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: 6530

2021 Romania National Olympiad, 2
Tags: inequalities
Prove that for all positive real numbers $a,b,c$ the following inequality holds: \[(a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\ge\frac{2(a^2+b^2+c^2)}{ab+bc+ca}+7\] and determine all cases of equality. [i]Lucian Petrescu[/i]

2015 China Northern MO, 8
Tags: algebra , inequalities
Given a positive integer $n \ge 3$. Find the smallest real number $k$ such that for any positive real number except $a_1, a_2,..,a_n$, $$\sum_{i=1}^{n-1}\frac{a_i}{ s-a_i}+\frac{ka_n}{s-a_n} \ge \frac{n-1}{n-2}$$ where, $s=a_1+a_2+..+a_n$

2011 Romania National Olympiad, 4
Tags: number theory , inequalities , floor
Let be a natural number $ n. $ Prove that there exists a number $ k\in\{ 0,1,2,\ldots n \} $ such that the floor of $ 2^{n+k}\sqrt 2 $ is even.

1976 Bulgaria National Olympiad, Problem 4
Tags: inequalities
Let $0<x_1\le x_2\le\ldots\le x_n$. Prove that $$\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\ge\frac{x_2}{x_1}+\frac{x_3}{x_2}+\ldots+\frac{x_n}{x_{n-1}}+\frac{x_1}{x_n}$$ [i]I. Tonov[/i]

2018 China Team Selection Test, 5
Tags: inequalities
Given positive integers $n, k$ such that $n\ge 4k$, find the minimal value $\lambda=\lambda(n,k)$ such that for any positive reals $a_1,a_2,\ldots,a_n$, we have \[ \sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}} \le \lambda\] Where $a_{n+i}=a_i,i=1,2,\ldots,k$

2014 AMC 12/AHSME, 20
Tags: inequalities , algebra , function , domain , calculus , integration , logarithm
For how many positive integers $x$ is $\log_{10}{(x-40)} + \log_{10}{(60-x)} < 2$? ${ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}}\ 20\qquad\textbf{(E)}\ \text{infinitely many} $

2014 Turkey EGMO TST, 4
Tags: inequalities
Let $x,y,z$ be positive real numbers such that $x+y+z \ge x^2+y^2+z^2$. Show that; $$\dfrac{x^2+3}{x^3+1}+\dfrac{y^2+3}{y^3+1}+\dfrac{z^2+3}{z^3+1}\ge6$$

2010 239 Open Mathematical Olympiad, 8
Tags: inequalities , algebra
For positive numbers $x$, $y$, and $z$, we know that $x + y^2 + z^3 = 1$. Prove that $$\frac{x}{2 + xy} + \frac{y}{2 + yz} + \frac{z}{2 + zx} > \frac{1}{2} .$$

2013 Math Prize for Girls Olympiad, 1
Tags: inequalities
Let $n$ be a positive integer. Let $a_1$, $a_2$, $\ldots\,$, $a_n$ be real numbers such that $-1 \le a_i \le 1$ (for all $1 \le i \le n$). Let $b_1$, $b_2$, $\ldots$, $b_n$ be real numbers such that $-1 \le b_i \le 1$ (for all $1 \le i \le n$). Prove that \[ \left| \prod_{i=1}^n a_i - \prod_{i=1}^n b_i \right| \le \sum_{i = 1}^n \left| a_i - b_i \right| \, . \]

1959 Kurschak Competition, 3
Tags: algebra , inequalities , permutation , combinatorics
What is the largest possible value of $|a_1 - 1| + |a_2-2|+...+ |a_n- n|$ where $a_1, a_2,..., a_n$ is a permutation of $1,2,..., n$?

2005 Switzerland - Final Round, 3
Tags: inequalities , algebra
Prove for all $a_1, ..., a_n > 0$ the following inequality and determine all cases in where the equaloty holds: $$\sum_{k=1}^{n}ka_k\le {n \choose 2}+\sum_{k=1}^{n}a_k^k.$$

1960 IMO Shortlist, 2
Tags: inequalities , algebra
For what values of the variable $x$ does the following inequality hold: \[ \dfrac{4x^2}{(1-\sqrt{2x+1})^2}<2x+9 \ ? \]

2017 Olympic Revenge, 4
Tags: algebra , inequalities
Let $f:\mathbb{R}_{+}^{*}$$\rightarrow$$\mathbb{R}_{+}^{*}$ such that $f'''(x)>0$ for all $x$ $\in$ $\mathbb{R}_{+}^{*}$. Prove that: $f(a^{2}+b^{2}+c^{2})+2f(ab+bc+ac)$ $\geq$ $f(a^{2}+2bc)+f(b^{2}+2ca)+f(c^{2}+2ab)$, for all $a,b,c$ $\in$ $\mathbb{R}_{+}^{*}$.

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

1982 Vietnam National Olympiad, 2
Tags: inequalities
Let $p$ be a positive integer and $q, z$ be real numbers with $0\le q\le 1$ and $q^{p+1}\le z\le 1$. Prove that \[\prod_{k=1}^p \left|\frac{z - q^k}{z + q^k}\right| \le\prod_{k=1}^p \left|\frac{1 - q^k}{1 + q^k}\right|.\]

2018 Taiwan TST Round 3, 1
Tags: inequalities
Suppose that $x,y$ are distinct positive reals, and $n>1$ is a positive integer. If \[x^n-y^n=x^{n+1}-y^{n+1},\] then show that \[1<x+y<\frac{2n}{n+1}.\]

1962 All Russian Mathematical Olympiad, 019
Tags: inequalities , algebra
Given a quartet of positive numbers $a,b,c,d$, and is known, that $abcd=1$. Prove that $$a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+dc \ge 10$$

2005 China Team Selection Test, 3
Tags: inequalities , algebra , polynomial , computer solutions , 4-variable inequality , symmetric inequality
Let $a,b,c,d >0$ and $abcd=1$. Prove that: \[ \frac{1}{(1+a)^2}+\frac{1}{(1+b)^2}+\frac{1}{(1+c)^2}+\frac{1}{(1+d)^2} \geq 1 \]

2023 All-Russian Olympiad Regional Round, 10.10
Tags: algebra , inequalities
Prove that for all positive reals $x, y, z$, the inequality $(x-y)\sqrt{3x^2+y^2}+(y-z)\sqrt{3y^2+z^2}+(z-x)\sqrt{3z^2+x^2} \geq 0$ is satisfied.

2017 Tournament Of Towns, 3
Tags: inequalities , algebra
From given positive numbers, the following infinite sequence is defined: $a_1$ is the sum of all original numbers, $a_2$ is the sum of the squares of all original numbers, $a_3$ is the sum of the cubes of all original numbers, and so on ($a_k$ is the sum of the $k$-th powers of all original numbers). a) Can it happen that $a_1 > a_2 > a_3 > a_4 > a_5$ and $a_5 < a_6 < a_7 < \ldots$? (4 points) b) Can it happen that $a_1 < a_2 < a_3 < a_4 < a_5$ and $a_5 > a_6 > a_7 > \ldots$? (4 points) [i](Alexey Tolpygo)[/i]

2007 Korea National Olympiad, 4
Tags: induction , inequalities , prime number , relatively prime , number theory
For all positive integer $ n\geq 2$, prove that product of all prime numbers less or equal than $ n$ is smaller than $ 4^{n}$.

2016 Kosovo National Mathematical Olympiad, 5
Tags: inequalities
If $a,b,c$ are sides of right triangle with $c$ hypothenuse then show that for every positive integer $n>2$ we have $c^n>a^n+b^n$ .

IV Soros Olympiad 1997 - 98 (Russia), 9.4
Tags: absolute value , inequalities , algebra
Find the smallest and largest values of the expression $$\frac{ \left| ...\left| |x-1|-1\right| ... -1\right| +1}{\left| |x-2|-1 \right|+1}$$ (The number of units in the numerator of a fraction, including the last one, is eleven, of which ten are under the absolute value sign.)

2016 Federal Competition For Advanced Students, P1, 1
Tags: inequalities , algebra
Determine the largest constant $C$ such that $$(x_1 + x_2 + \cdots + x_6)^2 \ge C \cdot (x_1(x_2 + x_3) + x_2(x_3 + x_4) + \cdots + x_6(x_1 + x_2))$$ holds for all real numbers $x_1, x_2, \cdots , x_6$. For this $C$, determine all $x_1, x_2, \cdots x_6$ such that equality holds. (Walther Janous)

2010 Victor Vâlcovici, 2
Tags: inequalities , am-gm , trigonometric inequality
$ \sum_{cyc}\frac{1}{\left(\text{tg} y+\text{tg} z\right) \text{cos}^2 x} \ge 3, $ for any $ x,y,z\in (0,\pi/2) $ [i]Carmen[/i] and [i]Viorel Botea[/i]