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

1993 IMO, 4
Tags: geometry , area , inequalities
For three points $A,B,C$ in the plane, we define $m(ABC)$ to be the smallest length of the three heights of the triangle $ABC$, where in the case $A$, $B$, $C$ are collinear, we set $m(ABC) = 0$. Let $A$, $B$, $C$ be given points in the plane. Prove that for any point $X$ in the plane, \[ m(ABC) \leq m(ABX) + m(AXC) + m(XBC). \]

2011 Singapore Senior Math Olympiad, 5
Tags: inequalities
Given $x_1,x_2,\dots,x_n>0,n\geq 5$, show that \[\frac{x_1x_2}{x_1^2+x_2^2+2x_3x_4}+\frac{x_2x_3}{x_2^2+x_3^2+2x_4x_5}+\cdots+\frac{x_nx_1}{x_n^2+x_1^2+2x_2x_3}\leq \frac{n-1}{2}\]

2015 Saudi Arabia JBMO TST, 2
Tags: inequalities , algebra
Let $a,b,c$ be positive real numbers. Prove that $$\frac{a}{\sqrt{(2a+b)(2a+c)}} +\frac{b}{\sqrt{(2b+c)(2b+a)}} +\frac{c}{\sqrt{(2c+a)(2c+b)}} \le 1 $$

2005 Alexandru Myller, 1
Tags: inequalities , algebra
Let $ x,y,z $ be numbers distinct from $ -1 $ that verify the equation $$ \frac{1}{1+a} +\frac{1}{1+b} +\frac{1}{1+c} =\frac{3}{2} . $$ Prove that if $ abc=1, $ then $ a $ or $ b $ or $ c $ is equal to $ 1. $

2016 EGMO, 1
Tags: inequality , algebra , inequalities , n-variable inequality , sequence
Let $n$ be an odd positive integer, and let $x_1,x_2,\cdots ,x_n$ be non-negative real numbers. Show that \[ \min_{i=1,\ldots,n} (x_i^2+x_{i+1}^2) \leq \max_{j=1,\ldots,n} (2x_jx_{j+1}) \]where $x_{n+1}=x_1$.

2023 Saint Petersburg Mathematical Olympiad, 1
Tags: inequalities
Let $a, b>1$ be reals such that $a+\frac{1}{a^2} \geq 5b-\frac{3}{b^2}$. Show that $a>5b-\frac{4}{b^2}$.

2006 China Team Selection Test, 3
Tags: function , inequalities
Given $n$ real numbers $a_1$, $a_2$ $\ldots$ $a_n$. ($n\geq 1$). Prove that there exists real numbers $b_1$, $b_2$ $\ldots$ $b_n$ satisfying: (a) For any $1 \leq i \leq n$, $a_i - b_i$ is a positive integer. (b)$\sum_{1 \leq i < j \leq n} (b_i - b_j)^2 \leq \frac{n^2-1}{12}$

2014 ISI Entrance Examination, 7
Tags: calculus , integration , inequalities , rearrangement inequality , real analysis , integral inequality
Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge (z-y)\int_{x}^{z}f(u)\,\mathrm{du} \end{align*}

1997 Poland - Second Round, 4
Tags: modular arithmetic , greatest common divisor , inequalities , number theory
There is a set with three elements: (2,3,5). It has got an interesting property: (2*3) mod 5=(2*5) mod 3=(3*5) mod 2. Prove that it is the only one set with such property.

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

2010 Balkan MO Shortlist, A1
Tags: rearrangement inequality , inequalities
Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]

2001 Hong kong National Olympiad, 2
Tags: inequalities , number theory
Find, with proof, all positive integers $n$ such that the equation $x^{3}+y^{3}+z^{3}=nx^{2}y^{2}z^{2}$ has a solution in positive integers.

2017 AMC 12/AHSME, 2
Tags: inequalities
Real numbers $x$, $y$, and $z$ satisfy the inequalities $$0<x<1,\qquad-1<y<0,\qquad\text{and}\qquad1<z<2.$$ Which of the following numbers is nessecarily positive? $\textbf{(A) } y+x^2 \qquad \textbf{(B) } y+xz \qquad \textbf{(C) }y+y^2 \qquad \textbf{(D) }y+2y^2 \qquad\\ \textbf{(E) } y+z$

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

2007 India National Olympiad, 5
Tags: inequalities , geometry
Let $ ABC$ be a triangle in which $ AB\equal{}AC$. Let $ D$ be the midpoint of $ BC$ and $ P$ be a point on $ AD$. Suppose $ E$ is the foot of perpendicular from $ P$ on $ AC$. Define \[ \frac{AP}{PD}\equal{}\frac{BP}{PE}\equal{}\lambda , \ \ \ \frac{BD}{AD}\equal{}m , \ \ \ z\equal{}m^2(1\plus{}\lambda)\] Prove that \[ z^2 \minus{} (\lambda^3 \minus{} \lambda^2 \minus{} 2)z \plus{} 1 \equal{} 0\] Hence show that $ \lambda \ge 2$ and $ \lambda \equal{} 2$ if and only if $ ABC$ is equilateral.

1960 AMC 12/AHSME, 26
Tags: inequalities
Find the set of $x$-values satisfying the inequality $|\frac{5-x}{3}|<2$. [The symbol $|a|$ means $+a$ if $a$ is positive, $-a$ if $a$ is negative, 0 if $a$ is zero. The notation $1<a<2$ means that $a$ can have any value between $1$ and $2$, excluding $1$ and $2$. ] $ \textbf{(A)}\ 1 < x < 11\qquad\textbf{(B)}\ -1 < x < 11\qquad\textbf{(C)}\ x< 11\qquad$ $\textbf{(D)}\ x>11\qquad\textbf{(E)}\ |x| < 6 $

2014 Estonia Team Selection Test, 3
Tags: max , area , inequalities , algebra
Three line segments, all of length $1$, form a connected figure in the plane. Any two different line segments can intersect only at their endpoints. Find the maximum area of the convex hull of the figure.

2020 DMO Stage 1, 1.
Tags: inequality , holder , inequalities
[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]

1994 North Macedonia National Olympiad, 3
Tags: sum , algebra , inequalities , product , max
a) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be negative real numbers and $ x_1 + x_2 + ... + x_n = m. $ Determine the maximum value of the sum $ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $ b) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be nonnegative natural numbers and $ x_1 + x_2 + ... + x_n = m. $ Determine the maximum value of the sum $ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $

2014 India Regional Mathematical Olympiad, 2
Tags: inequalities , inequality , 3-variable inequality
Let $x, y, z$ be positive real numbers. Prove that $\frac{y^2 + z^2}{x}+\frac{z^2 + x^2}{y}+\frac{x^2 + y^2}{z}\ge 2(x + y + z)$.

PEN P Problems, 12
Tags: function , floor function , calculus , integration , logarithm , inequalities
The positive function $p(n)$ is defined as the number of ways that the positive integer $n$ can be written as a sum of positive integers. Show that, for all positive integers $n \ge 2$, \[2^{\lfloor \sqrt{n}\rfloor}< p(n) < n^{3 \lfloor\sqrt{n}\rfloor }.\]

2010 Postal Coaching, 2
Tags: geometry , inequalities
Let $M$ be an interior point of a $\triangle ABC$ such that $\angle AM B = 150^{\circ} , \angle BM C = 120^{\circ}$. Let $P, Q, R$ be the circumcentres of the $\triangle AM B, \triangle BM C, \triangle CM A$ respectively. Prove that $[P QR] \ge [ABC]$.

2001 Brazil National Olympiad, 1
Tags: inequalities
Show that for any $a,b,c$ positive reals, \[ (a+b)(a+c) \geq 2 \sqrt{abc(a+b+c)} \]

2010 India IMO Training Camp, 9
Tags: geometry , rectangle , inequalities , algebra
Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$. Show that there exists $j<k$ and $l<m$ such that \[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]

1977 IMO Longlists, 58
Tags: inequalities , trigonometry , area of a triangle , heron's formula , geometry
Prove that for every triangle the following inequality holds: \[\frac{ab+bc+ca}{4S} \geq \cot \frac{\pi}{6}.\] where $a, b, c$ are lengths of the sides and $S$ is the area of the triangle.