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

2005 Bulgaria Team Selection Test, 4
Tags: function , quadratic , inequalities
Let $a_{i}$ and $b_{i}$, where $i \in \{1,2, \dots, 2005 \}$, be real numbers such that the inequality $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$ holds for all $x \in \mathbb{R}$ and all $i \in \{1,2, \dots, 2005 \}$. Find the maximum possible number of positive numbers amongst $a_{i}$ and $b_{i}$, $i \in \{1,2, \dots, 2005 \}$.

1989 Greece National Olympiad, 3
Tags: algebra , inequalities , polynomial
If $a\ge 0$ prove that $a^4+ a^3-10 a^2+9 a+4>0$.

2004 China Girls Math Olympiad, 5
Tags: inequalities , algebra
Let $ u, v, w$ be positive real numbers such that $ u\sqrt {vw} \plus{} v\sqrt {wu} \plus{} w\sqrt {uv} \geq 1$. Find the smallest value of $ u \plus{} v \plus{} w$.

2009 ISI B.Math Entrance Exam, 7
Tags: rectangle , parallelogram , geometric transformation , geometry , dilation , inequalities
Compute the maximum area of a rectangle which can be inscribed in a triangle of area $M$.

2003 Austrian-Polish Competition, 8
Tags: inequalities , algebra , sum
Given reals $x_1 \ge x_2 \ge ... \ge x_{2003} \ge 0$, show that $$x_1^n - x_2^n + x_2^n - ... - x_{2002}^n + x_{2003}^n \ge (x_1 - x_2 + x_3 - x_4 + ... - x_{2002} + x_{2003})^n$$ for any positive integer $n$.

2024 Polish MO Finals, 5
Tags: algebra , inequalities
We are given an integer $n \ge 2024$ and a sequence $a_1,a_2,\dots,a_{n^2}$ of real numbers satisfying \[\vert a_k-a_{k-1}\vert \le \frac{1}{k} \quad \text{and} \quad \vert a_1+a_2+\dots+a_k\vert \le 1\] for $k=2,3,\dots,n^2$. Show that $\vert a_{n(n-1)}\vert \le \frac{2}{n}$. [i]Note: Proving $\vert a_{n(n-1)}\vert \le \frac{75}{n}$ will be rewarded partial points.[/i]

PEN S Problems, 14
Tags: inequalities
Let $p$ be an odd prime. Determine positive integers $x$ and $y$ for which $x \le y$ and $\sqrt{2p}-\sqrt{x}-\sqrt{y}$ is nonnegative and as small as possible.

2010 Harvard-MIT Mathematics Tournament, 7
Tags: calculus , complex number , inequalities
Let $a_1$, $a_2$, and $a_3$ be nonzero complex numbers with non-negative real and imaginary parts. Find the minimum possible value of \[\dfrac{|a_1+a_2+a_3|}{\sqrt[3]{|a_1a_2a_3|}}.\]

2004 Mediterranean Mathematics Olympiad, 3
Tags: trigonometry , inequalities
Let $a,b,c>0$ and $ab+bc+ca+2abc=1$ then prove that \[2(a+b+c)+1\geq 32abc\]

2005 China Girls Math Olympiad, 7
Tags: inequalities , floor function
Let $ m$ and $ n$ be positive integers with $ m > n \geq 2.$ Set $ S \equal{} \{1, 2, \ldots, m\},$ and $ T \equal{} \{a_l, a_2, \ldots, a_n\}$ is a subset of S such that every number in $ S$ is not divisible by any two distinct numbers in $ T.$ Prove that \[ \sum^n_{i \equal{} 1} \frac {1}{a_i} < \frac {m \plus{} n}{m}. \]

2016 Regional Competition For Advanced Students, 2
Tags: algebra , inequalities
Let $a$, $b$, $c$ and $d$ be real numbers with $a^2 + b^2 + c^2 + d^2 = 4$. Prove that the inequality $$(a+2)(b+2) \ge cd$$ holds and give four numbers $a$, $b$, $c$ and $d$ such that equality holds. (Walther Janous)

2012 Kazakhstan National Olympiad, 3
Tags: inequalities , kanat satylkhanov
Let $ a,b,c,d>0$ for which the following conditions:: $a)$ $(a-c)(b-d)=-4$ $b)$ $\frac{a+c}{2}\geq\frac{a^{2}+b^{2}+c^{2}+d^{2}}{a+b+c+d}$ Find the minimum of expression $a+c$

2014 Contests, 1
Tags: algebra , inequalities , induction
Prove that for $n\ge 2$ the following inequality holds: $$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$

2019 Kazakhstan National Olympiad, 1
Tags: inequalities
Prove for any positives $a,b,c$ the inequality $$ \sqrt[3]{\dfrac{a}{b}}+\sqrt[5]{\dfrac{b}{c}}+\sqrt[7]{\dfrac{c}{a}}>\dfrac{5}{2}$$

2020 Centroamerican and Caribbean Math Olympiad, 5
Tags: inequalities , algebra , inequality , n-variable inequality , polynomial
Let $P(x)$ be a polynomial with real non-negative coefficients. Let $k$ be a positive integer and $x_1, x_2, \dots, x_k$ positive real numbers such that $x_1x_2\cdots x_k=1$. Prove that $$P(x_1)+P(x_2)+\cdots+P(x_k)\geq kP(1).$$

Russian TST 2021, P1
Tags: inequalities , inequality , 4-variable inequality , algebra
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2005 China Team Selection Test, 3
Tags: polynomial , inequalities , triangle inequality , induction , algebra , trigonometry
Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds. Prove that $\sum_{j=1}^n |a_j| \leq 3$.

2009 Hungary-Israel Binational, 3
Tags: function , induction , inequalities , algebra
Does there exist a pair $ (f; g)$ of strictly monotonic functions, both from $ \mathbb{N}$ to $ \mathbb{N}$, such that \[ f(g(g(n))) < g(f(n))\] for every $ n \in\mathbb{N}$?

2019 Turkey Team SeIection Test, 9
Tags: inequalities
Let $x, y, z$ be real numbers such that $y\geq 2z \geq 4x$ and $$ 2(x^3+y^3+z^3)+15(xy^2+yz^2+zx^2)\geq 16(x^2y+y^2z+z^2x)+2xyz.$$ Prove that: $4x+y\geq 4z$

1966 AMC 12/AHSME, 15
Tags: inequalities
If $x-y>x$ and $x+y<y$, then $\text{(A)} \ y<x \qquad \text{(B)} \ x<y \qquad \text{(C)} \ x<y<0 \qquad \text{(D)} \ x<0,y<0$ $\text{(E)} \ x<0,y>0$

2022 Bulgaria JBMO TST, 2
Tags: am-gm , titu's lemma , cauchy schwarz , inequalities
Let $a$, $b$ and $c$ be positive real numbers with $abc = 1$. Determine the minimum possible value of $$ \left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \cdot \left(\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}\right) $$ as well as all triples $(a,b,c)$ which attain the minimum.

2005 All-Russian Olympiad Regional Round, 10.2
Tags: inequalities
10.2 Prove for all $x>0$ and $n\in\mathbb{N}$ the following inequality \[1+x^{n+1}\geq \frac{(2x)^n}{(1+x)^{n-1}}.\] ([i]A. Khrabrov[/i])

2011 Mathcenter Contest + Longlist, 9 sl13
Tags: algebra , inequalities
Let $a,b,c\in\mathbb{R^+}$ If $3=a+b+c\le 3abc$ , prove that $$\frac{1}{\sqrt{2a+1}}+ \frac{1}{\sqrt{2b+1}}+\frac{1}{\sqrt{2c+1}}\le \left( \frac32\right)^{3/2}$$ [i](Real Matrik)[/i]

2014 Mexico National Olympiad, 5
Tags: holder , inequalities
Let $a, b, c$ be positive reals such that $a + b + c = 3$. Prove: \[ \frac{a^2}{a + \sqrt[3]{bc}} + \frac{b^2}{b + \sqrt[3]{ca}} + \frac{c^2}{c + \sqrt[3]{ab}} \geq \frac{3}{2} \] And determine when equality holds.

1979 Austrian-Polish Competition, 3
Tags: inequalities
Find all positive integers $n$ such that the inequality $$\left( \sum\limits_{i=1}^n a_i^2\right) \left(\sum\limits_{i=1}^n a_i \right) -\sum\limits_{i=1}^n a_i^3 \geq 6 \prod\limits_{i=1}^n a_i$$ holds for any $n$ positive numbers $a_1, \dots, a_n$.