_1, a_2, \ldots , a_{100}$ which achieve this maximum.

2017 IMEO, 4
Tags: algebra , inequalities
Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$\sqrt{\frac{a^3}{1+bc}}+\sqrt{\frac{b^3}{1+ac}}+\sqrt{\frac{c^3}{1+ab}}\geq 2$$ Are there any triples $(a,b,c)$, for which the equality holds? [i]Proposed by Konstantinos Metaxas.[/i]

2016 Costa Rica - Final Round, A3
Tags: inequalities , algebra
Let $x$ and $y$ be two positive real numbers, such that $x + y = 1$. Prove that $$\left(1 +\frac{1}{x}\right)\left(1 +\frac{1}{y}\right) \ge 9$$

2006 Moldova Team Selection Test, 3
Tags: algebra , polynomial , inequalities
Let $a,b,c$ be sides of the triangle. Prove that \[ a^2\left(\frac{b}{c}-1\right)+b^2\left(\frac{c}{a}-1\right)+c^2\left(\frac{a}{b}-1\right)\geq 0 . \]

2009 Jozsef Wildt International Math Competition, W. 27
Tags: number theory , inequalities , perfect number
Let $a$, $n$ be positive integers such that $a^n$ is a perfect number. Prove that $$a^{\frac{n}{\mu}}> \frac{\mu}{2}$$ where $\mu$ denotes the number of distinct prime divisors of $a^n$