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

2009 International Zhautykov Olympiad, 3
Tags: inequalities , trigonometry , trig identities , law of cosines , geometry
For a convex hexagon $ ABCDEF$ with an area $ S$, prove that: \[ AC\cdot(BD\plus{}BF\minus{}DF)\plus{}CE\cdot(BD\plus{}DF\minus{}BF)\plus{}AE\cdot(BF\plus{}DF\minus{}BD)\geq 2\sqrt{3}S \]

2006 Junior Balkan Team Selection Tests - Romania, 3
Tags: trigonometry , inequalities , algebra
Find all real numbers $ a$ and $ b$ such that \[ 2(a^2 \plus{} 1)(b^2 \plus{} 1) \equal{} (a \plus{} 1)(b \plus{} 1)(ab \plus{} 1). \] [i]Valentin Vornicu[/i]

2007 South East Mathematical Olympiad, 1
Tags: function , inequalities , algebra
Let $f(x)$ be a function satisfying $f(x+1)-f(x)=2x+1 (x \in \mathbb{R})$.In addition, $|f(x)|\le 1$ holds for $x\in [0,1]$. Prove that $|f(x)|\le 2+x^2$ holds for $x \in \mathbb{R}$.

2018 Thailand TST, 1
Tags: inequalities
Let $x, y, z$ be positive reals such that $xyz = 1$. Prove that $$\sum_{cyc} \frac{1}{\sqrt{x+2y+6}}\leq\sum_{cyc} \frac{x}{\sqrt{x^2+4\sqrt{y}+4\sqrt{z}}}.$$

2010 Bosnia Herzegovina Team Selection Test, 5
Tags: inequalities
Let $a$,$b$ and $c$ be sides of a triangle such that $a+b+c\le2$. Prove that $-3<{\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}-\frac{a^3}{c}-\frac{b^3}{a}-\frac{c^3}{b}}<3$

1996 Iran MO (3rd Round), 3
Tags: inequalities
Let $a_1 \geq a_2 \geq \cdots \geq a_n$ be $n$ real numbers such that $a_1^k +a_2^k + \cdots + a_n^k \geq 0$ for all positive integers $k$. Suppose that $p=\max\{|a_1|,|a_2|, \ldots,|a_n|\}$. Prove that $p=a_1$, and \[(x-a_1)(x-a_2)\cdots(x-a_n)\leq x^n-a_1^n \qquad \forall x>a_1.\]

2009 Jozsef Wildt International Math Competition, W. 18
Tags: inequalities
If $a$, $b$, $c>0$ and $abc=1$, then $$\sum \limits^{cyc} \frac{a+b+c^n}{a^{2n+3}+b^{2n+3}+ab} \leq a^{n+1}+b^{n+1}+c^{n+1}$$ for all $n\in \mathbb{N}$

2019 Dutch Mathematical Olympiad, 4
Tags: fibonacci sequence , fibonacci , sum , inequalities , algebra
The sequence of Fibonacci numbers $F_0, F_1, F_2, . . .$ is defined by $F_0 = F_1 = 1 $ and $F_{n+2} = F_n+F_{n+1}$ for all $n > 0$. For example, we have $F_2 = F_0 + F_1 = 2, F_3 = F_1 + F_2 = 3, F_4 = F_2 + F_3 = 5$, and $F_5 = F_3 + F_4 = 8$. The sequence $a_0, a_1, a_2, ...$ is defined by $a_n =\frac{1}{F_nF_{n+2}}$ for all $n \ge 0$. Prove that for all $m \ge 0$ we have: $a_0 + a_1 + a_2 + ... + a_m < 1$.

1994 Greece National Olympiad, 3
Tags: algebra , inequalities
If $a^2+b^2+c^2+d^2=1$, prove that $$(a-b)^2+(b-c)^2+(c-d)^2+(a-c)^2+(a-d)^2+(b-d)^2\leq 4$$ When does 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 $a_1, \dots, a_n$.

2004 Romania Team Selection Test, 2
Tags: rectangle , inequalities , analytic geometry , geometry
Let $\{R_i\}_{1\leq i\leq n}$ be a family of disjoint closed rectangular surfaces with total area 4 such that their projections of the $Ox$ axis is an interval. Prove that there exist a triangle with vertices in $\displaystyle \bigcup_{i=1}^n R_i$ which has an area of at least 1. [Thanks Grobber for the correction]

2010 Postal Coaching, 4
Tags: geometry , cauchy inequality , inequalities
$\triangle ABC$ has semiperimeter $s$ and area $F$ . A square $P QRS$ with side length $x$ is inscribed in $ABC$ with $P$ and $Q$ on $BC$, $R$ on $AC$, and $S$ on $AB$. Similarly, $y$ and $z$ are the sides of squares two vertices of which lie on $AC$ and $AB$, respectively. Prove that \[\frac 1x +\frac 1y + \frac 1z \le \frac{s(2+\sqrt3)}{2F}\]

2005 Junior Balkan Team Selection Tests - Romania, 2
Tags: inequalities , number theory
Find the largest positive integer $n>10$ such that the residue of $n$ when divided by each perfect square between $2$ and $\dfrac n2$ is an odd number.

2006 Victor Vâlcovici, 2
Tags: geometry , inequalities
Let $ ABC $ be a triangle with $ AB=AC $ and chose such that $ \angle BAC <120^{\circ } . $ On the altitude of $ ABC $ from $ A, $ consider the point $ O $ so that $ \angle BOC =120^{\circ } , $ and an arbitrary point $ M\neq O $ in the interior of $ ABC. $ Show that $ MA+MB+MC>OA+OB+OC. $ [i]Gheorghe Bucur[/i]

1995 Italy TST, 3
Tags: algebra , inequalities , function , functional equation , bounding
A function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the conditions \[\begin{cases}f(x+24)\le f(x)+24\\ f(x+77)\ge f(x)+77\end{cases}\quad\text{for all}\ x\in\mathbb{R}\] Prove that $f(x+1)=f(x)+1$ for all real $x$.

2009 Romanian Masters In Mathematics, 1
Tags: floor function , inequalities , number theory , greatest common divisor , least common multiple , triangle inequality , algebra
For $ a_i \in \mathbb{Z}^ \plus{}$, $ i \equal{} 1, \ldots, k$, and $ n \equal{} \sum^k_{i \equal{} 1} a_i$, let $ d \equal{} \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$. Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)}$ is an integer. [i]Dan Schwarz, Romania[/i]

2001 Romania National Olympiad, 2
Tags: trigonometry , inequalities , 3d geometry , tetrahedron , geometry
In the tetrahedron $OABC$ we denote by $\alpha,\beta,\gamma$ the measures of the angles $\angle BOC,\angle COA,$ and $\angle AOB$, respectively. Prove the inequality \[\cos^2\alpha+\cos^2\beta+\cos^2\gamma<1+2\cos\alpha\cos\beta\cos\gamma \]

2011 Costa Rica - Final Round, 4
Tags: algebra , inequalities
Let $p_1, p_2, ..., p_n$ be positive real numbers, such that $p_1 + p_2 +... + p_n = 1$. Let $x \in [0,1]$ and let $y_1, y_2, ..., y_n$ be such that $y^2_1 + y^2_2 +...+ y^2_n= x$. Prove that $$\left( \sum_{nx\le k \le n }y_k \sqrt{p_k} \right)^2 \le \sum_{k=1}^{n}\frac{k}{n} p_k$$

2011 German National Olympiad, 1
Tags: inequalities , trigonometry , algebra , sine double angle
Prove for each non-negative integer $n$ and real number $x$ the inequality \[ \sin{x} \cdot(n \sin{x}-\sin{nx}) \geq 0 \]

1999 Romania National Olympiad, 4
Tags: geometry , parallelogram , algebra , convexity , inequalities
a) Let $a,b\in R$, $a <b$. Prove that $x \in (a,b)$ if and only if there exists $\lambda \in (0,1)$ such that $x=\lambda a +(1-\lambda)b$. b) If the function $f: R \to R$ has the property: $$f (\lambda x+(1-\lambda) y) < \lambda f(x) + (1-\lambda)f(y), \forall x,y \in R, x\ne y, \forall \lambda \in (0,1), $$ prove that one cannot find four points on the function’s graph that are the vertices of a parallelogram

1986 Flanders Math Olympiad, 2
Tags: inequalities , induction
Prove that for integer $n$ we have: \[n! \le \left( \frac{n+1}{2} \right)^n\] [size=75][i](please note that the pupils in the competition never heard of AM-GM or alikes, it is intended to be solved without any knowledge on inequalities)[/i][/size]

V Soros Olympiad 1998 - 99 (Russia), 11.5
Tags: algebra , inequalities
Find the smallest value of the expression $$(x -y)^2 + (z - u)^2,$$ if $$(x -1)^2 + (y -4)^2 + (z-3)^2 + (u-2)^2 = 1.$$

1999 Austrian-Polish Competition, 2
Tags: inequalities
Find the best possible $k,k'$ such that \[k<\frac{v}{v+w}+\frac{w}{w+x}+\frac{x}{x+y}+\frac{y}{y+z}+\frac{z}{z+v}<k'\] for all positive reals $v,w,x,y,z$.

2019 Saudi Arabia JBMO TST, 4
Tags: inequalities
Prove that if $x, y, z$ are reals, then $x^2(3y^2+3z^2-2yz)=>yz(2xy+2xz-yz)$

2017 Irish Math Olympiad, 4
Tags: algebra , inequalities
Show that for all non-negative numbers $a,b$, $$ 1 + a^{2017} + b^{2017} \geq a^{10}b^{7} + a^{7}b^{2000} + a^{2000}b^{10} $$When is equality attained?