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: 592

2021 Azerbaijan Junior NMO, 3
Tags: inequality
$a,b,c $ are positive real numbers . Prove that $\sqrt[7]{\frac{a}{b+c}+\frac{b}{c+a}} +\sqrt[7]{\frac{b}{c+a}+\frac{c}{b+a}}+\sqrt[7]{\frac{c}{a+b}+\frac{a}{b+c}}\geq 3$

2014 Balkan MO, 1
Tags: algebra , inequality
Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds. [i]UK - David Monk[/i]

2020 Federal Competition For Advanced Students, P1, 1
Tags: algebra , inequality , inequalities
Let $x, y$ and $z$ be positive real numbers such that $x \geq y+z$. Proof that $$\frac{x+y}{z} + \frac{y+z}{x} +\frac{z+x}{y} \geq 7$$ When does equality occur? (Walther Janous)

1971 IMO Longlists, 21
Tags: algebra , inequality , n-variable inequality
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 $n>2$.

2023 Junior Balkan Team Selection Tests - Romania, P2
Tags: inequality , shortlist , algebra
Suppose that $a, b,$ and $c$ are positive real numbers such that $$a + b + c \ge \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$ Find the largest possible value of the expression $$\frac{a + b - c}{a^3 + b^3 + abc} + \frac{b + c - a}{b^3 + c^3 + abc} + \frac{c + a - b}{c^3 + a^3 + abc}.$$

1980 IMO Longlists, 2
Tags: algebra , recurrence relation , sequence , inequality
Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

2019 Balkan MO, 2
Tags: inequality
Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$ Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases. (Edit: Proposed by sir Leonard Giugiuc, Romania)

1969 IMO Shortlist, 67
Tags: inequality , algebra
Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.

2000 Moldova National Olympiad, Problem 2
Tags: inequality , inequalities
Show that if real numbers $x<1<y$ satisfy the inequality $$2\log x+\log(1-x)\ge3\log y+\log(y-1),$$then $x^3+y^3<2$.

1998 IMO, 2
Tags: matrix , combinatorics , inequality , set systems
In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that \[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]

2018 Thailand TSTST, 4
Tags: algebra , recurrence relation , sequence , inequality
Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

2024 Korea - Final Round, P3
Tags: algebra , inequality
Find the smallest real number $p(\leq 1)$ that satisfies the following condition. (Condition) For real numbers $x_1, x_2, \dots, x_{2024}, y_1, y_2, \dots, y_{2024}$, if [list] [*] $0 \leq x_1 \leq x_2 \leq \dots \leq x_{2024} \leq 1$, [*] $0 \leq y_1 \leq y_2 \leq \dots \leq y_{2024} \leq 1$, [*] $\displaystyle \sum_{i=1}^{2024}x_i = \displaystyle \sum_{i=1}^{2024}y_i = 2024p$, [/list] then the inequality $\displaystyle \sum_{i=1}^{2024}x_i(y_{2025-i}-y_{2024-i}) \geq 1 - p$ holds.

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

2021 Science ON grade VII, 3
Tags: algebra , inequality , symmetric sums , newton sums
Are there any real numbers $a,b,c$ such that $a+b+c=6$, $ab+bc+ca=9$ and $a^4+b^4+c^4=260$? What about if we let $a^4+b^4+c^4=210$? [i] (Andrei Bâra)[/i]

2002 Croatia National Olympiad, Problem 2
Tags: inequality , inequalities
Prove that for any positive number $a,b,c$ and any nonnegative integer $p$ $$a^{p+2}+b^{p+2}+c^{p+2}\ge a^pbc+b^pca+c^pab.$$

2021 Serbia JBMO TSTs, 1
Tags: algebra , inequality
Prove that for positive real numbers $a, b, c$ the following inequality holds: \begin{align*} \frac{a}{9bc+1}+\frac{b}{9ca+1}+\frac{c}{9ab+1}\geq \frac{a+b+c}{1+(a+b+c)^2} \end{align*} When does equality occur?

2020 Moldova Team Selection Test, 7
Tags: inequality , inequalities , algebra
Show that for any positive real numbers $a$, $b$, $c$ the following inequality takes place $$\frac{a}{\sqrt{7a^2+b^2+c^2}}+\frac{b}{\sqrt{a^2+7b^2+c^2}}+\frac{c}{\sqrt{a^2+b^2+7c^2}} \leq 1.$$

2016 JBMO Shortlist, 4
Tags: inequality
If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that $$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$ When does the equality occur? [i]Proposed by Dorlir Ahmeti, Albania[/i]

2001 China Team Selection Test, 2
Tags: inequality , sequence , inequalities , analysis , trigonometry
Let ${a_n}$ be a non-increasing sequence of positive numbers. Prove that if for $n \ge 2001$, $na_{n} \le 1$, then for any positive integer $m \ge 2001$ and $x \in \mathbb{R}$, the following inequality holds: $\left | \sum_{k=2001}^{m} a_{k} \sin kx \right | \le 1 + \pi$

1969 IMO Longlists, 35
Tags: inequality , series summation
$(HUN 2)$ Prove that $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{5}{4}$

2009 German National Olympiad, 4
Tags: inequalities , inequality , algebra , radical
Let $a$ and $b$ be two fixed positive real numbers. Find all real numbers $x$, such that inequality holds $$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$$

2019 Serbia JBMO TST, 2
Tags: algebra , inequality
If a b c positive reals smaller than 1, prove: a+b+c+2abc>ab+bc+ca+2(abc)^(1/2)

2009 Korea Junior Math Olympiad, 6
Tags: inequalities , algebra , four variables , inequality
If positive reals $a,b,c,d$ satisfy $abcd = 1.$ Prove the following inequality $$1<\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{d}{cd+d+1}+\frac{a}{da+a+1}<2.$$

2024 Belarusian National Olympiad, 11.7
Tags: algebra , inequality , inequalities
Positive real numbers $a_1,a_2,\ldots, a_n$ satisfy the equation $$2a_1+a_2+\ldots+a_{n-1}=a_n+\frac{n^2-3n+2}{2}$$ For every positive integer $n \geq 3$ find the smallest possible value of the sum $$\frac{(a_1+1)^2}{a_2}+\ldots+\frac{(a_{n-1}+1)^2}{a_n}$$ [i]M. Zorka[/i]

1988 IMO Shortlist, 16
Tags: algebra , polynomial , vieta , inequality , root
Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.