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.

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Found problems: 6530

2006 Taiwan National Olympiad, 2
Tags: inequalities , geometry
In triangle $ABC$, $D$ is the midpoint of side $AB$. $E$ and $F$ are points arbitrarily chosen on segments $AC$ and $BC$, respectively. Show that $[DEF] < [ADE] + [BDF]$.

2005 Indonesia MO, 6
Tags: symmetry , geometry , inequalities , number theory
Find all triples $ (x,y,z)$ of integers which satisfy $ x(y \plus{} z) \equal{} y^2 \plus{} z^2 \minus{} 2$ $ y(z \plus{} x) \equal{} z^2 \plus{} x^2 \minus{} 2$ $ z(x \plus{} y) \equal{} x^2 \plus{} y^2 \minus{} 2$.

2021 Thailand TSTST, 1
Tags: inequalities , algebra
Let $a,b,c$ be distinct positive real numbers such that $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\leq 1$. Prove that $$2\left(\sqrt{\frac{a+b}{ac}}+\sqrt{\frac{b+c}{ba}}+\sqrt{\frac{c+a}{cb}}\right)<\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-c)(b-a)}+\frac{c^3}{(c-a)(c-b)}.$$

2007 Gheorghe Vranceanu, 3
Tags: function , partition , algebra , inequalities
Let be a function $ s:\mathbb{N}^2\longrightarrow \mathbb{N} $ that sends $ (m,n) $ to the number of solutions in $ \mathbb{N}^n $ of the equation: $$ x_1+x_2+\cdots +x_n=m $$ [b]1)[/b] Prove that: $$ s(m+1,n+1)=s(m,n)+s(m,n+1) =\prod_{r=1}^n\frac{m-r+1}{r} ,\quad\forall m,n\in\mathbb{N} $$ [b]2)[/b] Find $ \max\left\{ a_1a_2\cdots a_{20}\bigg| a_1+a_2+\cdots +a_{20}=2007, a_1,a_2,\ldots a_{20}\in\mathbb{N} \right\} . $

2014 Contests, 2
Tags: inequalities , induction , number theory
Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

1996 Poland - Second Round, 3
Tags: function , inequalities
$a,b,c \geq-3/4$ and $a+b+c=1$. Show that: $\frac{a}{1+a^{2}}+\frac{b}{1+b^{2}}+\frac{c}{1+c^{2}}\leq \frac{9}{10}$

2005 Baltic Way, 5
Tags: logarithm , inequalities
Let $a$, $b$, $c$ be positive real numbers such that $abc=1$. Prove that \[\frac a{a^{2}+2}+\frac b{b^{2}+2}+\frac c{c^{2}+2}\leq 1 \]

2019 Bangladesh Mathematical Olympiad, 4
Tags: algebra , inequalities
$A$ is a positive real number.$n$ is positive integer number.Find the set of possible values of the infinite sum $x_0^n+x_1^n+x_2^n+...$ where $x_0,x_1,x_2...$ are all positive real numbers so that the infinite series $x_0+x_1+x_2+...$ has sum $A$.

2010 Contests, 4
Tags: inequalities , number theory
Find all positive integers $n$ which satisfy the following tow conditions: (a) $n$ has at least four different positive divisors; (b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)[/i]

2010 Contests, 3
Tags: inequalities
Prove that for all $n \in \mathbb{Z^+}$ and for all positive real numbers satisfying $a_1a_2...a_n=1$ \[ \displaystyle\sum_{i=1}^{n} \frac{a_i}{\sqrt{{a_i}^4+3}} \leq \frac{1}{2}\displaystyle\sum_{i=1}^{n} \frac{1}{a_i} \]

2009 Kyiv Mathematical Festival, 5
Tags: algebra , sequence , recurrence relation , recursion , inequalities
a) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $x_n-2x_{n+1}+x_{n+2} \le 0$ for any $n$ . Moreover $x_o=1,x_{20}=9,x_{200}=6$. What is the maximal value of $x_{2009}$ can be? b) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $2x_n-3x_{n+1}+x_{n+2} \le 0$ for any $n$. Moreover $x_o=1,x_1=2,x_3=1$. Can $x_{2009}$ be greater then $0,678$ ?

1987 Greece National Olympiad, 3
Tags: algebra , inequalities
Prova that for any real $a$, expresssion $A=(a-1)(a-3)(a-4)(a-6)+10$ is always positive. What is the minimum value that expression $A$ can take and for which values of $a$?

2004 Croatia National Olympiad, Problem 1
Tags: complex number , inequality , complex inequality , inequalities
Let $z_1,\ldots,z_n$ and $w_1,\ldots,w_n$ $(n\in\mathbb N)$ be complex numbers such that $$|\epsilon_1z_1+\ldots+\epsilon_nz_n|\le|\epsilon_1w_1+\ldots+\epsilon_nw_n|$$holds for every choice of $\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}$. Prove that $$|z_1|^2+\ldots+|z_n|^2\le|w_1|^2+\ldots+|w_n|^2.$$

2004 Greece Junior Math Olympiad, 3
Tags: inequalities
x,y,z positive real numbers such that $x^2+y^2+z^2=25$ Find the min price of $A=\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}$

MathLinks Contest 7th, 5.1
Tags: algebra , polynomial , inequalities , vieta , product of roots , absolute value
Find all real polynomials $ g(x)$ of degree at most $ n \minus{} 3$, $ n\geq 3$, knowing that all the roots of the polynomial $ f(x) \equal{} x^n \plus{} nx^{n \minus{} 1} \plus{} \frac {n(n \minus{} 1)}2 x^{n \minus{} 2} \plus{} g(x)$ are real.

2007 Romania National Olympiad, 1
Tags: calculus , integration , inequalities , function , derivative , trigonometry , logarithm
Let $\mathcal{F}$ be the set of functions $f: [0,1]\to\mathbb{R}$ that are differentiable, with continuous derivative, and $f(0)=0$, $f(1)=1$. Find the minimum of $\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx$ (where $f\in\mathcal{F}$) and find all functions $f\in\mathcal{F}$ for which this minimum is attained. [hide="Comment"] In the contest, this was the b) point of the problem. The a) point was simply ``Prove the Cauchy inequality in integral form''. [/hide]

2024-IMOC, A6
Tags: inequalities , calculus
Given positive real $a,b,c$ satisfying \[\frac{1}{\sqrt{a+1}}+\frac{3}{\sqrt{b+3}}+\frac{3}{\sqrt{c+3}}=\frac72\] Prove that $abc\leq 3$.\\ I was asked to propose a inequality for the condition of $abc<3$ inequality since <3 looks like a heart shape, then I construct a equality and with the help of wolfram, I gave the birth of this bad-looking inequality, I’m glad to see any method besides calculus.

2021 Flanders Math Olympiad, 4
Tags: inequalities , functional , algebra , functional equation
(a) Prove that for every $x \in R$ holds that $$-1 \le \frac{x}{x^2 + x + 1} \le \frac 13$$ (b) Determine all functions $f : R \to R$ for which for every $x \in R$ holds that $$f \left( \frac{x}{x^2 + x + 1} \right) = \frac{x^2}{x^4 + x^2 + 1}$$

2005 China Girls Math Olympiad, 5
Tags: inequalities
Let $ x$ and $ y$ be positive real numbers with $ x^3 \plus{} y^3 \equal{} x \minus{} y.$ Prove that \[ x^2 \plus{} 4y^2 < 1.\]

2008 Hong Kong TST, 1
Tags: inequalities , combinatorics
In a school there are $ 2008$ students. Students are members of certain committees. A committee has at most $ 1004$ members and every two students join a common committee. (i) Determine the smallest possible number of committees in the school. (ii) If it is further required that the union of any two committees consists of at most $ 1800$ students, will your answer in (i) still hold?

2010 Slovenia National Olympiad, 4
Tags: probability , function , inequalities
Let $x,y$ and $z$ be real numbers such that $0 \leq x,y,z \leq 1.$ Prove that \[xyz+(1-x)(1-y)(1-z) \leq 1.\] When does equality hold?

2011 Hanoi Open Mathematics Competitions, 12
Tags: inequalities
Suppose that $a > 0; b > 0$ and $a + b \leq 1$. Determine the minimum value of $M=\frac{1}{ab} +\frac{1}{a^2+ab}+\frac{1}{ab+b^2}+\frac{1}{a^2+b^2}$.

2011 NIMO Problems, 4
Tags: inequalities
Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality \[ 1 < a < b+2 < 10. \] [i]Proposed by Lewis Chen [/i]

1990 Tournament Of Towns, (263) 1
Tags: algebra , inequalities
Suppose two positive real numbers are given. Prove that if their sum is less than their product then their sum is greater than four. (N Vasiliev, Moscow)

2011 IberoAmerican, 2
Tags: inequalities , induction , strong induction , algebra
Let $x_1,\ldots ,x_n$ be positive real numbers. Show that there exist $a_1,\ldots ,a_n\in\{-1,1\}$ such that: \[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\ge (a_1x_1+a_2x_2+\ldots + a_n x_n)^2\]