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

2012 Mathcenter Contest + Longlist, 5
Tags: algebra , inequalities
Let $a,b,c>0$ and $a+b+c+abc=4$. Prove that $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{c+a}}+\frac{c}{\sqrt{a+b} }\ge \frac{1}{\sqrt{2}}(a+b+c).$$ [i](Zhuge Liang)[/i]

1999 Brazil Team Selection Test, Problem 2
Tags: geometric inequality , inequalities , geometry , ratio
In a triangle $ABC$, the bisector of the angle at $A$ of a triangle $ABC$ intersects the segment $BC$ and the circumcircle of $ABC$ at points $A_1$ and $A_2$, respectively. Points $B_1,B_2,C_1,C_2$ are analogously defined. Prove that $$\frac{A_1A_2}{BA_2+CA_2}+\frac{B_1B_2}{CB_2+AB_2}+\frac{C_1C_2}{AC_2+BC_2}\ge\frac34.$$

2000 Junior Balkan MO, 4
Tags: inequalities , combinatorics
At a tennis tournament there were $2n$ boys and $n$ girls participating. Every player played every other player. The boys won $\frac 75$ times as many matches as the girls. It is knowns that there were no draws. Find $n$. [i]Serbia[/i]

2003 China Team Selection Test, 1
Tags: inequalities
$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of: \[ x^7(yz-1)+y^7(zx-1)+z^7(xy-1) \]

2011 India IMO Training Camp, 1
Tags: trigonometry , inequalities
Let $ABC$ be an acute-angled triangle. Let $AD,BE,CF$ be internal bisectors with $D, E, F$ on $BC, CA, AB$ respectively. Prove that \[\frac{EF}{BC}+\frac{FD}{CA}+\frac{DE}{AB}\geq 1+\frac{r}{R}\]

2018-2019 Fall SDPC, 8
Tags: number theory , relatively prime , inequalities
Let $S(n)=1\varphi(1)+2\varphi(2) \ldots +n\varphi(n)$, where $\varphi(n)$ is the number of positive integers less than or equal to $n$ that are relatively prime to $n$. (For instance $\varphi(12)=4$ and $\varphi(20)=8$.) Prove that for all $n \geq 2018$, the following inequality holds: $$0.17n^3 \leq S(n) \leq 0.23n^3$$

2005 Germany Team Selection Test, 3
Tags: inequalities , modular arithmetic , algorithm , induction , number theory
We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.

2008 Bulgaria National Olympiad, 3
Tags: inequalities , trigonometry , modular arithmetic , induction , ceiling function , pigeonhole principle , algebra
Let $n\in\mathbb{N}$ and $0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi$ and $b_1,b_2,\ldots ,b_n$ are real numbers for which the following inequality is satisfied : \[\left|\sum_{i\equal{}1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\] for all $ k\in\mathbb{N}$. Prove that $ b_1\equal{}b_2\equal{}\ldots \equal{}b_n\equal{}0$.

2012 Abels Math Contest (Norwegian MO) Final, 4a
Tags: algebra , inequalities
Two positive numbers $x$ and $y$ are given. Show that $\left(1 +\frac{x}{y} \right)^3 + \left(1 +\frac{y}{x}\right)^3 \ge 16$.

2009 Korea National Olympiad, 2
Tags: inequalities
Let $ a,b,c$ be positive real numbers. Prove that \[ \frac{a^3}{c(a^2 + bc)} + \frac{b^3}{a(b^2 + ca)} + \frac{c^3}{b(c^2 +ab )} \ge \frac{3}{2} . \]

2017 Junior Balkan Team Selection Tests - Moldova, Problem 2
Tags: inequalities
Let $a,b,c$ be the sidelengths of a triangle. Prove that $$2<\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}<\sqrt{6}.$$

2005 IMAR Test, 2
Tags: function , inequalities
Let $n \geq 3$ be an integer and let $a,b\in\mathbb{R}$ such that $nb\geq a^2$. We consider the set \[ X = \left\{ (x_1,x_2,\ldots,x_n)\in\mathbb{R}^n \mid \sum_{k=1}^n x_k = a, \ \sum_{k=1}^n x_k^2 = b \right\} . \] Find the image of the function $M: X\to \mathbb{R}$ given by \[ M(x_1,x_2,\ldots,x_n) = \max_{1\leq k\leq n} x_k . \] [i]Dan Schwarz[/i]

2002 Flanders Junior Olympiad, 1
Tags: inequalities , algebra
Prove that for all $a,b,c \in \mathbb{R}^+_0$ we have \[\frac{a}{bc}+\frac{b}{ac}+\frac{c}{ab} \ge \frac2a+\frac2b-\frac2c\] and determine when equality occurs.

1998 Croatia National Olympiad, Problem 2
Tags: inequalities
If $a,b$ are nonnegative real numbers, prove the inequality $$\frac{a+\sqrt[3]{a^2b}+\sqrt[3]{ab^2}+b}4\le\frac{\sqrt{a+\sqrt{ab}+b}}3.$$

1999 Korea - Final Round, 3
Tags: inequalities
Let $a_1, a_2, ..., a_{1999}$ be nonnegative real numbers satisfying the following conditions: a. $a_1+a_2+...+a_{1999}=2$ b. $a_1a_2+a_2a_3+...+a_{1999}a_1=1$. Let $S=a_1^ 2+a_2 ^ 2+...+a_{1999}^2$. Find the maximum and minimum values of $S$.

2003 Junior Balkan Team Selection Tests - Moldova, 2
Tags: inequalities
Let $a, b, c>0$ such that $a^{2}+b^{2}+c^{2}=3abc.$ Prove the following inequality: \[ \frac{a}{b^{2}c^{2}}+\frac{b}{c^{2}a^{2}}+\frac{c}{a^{2}b^{2}}\geq\frac{9}{a+b+c} \]

2007 ITest, 35
Tags: inequalities
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits.

2010 Romania Team Selection Test, 1
Tags: induction , strong induction , inequalities
Given a positive integer number $n$, determine the minimum of \[\max \left\{\dfrac{x_1}{1 + x_1},\, \dfrac{x_2}{1 + x_1 + x_2},\, \cdots,\, \dfrac{x_n}{1 + x_1 + x_2 + \cdots + x_n}\right\},\] as $x_1, x_2, \ldots, x_n$ run through all non-negative real numbers which add up to $1$. [i]Kvant Magazine[/i]

1989 IMO Longlists, 5
Tags: inequalities , induction , trigonometry , limit , algebra
The sequences $ a_0, a_1, \ldots$ and $ b_0, b_1, \ldots$ are defined for $ n \equal{} 0, 1, 2, \ldots$ by the equalities \[ a_0 \equal{} \frac {\sqrt {2}}{2}, \quad a_{n \plus{} 1} \equal{} \frac {\sqrt {2}}{2} \cdot \sqrt {1 \minus{} \sqrt {1 \minus{} a^2_n}} \] and \[ b_0 \equal{} 1, \quad b_{n \plus{} 1} \equal{} \frac {\sqrt {1 \plus{} b^2_n} \minus{} 1}{b_n} \] Prove the inequalities for every $ n \equal{} 0, 1, 2, \ldots$ \[ 2^{n \plus{} 2} a_n < \pi < 2^{n \plus{} 2} b_n. \]

2018 Austria Beginners' Competition, 1
Tags: inequalities , algebra
Let $a, b$ and $c$ denote positive real numbers. Prove that $\frac{a}{c}+\frac{c}{b}\ge \frac{4a}{a + b}$ . When does equality hold? (Walther Janous)

2020 Silk Road, 1
Tags: inequalities , divisible , divide , number theory
Given a strictly increasing infinite sequence of natural numbers $ a_1, $ $ a_2, $ $ a_3, $ $ \ldots $. It is known that $ a_n \leq n + 2020 $ and the number $ n ^ 3 a_n - 1 $ is divisible by $ a_ {n + 1} $ for all natural numbers $ n $. Prove that $ a_n = n $ for all natural numbers $ n $.

2005 Germany Team Selection Test, 2
Tags: inequalities , n-variable inequality
Let $ n$ be a positive integer such that $ n\geq 3$. Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2n$ positive real numbers satisfying the equations \[ a_1 \plus{} a_2 \plus{} ... \plus{} a_n \equal{} 1, \quad \text{and} \quad b_1^2 \plus{} b_2^2 \plus{} ... \plus{} b_n^2 \equal{} 1.\] Prove the inequality \[a_1\left(b_1 \plus{} a_2\right) \plus{} a_2\left(b_2 \plus{} a_3\right) \plus{} ... \plus{} a_{n \minus{} 1}\left(b_{n \minus{} 1} \plus{} a_n\right) \plus{} a_n\left(b_n \plus{} a_1\right) < 1.\]

1969 All Soviet Union Mathematical Olympiad, 128
Tags: algebra , inequalities
Prove that for the arbitrary positive $a_1, a_2, ... , a_n$ the following inequality is held $$\frac{a_1}{a_2+a_3}+\frac{a_2}{a_3+a_4}+....+\frac{a_{n-1}}{a_n+a_1}+\frac{a_n}{a_1+a_2}>\frac{n}{4}$$

2015 Switzerland - Final Round, 10
Tags: algebra , inequalities
Find the largest natural number $n$ such that for all real numbers $a, b, c, d$ the following holds: $$(n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d)$$

2015 Denmark MO - Mohr Contest, 1
Tags: algebra , inequalities
The numbers $a, b, c, d$ and $e$ satisfy $$a + b < c + d < e + a < b + c < d + e .$$ Which of the numbers is the smallest, and which is the largest?