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

2015 Iran MO (3rd round), 1
Tags: inequalities
$x,y,z$ are three real numbers inequal to zero satisfying $x+y+z=xyz$. Prove that $$ \sum (\frac{x^2-1}{x})^2 \geq 4$$ [i]Proposed by Amin Fathpour[/i]

2014 USA Team Selection Test, 3
Tags: induction , inequalities , combinatorics , graph theory , extremal combinatorics
Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be [i]amicable[/i] if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable. [i]Zoltán Füredi (suggested by Po-Shen Loh)[/i]

2011 Dutch BxMO TST, 4
Tags: number theory , inequalities , algebra
Let $n \ge 2$ be an integer. Let $a$ be the greatest positive integer such that $2^a | 5^n - 3^n$. Let $b$ be the greatest positive integer such that $2^b \le n$. Prove that $a \le b + 3$.

1952 Miklós Schweitzer, 3
Tags: inequalities , number theory
Prove:If $ a\equal{}p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_{n}^{\alpha_n}$ is a perfect number, then $ 2<\prod_{i\equal{}1}^n\frac{p_i}{p_i\minus{}1}<4$ ; if moreover, $ a$ is odd, then the upper bound $ 4$ may be reduced to $ 2\sqrt[3]{2}$.

2013 All-Russian Olympiad, 2
Tags: inequalities
Let $a,b,c,d$ be positive real numbers such that $ 2(a+b+c+d)\ge abcd $. Prove that \[ a^2+b^2+c^2+d^2 \ge abcd .\]

2016 Baltic Way, 7
Tags: inequalities
Find all positive integers $n$ for which $$3x^n + n(x + 2) - 3 \geq nx^2$$ holds for all real numbers $x.$

2012 ISI Entrance Examination, 6
Tags: geometry , perimeter , calculus , ellipse , inequalities , perpendicular bisector , conic
[b]i)[/b] Let $0<a<b$.Prove that amongst all triangles having base $a$ and perimeter $a+b$ the triangle having two sides(other than the base) equal to $\frac {b}{2}$ has the maximum area. [b]ii)[/b]Using $i)$ or otherwise, prove that amongst all quadrilateral having give perimeter the square has the maximum area.

2016 Mediterranean Mathematics Olympiad, 2
Tags: inequalities
Let $a,b,c$ be positive real numbers with $a+b+c=3$. Prove that \[ \sqrt{\frac{b}{a^2+3}}+ \sqrt{\frac{c}{b^2+3}}+ \sqrt{\frac{a}{c^2+3}} ~\le~ \frac32\sqrt[4]{\frac{1}{abc}}\]

2014 China Northern MO, 6
Tags: inequalities
Let $x,y,z,w $ be real numbers such that $x+2y+3z+4w=1$. Find the minimum of $x^2+y^2+z^2+w^2+(x+y+z+w)^2$.

1970 Regional Competition For Advanced Students, 1
Tags: inequalities
Let $x,y,z$ be positive real numbers such that $x+y+z=1$ Prove that always $\left( 1+\frac1x\right)\times\left(1+\frac1y\right)\times\left(1 +\frac1z\right)\ge 64$ When does equality hold?

2021 China Team Selection Test, 4
Tags: inequality , number theory function , number theory , floor function , inequalities
Proof that $$ \sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .$$

1991 Vietnam National Olympiad, 3
Tags: inequalities
Prove that: $ \frac {x^{2}y}{z} \plus{} \frac {y^{2}z}{x} \plus{} \frac {z^{2}x}{y}\geq x^{2} \plus{} y^{2} \plus{} z^{2}$ where $ x;y;z$ are real numbers saisfying $ x \geq y \geq z \geq 0$

1965 German National Olympiad, 6
Tags: trigonometry , inequalities , geometric inequalities , geometry
Let $\alpha,\beta, \gamma$ be the angles of a triangle. Prove that $\cos\alpha, + \cos\beta + \cos\gamma \le \frac{3}{2} $ and find the cases of equality.

2002 All-Russian Olympiad, 1
Tags: inequalities
For positive real numbers $a, b, c$ such that $a+b+c=3$, show that: \[\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab+bc+ca.\]

1996 Czech and Slovak Match, 3
Tags: inequalities , solid geometry , pyramid , geometry , 3d geometry
The base of a regular quadrilateral pyramid $\pi$ is a square with side length $2a$ and its lateral edge has length a$\sqrt{17}$. Let $M$ be a point inside the pyramid. Consider the five pyramids which are similar to $\pi$ , whose top vertex is at $M$ and whose bases lie in the planes of the faces of $\pi$ . Show that the sum of the surface areas of these five pyramids is greater or equal to one fifth the surface of $\pi$ , and find for which $M$ equality holds.

2009 National Olympiad First Round, 31
Tags: inequalities
For all $ |x| \ge n$, the inequality $ |x^3 \plus{} 3x^2 \minus{} 33x \minus{} 3| \ge 2x^2$ holds. Integer $ n$ can be at least ? $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 5$

1988 Irish Math Olympiad, 10
Tags: inequalities
Let $0\le x\le 1$. Show that if $n$ is any positive integer, then $$(1+x)^n\ge (1-x)^n+2nx(1-x^2)^{\frac{n-1}{2}}$$.

2010 Mediterranean Mathematics Olympiad, 2
Tags: n-variable inequality , inequalities
Given the positive real numbers $a_{1},a_{2},\dots,a_{n},$ such that $n>2$ and $a_{1}+a_{2}+\dots+a_{n}=1,$ prove that the inequality \[ \frac{a_{2}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{1}+n-2}+\frac{a_{1}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{2}+n-2}+\dots+\frac{a_{1}\cdot a_{2}\cdot\dots\cdot a_{n-1}}{a_{n}+n-2}\leq\frac{1}{\left(n-1\right)^{2}}\] does holds.

2015 India IMO Training Camp, 3
Tags: trigonometry , inequalities
Prove that for any triangle $ABC$, the inequality $\displaystyle\sum_{\text{cyclic}}\cos A\le\sum_{\text{cyclic}}\sin (A/2)$ holds.

2018 Switzerland - Final Round, 8
Tags: inequalities
Let $a,b,c,d,e$ be positive real numbers. Find the largest possible value for the expression $$\frac{ab+bc+cd+de}{2a^2+b^2+2c^2+d^2+2e^2}.$$

1999 IMC, 4
Tags: function , inequalities , limit , real analysis
Prove that there's no function $f: \mathbb{R}^+\rightarrow\mathbb{R}^+$ such that $f(x)^2\ge f(x+y)\left(f(x)+y\right)$ for all $x,y>0$.

1991 Czech And Slovak Olympiad IIIA, 1
Tags: trigonometry , inequalities
Prove that for any real numbers $p,q,r,\phi$,: $$\cos^2\phi+q \sin \phi \cos \phi +r\sin^2 \phi \ge \frac12 (p+r-\sqrt{(p-r)^2+q^2})$$

1998 Denmark MO - Mohr Contest, 4
Tags: algebra , inequalities
Let $a$ and $b$ be positive real numbers with $a + b =1$. Show that $$\left(a+\frac{1}{a}\right)^2 + \left(b+\frac{1}{b}\right)^2 \ge \frac{25}{2}.$$

2016 Swedish Mathematical Competition, 2
Tags: algebra , inequalities
Determine whether the inequality $$ \left|\sqrt{x^2+2x+5}-\sqrt{x^2-4x+8}\right|<3$$ is valid for all real numbers $x$.

2007 Grigore Moisil Intercounty, 4
Tags: inequalities , real analysis , convergence , sequence
Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of positive real numbers, verifying the inequality $ x_n\le \frac{x_{n-1}+x_{n-2}}{2} , $ for any natural number $ n\ge 3. $ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent.