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 India Regional MathematicaI Olympiad, 8
Tags: inequalities , number theory , geometry
The length of each side of a convex quadrilateral $ABCD$ is a positive integer. If the sum of the lengths of any three sides is divisible by the length of the remaining side then prove that some two sides of the quadrilateral have the same length.

2008 Danube Mathematical Competition, 1
Tags: inequalities
$x,y,z,t \in \mathbb R_+^*$: \[ (xy)^{1/2}+(yz)^{1/2}+(zt)^{1/2}+(tx)^{1/2}+(xz)^{1/2}+(yt)^{1/2} \ge 3(xyz+xyt+xzt+yzt)^{\frac{1}{3}} \]

2011 Korea Junior Math Olympiad, 7
Tags: inequalities , n-variable inequality , convex function
For those real numbers $x_1 , x_2 , \ldots , x_{2011}$ where each of which satisfies $0 \le x_1 \le 1$ ($i = 1 , 2 , \ldots , 2011$), find the maximum of \[ x_1^3+x_2^3+ \cdots + x_{2011}^3 - \left( x_1x_2x_3 + x_2x_3x_4 + \cdots + x_{2011}x_1x_2 \right) \]

1997 APMO, 3
Tags: trigonometry , geometry , circumcircle , inequalities , ratio , angle bisector , geometric inequality
Let $ABC$ be a triangle inscribed in a circle and let \[ l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ , \] where $m_a$,$m_b$, $m_c$ are the lengths of the angle bisectors (internal to the triangle) and $M_a$, $M_b$, $M_c$ are the lengths of the angle bisectors extended until they meet the circle. Prove that \[ \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3 \] and that equality holds iff $ABC$ is an equilateral triangle.

2017 Estonia Team Selection Test, 8
Tags: inequalities , three variable inequality , ineqstd
Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$ [i]Proposed by Tigran Margaryan, Armenia[/i]

2014 Peru IMO TST, 15
Tags: algebra , inequalities , sequence
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]

2019 Jozsef Wildt International Math Competition, W. 65
Tags: inequalities
If $a$, $b$, $c \geq 1$; $y \geq x \geq 1$; $p$, $q$, $r > 0$ then$$\left(\frac{1+y\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}{1+x\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}\right)^{\frac{p+q+r}{\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}}\left(\frac{1+xa}{1+ya}\right)^{\frac{p}{a}}\left(\frac{1+xb}{1+yb}\right)^{\frac{q}{b}}\left(\frac{1+xc}{1+yc}\right)^{\frac{r}{c}}$$ $$\geq \prod \limits_{cyc}\left(\frac{1+y\left(a^pb^q\right)^{\frac{1}{p+q}}}{1+x\left(a^pb^q\right)^{\frac{1}{p+q}}}\right)^{\frac{p+q}{\left(a^pb^q\right)^{\frac{1}{p+q}}}}$$

1982 Poland - Second Round, 3
Tags: logarithm , algebra , inequalities
Prove that for every natural number $ n \geq 2 $ the inequality holds $$ \log_n 2 \cdot \log_n 4 \cdot \log_n 6 \ldots \log_n (2n - 2) \leq 1.$$

2012 ELMO Shortlist, 2
Tags: inequalities
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]

III Soros Olympiad 1996 - 97 (Russia), 11.4
Tags: inequalities , algebra , trigonometry
Find the smallest value of a function $$y = \cos 8x + 3\cos 4x +3\cos2x + 2\cos x.$$

2005 All-Russian Olympiad, 1
Tags: linear algebra , matrix , inequalities , combinatorics
We select $16$ cells on an $8\times 8$ chessboard. What is the minimal number of pairs of selected cells in the same row or column?

2017 Saudi Arabia IMO TST, 3
Tags: sequence , inequalities
Find the greatest positive real number $M$ such that for all positive real sequence $(a_n)$ and for all real number $m < M$, it is possible to find some index $n \ge 1$ that satisfies the inequality $a_1 + a_2 + a_3 + ...+ a_n +a_{n+1} > m a_n$.

1988 Tournament Of Towns, (199) 2
Tags: geometric inequality , inequalities
Prove that $a^2pq + b^2qr + c^2rp \le 0$, whenever $a, b$ and $c$ are the lengths of the sides of a triangle and $p + q + r = 0$ . ( J. Mustafaev , year 12 student, Baku)

1986 Tournament Of Towns, (114) 1
Tags: maximum , algebra , inequalities
For which natural number $k$ does $\frac{k^2}{1.001^k}$ attain its maximum value?

2012 Junior Balkan Team Selection Tests - Romania, 1
Tags: inequalities , sequence , algebra
Let $a_1, a_2, ..., a_n$ be real numbers such that $a_1 = a_n = a$ and $a_{k+1} \le \frac{a_k + a_{k+2}}{2} $, for all $k = 1, 2, ..., n - 2$. Prove that $a_k \le a,$ for all $k = 1, 2, ..., n.$

1987 Bundeswettbewerb Mathematik, 4
Tags: inequality , equality , algebra , inequalities
Let $1<k\leq n$ be positive integers and $x_1 , x_2 , \ldots , x_k$ be positive real numbers such that $x_1 \cdot x_2 \cdot \ldots \cdot x_k = x_1 + x_2 + \ldots +x_k.$ a) Show that $x_{1}^{n-1} +x_{2}^{n-1} + \ldots +x_{k}^{n-1} \geq kn.$ b) Find all numbers $k,n$ and $x_1, x_2 ,\ldots , x_k$ for which equality holds.

2008 Ukraine Team Selection Test, 2
Tags: inequalities , combinatorics , algebra
There is a row that consists of digits from $ 0$ to $ 9$ and Ukrainian letters (there are $ 33$ of them) with following properties: there aren’t two distinct digits or letters $ a_i$, $ a_j$ such that $ a_i > a_j$ and $ i < j$ (if $ a_i$, $ a_j$ are letters $ a_i > a_j$ means that $ a_i$ has greater then $ a_j$ position in alphabet) and there aren’t two equal consecutive symbols or two equal symbols having exactly one symbol between them. Find the greatest possible number of symbols in such row.

2009 Singapore Team Selection Test, 2
Tags: inequalities
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

2016 India IMO Training Camp, 3
Tags: inequalities
Let a,b,c,d be real numbers satisfying $|a|,|b|,|c|,|d|>1$ and $abc+abd+acd+bcd+a+b+c+d=0$. Prove that $\frac {1} {a-1}+\frac {1} {b-1}+ \frac {1} {c-1}+ \frac {1} {d-1} >0$

1969 Yugoslav Team Selection Test, Problem 1
Tags: inequalities
Given real numbers $a_i,b_i~(i=1,2,\ldots,n)$ such that \begin{align*} &a_1\ge a_2\ge\ldots\ge a_n>0,\\ &b_1\ge a_1,\\ &b_1b_2\ge a_1a_2,\\ &\vdots\\ &b_1b_2\cdots b_n\ge a_1a_2\cdots a_n, \end{align*}prove that $b_1+b_2+\ldots+b_n\ge a_1+a_2+\ldots+a_n$.

1996 India National Olympiad, 2
Tags: inequalities , analytic geometry , vector , circumcircle , function , trigonometry , geometry
Let $C_1$ and $C_2$ be two concentric circles in the plane with radii $R$ and $3R$ respectively. Show that the orthocenter of any triangle inscribed in circle $C_1$ lies in the interior of circle $C_2$. Conversely, show that every point in the interior of $C_2$ is the orthocenter of some triangle inscribed in $C_1$.

1998 Estonia National Olympiad, 1
Tags: inequalities , algebra
Prove that for any reals $a> b> c$, the inequality $a^2(b - c) + b^2(c - a) + c^2(a - b)> 0$.

2022 Kosovo National Mathematical Olympiad, 2
Tags: inequalities
Show that for any positive real numbers $a$ and $b$ the following inequality hold, $$\frac{a(a+1)}{b+1}+\frac{b(b+1)}{a+1}\geq a+b.$$

2010 Contests, 4
Tags: rearrangement inequality , inequalities
Prove that \[ a^2b^2(a^2+b^2-2) \geq (a+b)(ab-1) \] for all positive real numbers $a$ and $b.$

2008 Bosnia Herzegovina Team Selection Test, 1
Tags: inequalities , trigonometry , calculus , derivative , geometry
Prove that in an isosceles triangle $ \triangle ABC$ with $ AC\equal{}BC\equal{}b$ following inequality holds $ b> \pi r$, where $ r$ is inradius.