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.

AND:
OR:
NO:

Found problems: 583

1959 AMC 12/AHSME, 25
Tags: inequalities , AMC , absolute value , Inequality , AMC 12
The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if $a$ is less than or equal to zero; the symbol $<$ means "less than"; the symbol $>$ means "greater than." The set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that: $ \textbf{(A)}\ x^2<49 \qquad\textbf{(B)}\ x^2>1 \qquad\textbf{(C)}\ 1<x^2<49\qquad\textbf{(D)}\ -1<x<7\qquad\textbf{(E)}\ -7<x<1 $

2015 JBMO TST - Turkey, 4
Tags: Inequality
Prove that $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \ge \dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+2(a+b+c)$$ for the all $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2+2abc \le 1$.

2023 Romania National Olympiad, 1
Tags: fractional part , algebra , Inequality
For natural number $n$ we define \[ a_n = \{ \sqrt{n} \} - \{ \sqrt{n + 1} \} + \{ \sqrt{n + 2} \} - \{ \sqrt{n + 3} \}. \] a) Show that $a_1 > 0,2$. b) Show that $a_n < 0$ for infinity many values of $n$ and $a_n > 0$ for infinity values of natural numbers of $n$ as well. ( We denote by $\{ x \} $ the fractional part of $x.$)

2020 OMMock - Mexico National Olympiad Mock Exam, 1
Tags: algebra , Inequality , square root , inequalities
Let $a$, $b$, $c$ and $d$ positive real numbers with $a > c$ and $b < d$. Assume that \[a + \sqrt{b} \ge c + \sqrt{d} \qquad \text{and} \qquad \sqrt{a} + b \le \sqrt{c} + d\] Prove that $a + b + c + d > 1$. [i]Proposed by Victor Domínguez[/i]

2021-IMOC, A7
Tags: algebra , Inequality , inequalities
For any positive reals $a,b,c,d$ that satisfy $a^2 + b^2 + c^2 + d^2 = 4,$ show that $$\frac{a^3}{a+b} + \frac{b^3}{b+c} + \frac{c^3}{c+d} + \frac{d^3}{d+a} + 4abcd \leq 6.$$

1984 IMO Shortlist, 5
Tags: Inequality , three variable inequality , calculus , maximization , IMO , IMO 1984 , Hi
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

2005 China Second Round Olympiad, 2
Tags: function , inequalities unsolved , inequalities , China , algebra , inequalities proposed , Inequality
Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function \[ f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}. \]

2022 JBMO Shortlist, N2
Tags: number theory , LCM , Junior , Balkan , shortlist , AZE JBMO TST , Inequality
Let $a < b < c < d < e$ be positive integers. Prove that $$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$ where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?

2009 Iran Team Selection Test, 3
Tags: calculus , function , three variable inequality , algebra , Inequality
Suppose that $ a$,$ b$,$ c$ be three positive real numbers such that $ a\plus{}b\plus{}c\equal{}3$ . Prove that : $ \frac{1}{2\plus{}a^{2}\plus{}b^{2}}\plus{}\frac{1}{2\plus{}b^{2}\plus{}c^{2}}\plus{}\frac{1}{2\plus{}c^{2}\plus{}a^{2}} \leq \frac{3}{4}$

2004 Bulgaria Team Selection Test, 2
Tags: inequalities , algebra , Inequality
Prove that if $a,b,c \ge 1$ and $a+b+c=9$, then $\sqrt{ab+bc+ca} \le \sqrt{a} +\sqrt{b} + \sqrt{c}$

2000 239 Open Mathematical Olympiad, 3
Tags: Inequality , algebra
For all positive real numbers $a_1, a_2, \dots, a_n$, prove that $$ \frac{a_1\! +\! a_2}{2} \cdot \frac{a_2\! +\! a_3}{2} \cdot \dots \cdot \frac{a_n\! +\! a_1}{2} \leq \frac{a_1\!+\!a_2\!+\!a_3}{2 \sqrt{2}} \cdot \frac{a_2\!+\!a_3\!+\!a_4}{2 \sqrt{2}} \cdot \dots \cdot \frac{a_n\!+\!a_1\!+\!a_2}{2 \sqrt{2}}.$$

2015 Canadian Mathematical Olympiad Qualification, 5
Tags: algebra , Inequality , inequalities
Let $x$ and $y$ be positive real numbers such that $x + y = 1$. Show that $$\left( \frac{x+1}{x} \right)^2 + \left( \frac{y+1}{y} \right)^2 \geq 18.$$

1971 IMO, 1
Tags: algebra , Inequality , n-variable inequality , IMO , IMO 1971
Let \[ E_n=(a_1-a_2)(a_1-a_3)\ldots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\ldots(a_2-a_n)+\ldots+(a_n-a_1)(a_n-a_2)\ldots(a_n-a_{n-1}). \] Let $S_n$ be the proposition that $E_n\ge0$ for all real $a_i$. Prove that $S_n$ is true for $n=3$ and $5$, but for no other $n>2$.

JOM 2015 Shortlist, A2
Tags: Inequality , inequalities
Let $ a, b, c $ be positive real numbers greater or equal to $ 3 $. Prove that $$ 3(abc+b+2c)\ge 2(ab+2ac+3bc) $$ and determine all equality cases.

2004 Croatia National Olympiad, Problem 1
Tags: Inequality , Complex number , complex inequality , inequalities
Let $z_1,\ldots,z_n$ and $w_1,\ldots,w_n$ $(n\in\mathbb N)$ be complex numbers such that $$|\epsilon_1z_1+\ldots+\epsilon_nz_n|\le|\epsilon_1w_1+\ldots+\epsilon_nw_n|$$holds for every choice of $\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}$. Prove that $$|z_1|^2+\ldots+|z_n|^2\le|w_1|^2+\ldots+|w_n|^2.$$

1984 IMO Longlists, 20
Tags: Inequality , three variable inequality , calculus , maximization , IMO , IMO 1984 , Hi
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

1982 IMO Longlists, 19
Tags: Inequality , calculus , geometric progression , IMO Shortlist , IMO Longlist
Show that \[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\] holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.

1967 IMO Shortlist, 3
Tags: trigonometry , Trigonometric inequality , Inequality , algebra , IMO Shortlist , IMO Longlist
Find all $x$ for which, for all $n,$ \[\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.\]

2002 Iran MO (3rd Round), 16
Tags: quadratics , Iran , Inequality , algebra
For positive $a,b,c$, \[a^{2}+b^{2}+c^{2}+abc=4\] Prove $a+b+c \leq3$

2012 Kosovo National Mathematical Olympiad, 2
Tags: geometry , Inequality
Let $a,b,c$ be the lengths of the sides of a triangle. Prove that, $\left|\frac {a}{b}+\frac {b}{c}+\frac {c}{a}-\frac {b}{a}-\frac {c}{b}-\frac {a}{c}\right|<1$

2004 Mexico National Olympiad, 2
Tags: number theory , Integers , Inequality
Find the maximum number of positive integers such that any two of them $a, b$ (with $a \ne b$) satisfy that$ |a - b| \ge \frac{ab}{100} .$

2013 Israel National Olympiad, 6
Tags: algebra , Inequality , Fractions , inequalities
Let $x_1,...,x_n$ be positive real numbers, satisfying $x_1+\dots+x_n=n$. Prove that $\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\leq\frac{4}{x_1\cdot x_2\cdot\dots\cdot x_n}+n-4$.

2012 India Regional Mathematical Olympiad, 8
Tags: inequalities , Inequality , 3-variable inequality
Let $x, y, z$ be positive real numbers such that $2(xy + yz + zx) = xyz$. Prove that $\frac{1}{(x-2)(y-2)(z-2)} + \frac{8}{(x+2)(y+2)(z+2)} \le \frac{1}{32}$

2018 Balkan MO Shortlist, A1
Tags: algebra , Inequality , TST , Balkan MO Shortlist , AZE JBMO TST
Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

2018 South East Mathematical Olympiad, 1
Tags: algebra , Inequality , inequalities
Assume $c$ is a real number. If there exists $x\in[1,2]$ such that $\max\left\{\left |x+\frac cx\right |, \left |x+\frac cx + 2\right |\right\}\geq 5$, please find the value range of $c$.