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: 787

2018 IFYM, Sozopol, 3
Tags: algebra , inequalities proposed
The number 1 is a solution of the equation $(x + a)(x + b)(x + c)(x + d) = 16$, where $a, b, c, d$ are positive real numbers. Find the largest value of $abcd$.

2009 Indonesia TST, 3
Tags: calculus , derivative , geometry , 3D geometry , function , inequalities proposed , inequalities
Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.

1997 Turkey MO (2nd round), 1
Tags: inequalities , inequalities proposed
Let $e > 0$ be a given real number. Find the least value of $f(e)$ (in terms of $e$ only) such that the inequality $a^{3}+ b^{3}+ c^{3}+ d^{3} \leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})$ holds for all real numbers $a, b, c, d$.

2001 India IMO Training Camp, 1
Tags: inequalities , inequalities proposed
Let $x$ , $y$ , $z>0$. Prove that if $xyz\geq xy+yz+zx$, then $xyz \geq 3(x+ y+z)$.

2013 China Western Mathematical Olympiad, 2
Tags: induction , inequalities proposed , inequalities
Let the integer $n \ge 2$, and the real numbers $x_1,x_2,\cdots,x_n\in \left[0,1\right] $.Prove that\[\sum_{1\le k<j\le n} kx_kx_j\le \frac{n-1}{3}\sum_{k=1}^n kx_k.\]

2011 Morocco National Olympiad, 2
Tags: inequalities , geometry , perimeter , trigonometry , inequalities proposed
Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle. $(a)$ Prove that \[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\] $(b)$ When do we have equality?

2010 Baltic Way, 3
Tags: inequalities , induction , inequalities proposed
Let $x_1, x_2, \ldots ,x_n(n\ge 2)$ be real numbers greater than $1$. Suppose that $|x_i-x_{i+1}|<1$ for $i=1, 2,\ldots ,n-1$. Prove that \[\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots +\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}<2n-1\]

2012 Czech-Polish-Slovak Match, 3
Tags: inequalities , inequalities proposed
Let $a,b,c,d$ be positive real numbers such that $abcd=4$ and \[a^2+b^2+c^2+d^2=10.\] Find the maximum possible value of $ab+bc+cd+da$.

1989 Federal Competition For Advanced Students, 2
Tags: inequalities , inequalities proposed
If $ a$ and $ b$ are nonnegative real numbers with $ a^2\plus{}b^2\equal{}4$, show that: $ \frac{ab}{a\plus{}b\plus{}2} \le \sqrt{2}\minus{}1$ and determine when equality occurs.

2005 Nordic, 2
Tags: inequalities , calculus , inequalities proposed , algebra , High school olympiad
Let $a,b,c$ be positive real numbers. Prove that \[\frac{2a^2}{b+c} + \frac{2b^2}{c+a} + \frac{2c^2}{a+b} \geq a+b+c\](this is, of course, a joke!) [b]EDITED with exponent 2 over c[/b]

2012 ELMO Shortlist, 2
Tags: inequalities , inequalities proposed
Let $a,b,c$ be three positive real numbers such that $ a \le b \le c$ and $a+b+c=1$. Prove that \[\frac{a+c}{\sqrt{a^2+c^2}}+\frac{b+c}{\sqrt{b^2+c^2}}+\frac{a+b}{\sqrt{a^2+b^2}} \le \frac{3\sqrt{6}(b+c)^2}{\sqrt{(a^2+b^2)(b^2+c^2)(c^2+a^2)}}.\] [i]Owen Goff.[/i]

2007 China Northern MO, 1
Tags: inequalities , trigonometry , inequalities proposed
Let $ \alpha$, $ \beta$ be acute angles. Find the maximum value of \[ \frac{\left(1-\sqrt{\tan\alpha\tan\beta}\right)^{2}}{\cot\alpha+\cot\beta}\]

2012 JBMO TST - Turkey, 4
Tags: inequalities , function , inequalities proposed , 3-variable inequality
Find the greatest real number $M$ for which \[ a^2+b^2+c^2+3abc \geq M(ab+bc+ca) \] for all non-negative real numbers $a,b,c$ satisfying $a+b+c=4.$

2013 China Girls Math Olympiad, 5
Tags: inequalities , inequalities proposed
For any given positive numbers $a_1,a_2,\ldots,a_n$, prove that there exist positive numbers $x_1,x_2,\ldots,x_n$ satisfying $\sum_{i=1}^n x_i=1$, such that for any positive numbers $y_1,y_2,\ldots,y_n$ with $\sum_{i=1}^n y_i=1$, the inequality $\sum_{i=1}^n \frac{a_ix_i}{x_i+y_i}\ge \frac{1}{2}\sum_{i=1}^n a_i$ holds.

1974 IMO Longlists, 32
Tags: inequalities , inequalities proposed
Let $a_1,a_2,\ldots ,a_n$ be $n$ real numbers such that $0<a\le a_k\le b$ for $k=1,2,\ldots ,n$. If $m_1=\frac{1}{n}(a_1+a_2+\cdots+a_n)$ and $m_2=\frac{1}{n}(a_1^2+a_2^2+\cdots + a_n^2)$, prove that $m_2\le\frac{(a+b)^2}{4ab}m_1^2$ and find a necessary and sufficient condition for equality.

1998 Irish Math Olympiad, 2
Tags: inequalities , inequalities proposed
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}.$

2013 Albania Team Selection Test, 2
Tags: inequalities , rearrangement inequality , inequalities proposed
Let $a,b,c,d$ be positive real numbers such that $abcd=1$.Find with proof that $x=3 $ is the minimal value for which the following inequality holds: \[a^x+b^x+c^x+d^x\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}\]

1974 IMO Longlists, 33
Tags: inequalities proposed , inequalities
Let a be a real number such that $0 < a < 1$, and let $n$ be a positive integer. Define the sequence $a_0, a_1, a_2, \ldots, a_n$ an recursively by \[a_0 = a, \quad a_{k+1} = a_k +\frac 1n a_k^2 \quad \text{ for } k = 0, 1, \ldots, n - 1.\] Prove that there exists a real number $A$, depending on $a$ but independent of $n$, such that \[0 < n(A - a_n) < A^3.\]

2004 Regional Olympiad - Republic of Srpska, 2
Tags: inequalities proposed , inequalities
The positive real numbers $x,y,z$ satisfy $x+y+z=1$. Show that \[\sqrt{3xyz}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{1-x}+\frac{1}{1-y}+\frac{1}{1-z}\right)\geq4+ \frac{4xyz}{(1-x)(1-y)(1-z)}.\]

2010 Contests, 2
Tags: inequalities , trigonometry , inequalities proposed
Let $x$ be a real number such that $0<x<\frac{\pi}{2}$. Prove that \[\cos^2(x)\cot (x)+\sin^2(x)\tan (x)\ge 1\]

2008 Balkan MO, 2
Tags: inequalities , induction , function , Cauchy Inequality , inequalities proposed
Is there a sequence $ a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $ n$: a) $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le n^2$ b) $ \frac1{a_1}\plus{}\frac1{a_2}\plus{}\ldots\plus{}\frac1{a_n}\le2008$?

2007 ISI B.Stat Entrance Exam, 5
Tags: trigonometry , inequalities proposed , inequalities
Show that \[-2 \le \cos \theta\left(\sin \theta + \sqrt{\sin ^2 \theta +3}\right) \le 2\] for all values of $\theta$.

2006 Moldova Team Selection Test, 3
Tags: inequalities , algebra , polynomial , inequalities proposed
Let $a,b,c$ be sides of the triangle. Prove that \[ a^2\left(\frac{b}{c}-1\right)+b^2\left(\frac{c}{a}-1\right)+c^2\left(\frac{a}{b}-1\right)\geq 0 . \]

Russian TST 2014, P3
Tags: inequalities , inequalities proposed
Find the maximum value of real number $k$ such that \[\frac{a}{1+9bc+k(b-c)^2}+\frac{b}{1+9ca+k(c-a)^2}+\frac{c}{1+9ab+k(a-b)^2}\geq \frac{1}{2}\] holds for all non-negative real numbers $a,\ b,\ c$ satisfying $a+b+c=1$.

2010 Romania National Olympiad, 1
Tags: inequalities proposed , inequalities
Let $a,b,c$ be integers larger than $1$. Prove that \[a(a-1)+b(b-1)+c(c-1)\le (a+b+c-4)(a+b+c-5)+4.\]