This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 592

2013 Taiwan TST Round 1, 2
Tags: inequality , algebra
Prove that for positive reals $a,b,c$, \[\frac{8a^2+2ab}{(b+\sqrt{6ac}+3c)^2}+\frac{2b^2+3bc}{(3c+\sqrt{2ab}+2a)^2}+\frac{18c^2+6ac}{(2a+\sqrt{3bc}+b})^2\geq 1\]

1967 IMO Longlists, 37
Tags: inequality , three variable inequality , muirhead , algebra , inequalities
Prove that for arbitrary positive numbers the following inequality holds \[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]

2017 Junior Balkan Team Selection Tests - Romania, 2
Tags: algebra , inequality
a) Find : $A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$ b) Prove that for any $(a,b,c) \in A$ next inequality hold : \begin{align*} \frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8} \end{align*}

2019 District Olympiad, 2
Tags: determinant , characteristic polynomial , inequality , linear algebra
Let $n \in \mathbb{N},n \ge 2,$ and $A,B \in \mathcal{M}_n(\mathbb{R}).$ Prove that there exists a complex number $z,$ such that $|z|=1$ and $$\Re \left( {\det(A+zB)} \right) \ge \det(A)+\det(B),$$ where $\Re(w)$ is the real part of the complex number $w.$

2008 Greece JBMO TST, 2
Tags: algebra , inequality
If $a,b,c$ are positive real numbers, prove that $\frac{a^2b^2}{a+b}+\frac{b^2c^2}{b+c}+\frac{c^2a^2}{c+a}\le \frac{a^3+b^3+c^3}{2}$

2020 Junior Macedonian National Olympiad, 2
Tags: inequality , algebra
Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 27$. Prove that $x + y + z \ge \sqrt{3xyz}$. When does equality hold?

1967 IMO Shortlist, 5
Tags: geometry , function , inequality , linear function , algebra
A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$ \[|l(z)| \leq M \rho,\] where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$

2012 Korea Junior Math Olympiad, 1
Tags: inequality , algebra , ineqstd
Prove the following inequality where positive reals $a$, $b$, $c$ satisfies $ab+bc+ca=1$. \[ \frac{a+b}{\sqrt{ab(1-ab)}} + \frac{b+c}{\sqrt{bc(1-bc)}} + \frac{c+a}{\sqrt{ca(1-ca)}} \le \frac{\sqrt{2}}{abc} \]

2016 Junior Balkan Team Selection Tests - Romania, 2
Tags: algebra , inequality
a,b,c>0 and $abc\ge 1$.Prove that: $\dfrac{1}{a^3+2b^3+6}+\dfrac{1}{b^3+2c^3+6}+\dfrac{1}{c^3+2a^3+6} \le \dfrac{1}{3}$

2013 Junior Balkan MO, 3
Tags: algebra , inequality
Show that \[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\] for all positive real numbers $a$ and $b$ such that $ab\geq 1$.

2024 Junior Balkan Team Selection Tests - Moldova, 10
Tags: algebra , inequality
Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that $$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$

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

1977 IMO Shortlist, 7
Tags: function , trigonometry , trigonometric inequality , inequality
Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$

2017 District Olympiad, 4
Tags: algebra , inequality , inequalities
If $ a,b,c>0 $ and $ ab+bc+ca+abc=4, $ then $ \sqrt{ab} +\sqrt{bc} +\sqrt{ca} \le 3\le a+b+c. $

2021 Austrian MO National Competition, 1
Tags: algebra , inequality
Let $a,b,c\geq 0$ and $a+b+c=1.$ Prove that$$\frac{a}{2a+1}+\frac{b}{3b+1}+\frac{c}{6c+1}\leq \frac{1}{2}.$$ [size=50](Marian Dinca)[/size]

2021 Polish MO Finals, 4
Tags: inequality , algebra
Prove that for every pair of positive real numbers $a, b$ and for every positive integer $n$, $$(a+b)^n-a^n-b^n \ge \frac{2^n-2}{2^{n-2}} \cdot ab(a+b)^{n-2}.$$

2004 Brazil Team Selection Test, Problem 1
Tags: inequality , inequalities , algebra
Let $x,y,z$ be positive numbers such that $x^2+y^2+z^2=1$. Prove that $$\frac x{1-x^2}+\frac y{1-y^2}+\frac z{1-z^2}\ge\frac{3\sqrt3}2$$

2021 Science ON all problems, 2
Tags: inequality , algebra
$a,b,c$ are nonnegative integers that satisfy $a^2+b^2+c^2=3$. Find the minimum and maximum value the sum $$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a}$$ may achieve and find all $a,b,c$ for which equality occurs.\\ \\ [i](Andrei Bâra)[/i]

2023 Mongolian Mathematical Olympiad, 1
Tags: algebra , inequality , inequalities
Let $u, v$ be arbitrary positive real numbers. Prove that \[\min{(u, \frac{100}{v}, v+\frac{2023}{u})} \leq \sqrt{2123}.\]

1979 IMO Shortlist, 11
Tags: inequality , maximization , calculus , optimization , lagrange
Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.

2021 Indonesia TST, A
Tags: inequality , inequalities , algebra
A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying $$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$ then the following inequality holds: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$ (a) Prove that $M=20-\frac{1}{20}$ is not $strong$. (b) Prove that $M=20-\frac{1}{21}$ is $strong$.

2006 Singapore Team Selection Test, 2
Tags: inequality , rearrangement inequality
Let n be an integer greater than 1 and let $x_1, x_2, . . . , x_n$ be real numbers such that $|x_1| + |x_2| + ... + |x_n| = 1$ and $x_1 + x_2 + ... + x_n = 0$ Prove that $\left| \frac{x_1}{1}+\frac{x_2}{2}+\cdots+\frac{x_n}{n} \right| \leq \frac{1}{2} \left(1-\frac{1}{n}\right)$

1977 IMO, 1
Tags: function , trigonometry , trigonometric inequality , inequality
Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$

2013 Kyiv Mathematical Festival, 1
Tags: inequality , 4-variable inequality , inequalities , algebra
For every positive $a, b, c, d$ such that $a + c\le ac$ and $b + d \le bd$ prove that $ab + cd \ge 8$.

2017 China National Olympiad, 6
Tags: prc , inequalities , convex-concave inequalities , sqing orz , inequality
Given an integer $n \geq2$ and real numbers $a,b$ such that $0<a<b$. Let $x_1,x_2,\ldots, x_n\in [a,b]$ be real numbers. Find the maximum value of $$\frac{\frac{x^2_1}{x_2}+\frac{x^2_2}{x_3}+\cdots+\frac{x^2_{n-1}}{x_n}+\frac{x^2_n}{x_1}}{x_1+x_2+\cdots +x_{n-1}+x_n}.$$