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

2002 Federal Math Competition of S&M, Problem 1
Tags: inequalities
Real numbers $x,y,z$ satisfy the inequalities $$x^2\le y+z,\qquad y^2\le z+x\qquad z^2\le x+y.$$Find the minimum and maximum possible values of $z$.

1935 Eotvos Mathematical Competition, 1
Tags: algebra , inequalities
Let $n$ be a positive integer. Prove that $$\frac{a_1}{b_1}+ \frac{a_2}{b_2}+ ...+\frac{a_n}{b_n} \ge n $$ where $(b_1, b_2, ..., b_n)$ is any permutation of the positive real numbers $a_1, a_2, ..., a_n$.

1971 Bundeswettbewerb Mathematik, 4
Tags: probability , inequalities , expected value , triangle inequality , combinatorics
Inside a square with side lengths $1$ a broken line of length $>1000$ without selfintersection is drawn. Show that there is a line parallel to a side of the square that intersects the broken line in at least $501$ points.

2000 Baltic Way, 16
Tags: inequalities , analytic geometry , three variable inequality
Prove that for all positive real numbers $a,b,c$ we have \[\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}\ge\sqrt{a^2+ac+c^2} \]

2012 239 Open Mathematical Olympiad, 4
Tags: inequalities , algebra
For some positive numbers $a$, $b$, $c$ and $d$, we know that $$ \frac{1}{a^3 + 1}+ \frac{1}{b^3 + 1}+ \frac{1}{c^3 + 1} + \frac{1}{d^3 + 1} = 2. $$ Prove that $$ \frac{1 - a}{a^2 - a + 1} + \frac{1-b}{b^2 - b + 1} + \frac{1-c}{c^2 - c + 1} +\frac{1-d}{d^2 - d + 1} \geq 0. $$

2017 Iran MO (3rd round), 3
Tags: inequality , am-gm , inequalities , algebra
Let $a,b$ and $c$ be positive real numbers. Prove that $$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$

1994 Taiwan National Olympiad, 2
Tags: inequalities
Let $a,b,c$ are positive real numbers and $\alpha$ be any real number. Denote $f(\alpha)=abc(a^{\alpha}+b^{\alpha}+c^{\alpha}), g(\alpha)=a^{2+\alpha}(b+c-a)+b^{2+\alpha}(-b+c+a)+c^{2+\alpha}(b-c+a)$. Determine $\min{|f(\alpha)-g(\alpha)|}$ and $\max{|f(\alpha)-g(\alpha)|}$, if they are exists.

2012 Junior Balkan Team Selection Tests - Moldova, 2
Tags: floor function , inequalities
Let $ a,b,c,d$ be positive real numbers and $cd=1$. Prove that there exists a positive integer $n$ such that $ab\leq n^2\leq (a+c)(b+d)$

2005 Thailand Mathematical Olympiad, 6
Tags: algebra , inequalities
Let $a, b, c$ be distinct real numbers. Prove that $$\left(\frac{2a - b}{a -b} \right)^2+\left(\frac{2b - c}{b - c} \right)^2+\left(\frac{2c - a}{c - a} \right)^2 \ge 5$$

2017 Puerto Rico Team Selection Test, 5
Tags: algebra , inequalities , maximum , max , minimum , min
Let $a, b$ be two real numbers that satisfy $a^3 + b^3 = 8-6ab$. Find the maximum value and the minimum value that $a + b$ can take.

2014 Contests, 1
Tags: inequalities , trigonometry , calculus , topology , three variable inequality
In a non-obtuse triangle $ABC$, prove that \[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]

2002 Tournament Of Towns, 4
Tags: trigonometry , inequalities
$x,y,z\in\left(0,\frac{\pi}{2}\right)$ are given. Prove that: \[ \frac{x\cos x+y\cos y+z\cos z}{x+y+z}\le \frac{\cos x+\cos y+\cos z}{3} \]

2022 New Zealand MO, 2
Tags: algebra , inequalities
Find all triples $(a, b, c) $ of real numbers such that $a^2 + b^2 + c^2 = 1$ and $a(2b - 2a - c) \ge \frac12$.

PEN C Problems, 4
Tags: floor function , quadratic , inequalities , number theory , prime number , quadratic residue
Let $M$ be an integer, and let $p$ be a prime with $p>25$. Show that the set $\{M, M+1, \cdots, M+ 3\lfloor \sqrt{p} \rfloor -1\}$ contains a quadratic non-residue to modulus $p$.

2016 Taiwan TST Round 1, 2
Tags: inequalities , algebra , multivariate polynomial , maximization , n-variable inequality
Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.

2011 AMC 12/AHSME, 23
Tags: function , linear algebra , matrix , algebra , polynomial , induction , inequalities
Let $f(z)=\frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $|a|=1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $|b|$? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \sqrt{2}-1 \qquad \textbf{(C)}\ \sqrt{3}-1 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

2016 Singapore Senior Math Olympiad, 3
Tags: algebra , inequalities
For any integer $n \ge 1$, show that $$\sum_{k=1}^{n} \frac{2^k}{\sqrt{k+0.5}} \le 2^{n+1}\sqrt{n+1}-\frac{4n^{3/2}}{3}$$

2014 Federal Competition For Advanced Students, P2, 3
Tags: inequalities , triangle inequality , geometric inequality , exponential
(i) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also the inequalities $a^2 + b^2> c^2, b^2 + c^2> a^2$ and $a^2 + c^2> b^2$ ? (ii) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also for all positive natural $n$ the inequalities $a^n + b^n> c^n, b^n + c^n> a^n$ and $a^n + c^n> b^n$ ?

2024 Ukraine National Mathematical Olympiad, Problem 5
Tags: algebra , inequality , inequalities
For real numbers $a, b, c, d \in [0, 1]$, find the largest possible value of the following expression: $$a^2+b^2+c^2+d^2-ab-bc-cd-da$$ [i]Proposed by Mykhailo Shtandenko[/i]

1994 Baltic Way, 2
Tags: trigonometry , algebra , polynomial , inequalities
Let $a_1,a_2,\ldots ,a_9$ be any non-negative numbers such that $a_1=a_9=0$ and at least one of the numbers is non-zero. Prove that for some $i$, $2\le i\le 8$, the inequality $a_{i-1}+a_{i+1}<2a_i$ holds. Will the statement remain true if we change the number $2$ in the last inequality to $1.9$?

2005 France Pre-TST, 4
Tags: inequalities
Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2 = 25.$ Find the minimum of $\frac {xy} z + \frac {yz} x + \frac {zx} y .$ Pierre.

2002 Junior Balkan Team Selection Tests - Moldova, 9
Tags: inequalities , algebra
The real numbers $a$ and $b$ satisfy the relation $a + b \ge 1$. Show that $8 (a^4 + b^4) \ge 1$.

1981 All Soviet Union Mathematical Olympiad, 325
Tags: polynomial , minimum , algebra , inequalities
a) Find the minimal value of the polynomial $$P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2$$ b) Prove that it cannot be represented as a sum of the squares of some polynomials of $x,y$.

2011 Kosovo Team Selection Test, 1
Tags: inequalities
Let $a,b,c$ be real positive numbers. Prove that the following inequality holds: \[{ \sum_{\rm cyc}\sqrt{5a^2+5c^2+8b^2\over 4ac}\ge 3\cdot \root 9 \of{8(a+b)^2(b+c)^2(c+a)^2\over (abc)^2} }\]

2010 Indonesia TST, 1
Tags: inequalities
Given $ a,b, c $ positive real numbers satisfying $ a+b+c=1 $. Prove that \[ \dfrac{1}{\sqrt{ab+bc+ca}}\ge \sqrt{\dfrac{2a}{3(b+c)}} +\sqrt{\dfrac{2b}{3(c+a)}} + \sqrt{\dfrac{2c}{3(a+b)}} \ge \sqrt{a} +\sqrt{b}+\sqrt{c} \]