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: 6530

2006 Korea National Olympiad, 6
Tags: inequalities , algebra
Prove that for any positive real numbers $x,y$ and $z,$ $xyz(x+2)(y+2)(z+2)\le(1+\frac{2(xy+yz+zx)}{3})^3$

2022 Thailand TST, 2
Tags: real analysis , inequalities , algebra
Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

MathLinks Contest 2nd, 1.1
Tags: inequalities
Let $x, y, z$ be positive numbers such that $xyz \le 2$ and $\frac{1}{x^2}+ \frac{1}{y^2}+ \frac{1}{z^2}< k$, for some real $k \ge 2$. Find all values of $k$ such that the conditions above imply that there exist a triangle having the side-lengths $x, y, z$.

2001 China Western Mathematical Olympiad, 4
Tags: inequalities
Let $ x, y, z$ be real numbers such that $ x \plus{} y \plus{} z \geq xyz$. Find the smallest possible value of $ \frac {x^2 \plus{} y^2 \plus{} z^2}{xyz}$.

2014 India IMO Training Camp, 2
Tags: triangle inequality , inequalities , algebra
For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold: $x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$ $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$ $v_{1},v_{2},v_{3}$ satisfy triangle inequality $v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality. Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.

2005 India IMO Training Camp, 3
Tags: inequalities
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

2007 Hong Kong TST, 3
Tags: trigonometry , inequalities
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 3 Let $A$, $B$ and $C$ be real numbers such that (i) $\sin A \cos B+|\cos A \sin B|=\sin A |\cos A|+|\sin B|\cos B$, (ii) $\tan C$ and $\cot C$ are defined. Find the minimum value of $(\tan C-\sin A)^{2}+(\cot C-\cos B)^{2}$.

Indonesia MO Shortlist - geometry, g7
Tags: inequalities , geometric inequality , trigonometry , geometry
In triangle $ABC$, find the smallest possible value of $$|(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)|$$

1997 Tournament Of Towns, (544) 5
Tags: algebra , inequalities , number theory
Prove that $$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a} <1$$ where $a, b$ and $c$ are positive numbers such that $abc = 1$. (G Galperin)

2021 Israel TST, 2
Tags: inequalities
Suppose $x,y,z\in \mathbb R^+$. Prove that \[\frac {x}{\sqrt{yz+4xy+4xz}}+\frac {y}{\sqrt{zx+4yz+4yx}}+\frac {z}{\sqrt{xy+4zx+4zy}}\geq 1\].

2020 DMO Stage 1, 1.
Tags: inequalities , inequality , holder
[b]Q.[/b] Find the minimum value of the expression for $x,y,z\in \mathbb{R}^{+}$ $$\sum_{\text{cyc}}\frac{(x+1)^{4}+2(y+1)^{6}-(y+1)^{4}}{(y+1)^{6}}$$ [i]Proposed by Aritra12[/i]

2018-2019 Fall SDPC, 8
Tags: number theory , inequalities , relatively prime
Let $S(n)=1\varphi(1)+2\varphi(2) \ldots +n\varphi(n)$, where $\varphi(n)$ is the number of positive integers less than or equal to $n$ that are relatively prime to $n$. (For instance $\varphi(12)=4$ and $\varphi(20)=8$.) Prove that for all $n \geq 2018$, the following inequality holds: $$0.17n^3 \leq S(n) \leq 0.23n^3$$

1975 Polish MO Finals, 3
Tags: inequalities , limit
consider $0<u<1$. find $\alpha > 0$ minimum such that there exists $\beta > 0$ satisfying $(1+x)^u +(1-x)^u \leq 2 - \frac{x^\alpha}{\beta} \forall 0<x<1$

2006 Putnam, B5
Tags: function , algebra , integration , quadratic , calculus , inequalities
For each continuous function $f: [0,1]\to\mathbb{R},$ let $I(f)=\int_{0}^{1}x^{2}f(x)\,dx$ and $J(f)=\int_{0}^{1}x\left(f(x)\right)^{2}\,dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$

2005 China Team Selection Test, 2
Tags: inequalities
Let $a$, $b$, $c$ be nonnegative reals such that $ab+bc+ca = \frac{1}{3}$. Prove that \[\frac{1}{a^{2}-bc+1}+\frac{1}{b^{2}-ca+1}+\frac{1}{c^{2}-ab+1}\leq 3 \]

2013 Kazakhstan National Olympiad, 1
Tags: algebra , inequalities
Find maximum value of $|a^2-bc+1|+|b^2-ac+1|+|c^2-ba+1|$ when $a,b,c$ are reals in $[-2;2]$.

1979 Brazil National Olympiad, 1
Tags: algebra , inequalities , trigonometry , function
Show that if $a < b$ are in the interval $\left[0, \frac{\pi}{2}\right]$ then $a - \sin a < b - \sin b$. Is this true for $a < b$ in the interval $\left[\pi,\frac{3\pi}{2}\right]$?

2024 SG Originals, Q4
Tags: sequence , inequalities
Alice and Bob play a game. Bob starts by picking a set $S$ consisting of $M$ vectors of length $n$ with entries either $0$ or $1$. Alice picks a sequence of numbers $y_1\le y_2\le\dots\le y_n$ from the interval $[0,1]$, and a choice of real numbers $x_1,x_2\dots,x_n\in \mathbb{R}$. Bob wins if he can pick a vector $(z_1,z_2,\dots,z_n)\in S$ such that $$\sum_{i=1}^n x_iy_i\le \sum_{i=1}^n x_iz_i,$$otherwise Alice wins. Determine the minimum value of $M$ so that Bob can guarantee a win. [i]Proposed by DVDthe1st[/i]

2015 China National Olympiad, 1
Tags: complex number , inequalities , trigonometry
Let $z_1,z_2,...,z_n$ be complex numbers satisfying $|z_i - 1| \leq r$ for some $r$ in $(0,1)$. Show that \[ \left | \sum_{i=1}^n z_i \right | \cdot \left | \sum_{i=1}^n \frac{1}{z_i} \right | \geq n^2(1-r^2).\]

2015 Korea Junior Math Olympiad, 4
Tags: inequalities , algebra
Reals $a,b,c,x,y$ satisfy $a^2+b^2+c^2=x^2+y^2=1$. Find the maximum value of $$(ax+by)^2+(bx+cy)^2$$

2022 Federal Competition For Advanced Students, P1, 1
Tags: inequalities , algebra
Prove that for all positive real numbers $x, y$ and $z$, the double inequality $$0 < \frac{1}{x + y + z + 1} -\frac{1}{(x + 1)(y + 1)(z + 1)} \le \frac18$$ holds. When does equality hold in the right inequality? [i](Walther Janous)[/i]

2019 Saudi Arabia JBMO TST, 2
Tags: inequalities
Let $a, b, c$ be non-negative reals which satisfy $a+b+c=1$. Prove that $\frac{\sqrt{a}}{b+1}+\frac{\sqrt{b}}{c+1}+\frac{\sqrt{c}}{a+1}>\frac{1}{2}(\sqrt{a}+\sqrt{b}+\sqrt{c})$

2002 Tuymaada Olympiad, 2
Tags: inequalities
Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove that \[ \frac{1+ab}{1+a} + \frac{1+bc}{1+b} + \frac{1+cd}{1+c} + \frac{1+da}{1+d} \geq 4 . \] [i]Proposed by A. Khrabrov[/i]

2010 IMO Shortlist, 6
Tags: circumcircle , inequalities , incenter , geometry
The vertices $X, Y , Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC, CA, AB$ of an acute-angled triangle $ABC.$ Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ.$ [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

2023 NMTC Junior, P5
Tags: inequalities , algebra
$a,b,c$ are positive reals satisfying $\frac{2}{5} \leq c \leq \min{a,b}$ ; $ac \geq \frac{4}{15}$ and $bc \geq \frac{1}{5}$ Find the maximum value of $\left(\frac{1}{a}+\frac{2}{b}+\frac{3}{c}\right)$.