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: 6530

2011 Putnam, B1
Tags: integration , inequalities , floor function , absolute value , putnam analysis
Let $h$ and $k$ be positive integers. Prove that for every $\varepsilon >0,$ there are positive integers $m$ and $n$ such that \[\varepsilon < \left|h\sqrt{m}-k\sqrt{n}\right|<2\varepsilon.\]

1993 Moldova Team Selection Test, 7
Tags: inequalities
If $x_1 + x_2 + \cdots + x_n = \sum_{i=1}^{n} x_i = \frac{1}{2}$ and $x_i > 0$ ; then prove that: $ \frac{1-x_1}{1+x_1} \cdot \frac{1-x_2}{1+x_2} \cdots \frac{1-x_n}{1+x_n} = \prod_{i=1}^{n} \frac{1-x_i}{1+x_i} \geq \frac{1}{3}$

2011 Moldova Team Selection Test, 1
Tags: calculus , derivative , inequalities , algebra
Find all real numbers $x, y$ such that: $y+3\sqrt{x+2}=\frac{23}2+y^2-\sqrt{49-16x}$

2009 China Team Selection Test, 3
Tags: modular arithmetic , inequalities , number theory
Prove that for any odd prime number $ p,$ the number of positive integer $ n$ satisfying $ p|n! \plus{} 1$ is less than or equal to $ cp^\frac{2}{3}.$ where $ c$ is a constant independent of $ p.$

2019 CHKMO, 1
Tags: inequalities , algebra
Given that $a,b$, and $c$ are positive real numbers such that $ab + bc + ca \geq 1$, prove that \[ \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} \geq \frac{\sqrt{3}}{abc} .\]

2024 Bangladesh Mathematical Olympiad, P6
Tags: inequalities , algebra , permutation
Let $a_1, a_2, \ldots, a_{2024}$ be a permutation of $1, 2, \ldots, 2024$. Find the minimum possible value of\[\sum_{i=1} ^{2023} \Big[(a_i+a_{i+1})\Big(\frac{1}{a_i}+\frac{1}{a_{i+1}}\Big)+\frac{1}{a_ia_{i+1}}\Big]\] [i]Proposed by Md. Ashraful Islam Fahim[/i]

1991 Austrian-Polish Competition, 5
Tags: inequalities , algebra
If $x,y, z$ are arbitrary positive numbers with $xyz = 1$, prove the inequality $$x^2+y^2+z^2 + xy+yz + zx \ge 2(\sqrt{x} +\sqrt{y}+ \sqrt{z})$$.

2004 China Team Selection Test, 2
Tags: combinatorics , inequalities
Let $ k$ be a positive integer. Set $ A \subseteq \mathbb{Z}$ is called a $ \textbf{k \minus{} set}$ if there exists $ x_1, x_2, \cdots, x_k \in \mathbb{Z}$ such that for any $ i \neq j$, $ (x_i \plus{} A) \cap (x_j \plus{} A) \equal{} \emptyset$, where $ x \plus{} A \equal{} \{ x \plus{} a \mid a \in A \}$. Prove that if $ A_i$ is $ \textbf{k}_i\textbf{ \minus{} set}$($ i \equal{} 1,2, \cdots, t$), and $ A_1 \cup A_2 \cup \cdots \cup A_t \equal{} \mathbb{Z}$, then $ \displaystyle \frac {1}{k_1} \plus{} \frac {1}{k_2} \plus{} \cdots \plus{} \frac {1}{k_t} \geq 1$.

2017 Iberoamerican, 6
Tags: inequalities
Let $n > 2$ be an even positive integer and let $a_1 < a_2 < \dots < a_n$ be real numbers such that $a_{k + 1} - a_k \leq 1$ for each $1 \leq k \leq n - 1$. Let $A$ be the set of ordered pairs $(i, j)$ with $1 \leq i < j \leq n$ such that $j - i$ is even, and let $B$ the set of ordered pairs $(i, j)$ with $1 \leq i < j \leq n$ such that $j - i$ is odd. Show that $$\prod_{(i, j) \in A} (a_j - a_i) > \prod_{(i, j) \in B} (a_j - a_i)$$

2012 Online Math Open Problems, 20
Tags: algorithm , inequalities , induction , pigeonhole principle , strong induction
The numbers $1, 2, \ldots, 2012$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number $N$ remains. Find the remainder when the maximum possible value of $N$ is divided by 1000. [i]Victor Wang.[/i]

2007 Austria Beginners' Competition, 3
Tags: inequalities , algebra
For real numbers $x \ge 0$ and $y \ge 0$, write $A= \frac{x+y}{2}$ for the arithmetic mean and $G=\sqrt{xy}$ for the geometric mean of $x$ and $y$. Furthermore, let $W= \frac{\sqrt{x}+\sqrt{y}}{2}$ be the arithmetic mean of $\sqrt{x}$ and $\sqrt{y}$. Prove that $$G\le W^2 \le A.$$ Determine all $x$ and $y$ such that $G= W^2 = A$

1972 USAMO, 2
Tags: geometry , 3d geometry , tetrahedron , inequalities , parallelogram , triangle inequality
A given tetrahedron $ ABCD$ is isoceles, that is, $ AB\equal{}CD$, $ AC\equal{}BD$, $ AD\equal{}BC$. Show that the faces of the tetrahedron are acute-angled triangles.

2014 Bosnia Herzegovina Team Selection Test, 2
Tags: inequalities
Let $a$ ,$b$ and $c$ be distinct real numbers. $a)$ Determine value of $ \frac{1+ab }{a-b} \cdot \frac{1+bc }{b-c} + \frac{1+bc }{b-c} \cdot \frac{1+ca }{c-a} + \frac{1+ca }{c-a} \cdot \frac{1+ab}{a-b} $ $b)$ Determine value of $ \frac{1-ab }{a-b} \cdot \frac{1-bc }{b-c} + \frac{1-bc }{b-c} \cdot \frac{1-ca }{c-a} + \frac{1-ca }{c-a} \cdot \frac{1-ab}{a-b} $ $c)$ Prove the following ineqaulity $ \frac{1+a^2b^2 }{(a-b)^2} + \frac{1+b^2c^2 }{(b-c)^2} + \frac{1+c^2a^2 }{(c-a)^2} \geq \frac{3}{2} $ When does eqaulity holds?

1998 Irish Math Olympiad, 2
Tags: inequalities
Prove that if $ a,b,c$ are positive real numbers, then: $ \frac{9}{a\plus{}b\plus{}c} \le 2 \left( \frac{1}{a\plus{}b}\plus{}\frac{1}{b\plus{}c}\plus{}\frac{1}{c\plus{}a} \right) \le \frac{1}{a}\plus{}\frac{1}{b}\plus{}\frac{1}{c}.$

2019 LIMIT Category B, Problem 5
Tags: inequalities , parameterization
The set of values of $m$ for which $mx^2-6mx+5m+1>0$ for all real $x$ is $\textbf{(A)}~m<\frac14$ $\textbf{(B)}~m\ge0$ $\textbf{(C)}~0\le m\le\frac14$ $\textbf{(D)}~0\le m<\frac14$

2011 Morocco National Olympiad, 3
Tags: inequalities
Let $a$ and $b$ be two real numbers and let$M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$. Find the values of $a$ and $b$ for which $M(a,b)$ is minimal.

VI Soros Olympiad 1999 - 2000 (Russia), 8.1
Tags: number theory , inequalities
Let $p,q,r$ be prime numbers such that $2p>q$, $q > 2r$ and $q>p+r$. Prove that $p+q+r\ge 20$.

2012 Tuymaada Olympiad, 3
Tags: inequalities
Prove that for any real numbers $a,b,c$ satisfying $abc = 1$ the following inequality holds \[\dfrac{1} {2a^2+b^2+3}+\dfrac {1} {2b^2+c^2+3}+\dfrac{1} {2c^2+a^2+3}\leq \dfrac {1} {2}.\] [i]Proposed by V. Aksenov[/i]

2008 District Olympiad, 4
Tags: algebra , inequalities
Determine $ x,y,z>0$ for which $ x^3y\plus{}3<\equal{}4z, y^3z\plus{}3<\equal{}4x,z^3x\plus{}3<\equal{}4y.$

2007 Today's Calculation Of Integral, 225
Tags: calculus , integration , geometry , ratio , inequalities , function , logarithm
2 Points $ P\left(a,\ \frac{1}{a}\right),\ Q\left(2a,\ \frac{1}{2a}\right)\ (a > 0)$ are on the curve $ C: y \equal{}\frac{1}{x}$. Let $ l,\ m$ be the tangent lines at $ P,\ Q$ respectively. Find the area of the figure surrounded by $ l,\ m$ and $ C$.

2005 Indonesia MO, 1
Tags: calculus , integration , inequalities , combinatorics
Let $ n$ be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is $ n$.

2023 Thailand October Camp, 3
Tags: inequalities
Let $n>3$ be an integer. If $x_1<x_2<\ldots<x_{n+2}$ are reals with $x_1=0$, $x_2=1$ and $x_3>2$, what is the maximal value of $$(\frac{x_{n+1}+x_{n+2}-1}{x_{n+1}(x_{n+2}-1)})\cdot (\sum_{i=1}^{n}\frac{(x_{i+2}-x_{i+1})(x_{i+1}-x_i)}{x_{i+2}-x_i})?$$

2008 Hong Kong TST, 2
Tags: limit , inequalities
Let $ a$, $ b$, $ c$ be the three sides of a triangle. Determine all possible values of \[ \frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}\]

1988 Swedish Mathematical Competition, 6
Tags: inequalities , algebra , sequence , recurrence relation , radical
The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}$ for $n \ge 1$. Prove that there exists $a$ such that $\frac{1}{2} \le \frac{a_n}{n^a} \le 2$ for $n \ge 1$.

1986 IMO Longlists, 51
Tags: inequalities , rectangle , trigonometry , cyclic quadrilateral , geometry
Let $a, b, c, d$ be the lengths of the sides of a quadrilateral circumscribed about a circle and let $S$ be its area. Prove that $S \leq \sqrt{abcd}$ and find conditions for equality.