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

2003 Brazil National Olympiad, 1
Tags: trigonometry , inequalities , triangle inequality , geometry
Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.

2004 Polish MO Finals, 3
Tags: inequalities , combinatorics
On a tournament with $ n \ge 3$ participants, every two participants played exactly one match and there were no draws. A three-element set of participants is called a [i]draw-triple[/i] if they can be enumerated so that the first defeated the second, the second defeated the third, and the third defeated the first. Determine the largest possible number of draw-triples on such a tournament.

2006 District Olympiad, 1
Tags: inequalities , calculus , integration , function , real analysis
Let $f_1,f_2,\ldots,f_n : [0,1]\to (0,\infty)$ be $n$ continuous functions, $n\geq 1$, and let $\sigma$ be a permutation of the set $\{1,2,\ldots, n\}$. Prove that \[ \prod^n_{i=1} \int^1_0 \frac{ f_i^2(x) }{ f_{\sigma(i)}(x) } dx \geq \prod^n_{i=1} \int^1_0 f_i(x) dx. \]

PEN P Problems, 42
Tags: pigeonhole principle , inequalities , additive number theory
Prove that for each positive integer $K$ there exist infinitely many even positive integers which can be written in more than $K$ ways as the sum of two odd primes.

2017 Morocco TST-, 1
Tags: inequalities
Let $a,b,c$ be non-negative real numbers such that $a^2+b^2+c^2 \le 3$ then prove that; $$(a+b+c)(a+b+c-abc)\ge2(a^2b+b^2c+c^2a)$$

2001 India IMO Training Camp, 2
Tags: inequalities , combinatorics
Two symbols $A$ and $B$ obey the rule $ABBB = B$. Given a word $x_1x_2\ldots x_{3n+1}$ consisting of $n$ letters $A$ and $2n+1$ letters $B$, show that there is a unique cyclic permutation of this word which reduces to $B$.

1987 AIME Problems, 12
Tags: inequalities , algebra , binomial theorem
Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$.

1981 All Soviet Union Mathematical Olympiad, 318
Tags: geometry , perimeter , inequalities , geometric inequality , ratio
The points $C_1, A_1, B_1$ belong to $[AB], [BC], [CA]$ sides, respectively, of the triangle $ABC$ . $$\frac{|AC_1|}{|C_1B| }=\frac{ |BA_1|}{|A_1C| }= \frac{|CB_1|}{|B_1A| }= \frac{1}{3}$$ Prove that the perimeter $P$ of the triangle $ABC$ and the perimeter $p$ of the triangle $A_1B_1C_1$ , satisfy inequality $$\frac{P}{2} < p < \frac{3P}{4}$$

2012 IMO, 2
Tags: n-variable inequality , ineqstd , inequalities
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that \[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2017 Polish MO Finals, 6
Tags: inequalities
Three sequences $(a_0, a_1, \ldots, a_n)$, $(b_0, b_1, \ldots, b_{n})$, $(c_0, c_1, \ldots, c_{2n})$ of non-negative real numbers are given such that for all $0\leq i,j\leq n$ we have $a_ib_j\leq (c_{i+j})^2$. Prove that $$\sum_{i=0}^n a_i\cdot\sum_{j=0}^n b_j\leq \left( \sum_{k=0}^{2n} c_k\right)^2.$$

2009 Brazil Team Selection Test, 2
Tags: algebra , inequalities
Be $x_1, x_2, x_3, x_4, x_5$ be positive reais with $x_1x_2x_3x_4x_5=1$. Prove that $$\frac{x_1+x_1x_2x_3}{1+x_1x_2+x_1x_2x_3x_4}+\frac{x_2+x_2x_3x_4}{1+x_2x_3+x_2x_3x_4x_5}+\frac{x_3+x_3x_4x_5}{1+x_3x_4+x_3x_4x_5x_1}+\frac{x_4+x_4x_5x_1}{1+x_4x_5+x_4x_5x_1x_2}+\frac{x_5+x_5x_1x_2}{1+x_5x_1+x_5x_1x_2x_3} \ge \frac{10}{3}$$

2009 Korea - Final Round, 1
Tags: inequalities
$a,b,c$ are the length of three sides of a triangle. Let $A= \frac{a^2 +bc}{b+c}+\frac{b^2 +ca}{c+a}+\frac{c^2 +ab}{a+b}$, $B=\frac{1}{\sqrt{(a+b-c)(b+c-a)}}+\frac{1}{\sqrt{(b+c-a)(c+a-b)}}$$+\frac{1}{\sqrt{(c+a-b)(a+b-c)}}$. Prove that $AB \ge 9$.

2020 Turkey Team Selection Test, 4
Tags: function , inequality , inequalities
Let $Z^+$ be positive integers set. $f:\mathbb{Z^+}\to\mathbb{Z^+}$ is a function and we show $ f \circ f \circ ...\circ f $ with $f_l$ for all $l\in \mathbb{Z^+}$ where $f$ is repeated $l$ times. Find all $f:\mathbb{Z^+}\to\mathbb{Z^+}$ functions such that $$ (n-1)^{2020}< \prod _{l=1}^{2020} {f_l}(n)< n^{2020}+n^{2019} $$ for all $n\in \mathbb{Z^+}$

2014 Contests, 3
Tags: number theory , combinatorics , algebra , inequalities
$N$ in natural. There are natural numbers from $N^3$ to $N^3+N$ on the board. $a$ numbers was colored in red, $b$ numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that $b|a$

2015 China Western Mathematical Olympiad, 3
Tags: inequalities
Let the integer $n \ge 2$ , and $x_1,x_2,\cdots,x_n $ be positive real numbers such that $\sum_{i=1}^nx_i=1$ .Prove that$$\left(\sum_{i=1}^n\frac{1}{1-x_i}\right)\left(\sum_{1\le i<j\le n} x_ix_j\right)\le \frac{n}{2}.$$

1995 IMO, 5
Tags: geometry , inequalities , hexagon , geometric inequality
Let $ ABCDEF$ be a convex hexagon with $ AB \equal{} BC \equal{} CD$ and $ DE \equal{} EF \equal{} FA$, such that $ \angle BCD \equal{} \angle EFA \equal{} \frac {\pi}{3}$. Suppose $ G$ and $ H$ are points in the interior of the hexagon such that $ \angle AGB \equal{} \angle DHE \equal{} \frac {2\pi}{3}$. Prove that $ AG \plus{} GB \plus{} GH \plus{} DH \plus{} HE \geq CF$.

2018 China Team Selection Test, 6
Tags: combinatorics , inequalities
Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ subsets of a set of size $n$. Prove that $$ \sum_{i=1}^{m} \sum_{j=1}^{m}|A_i|\cdot |A_i \cap A_j|\geq \frac{1}{mn}\left(\sum_{i=1}^{m}|A_i|\right)^3.$$

1962 Vietnam National Olympiad, 1
Tags: inequalities
Prove that for positive real numbers $ a$, $ b$, $ c$, $ d$, we have \[ \frac{1}{\frac{1}{a}\plus{}\frac{1}{b}}\plus{}\frac{1}{\frac{1}{c}\plus{}\frac{1}{d}}\le\frac{1}{\frac{1}{a\plus{}c}\plus{}\frac{1}{b\plus{}d}}\]

2015 IFYM, Sozopol, 4
Tags: algebra , inequality , inequalities
For all real numbers $a,b,c>0$ such that $abc=1$, prove that $\frac{a}{1+b^3}+\frac{b}{1+c^3}+\frac{c}{1+a^3}\geq \frac{3}{2}$.

2008 India Regional Mathematical Olympiad, 3
Tags: inequalities , vieta , calculus , derivative , trigonometry , algebra
Suppose $ a$ and $ b$ are real numbers such that the roots of the cubic equation $ ax^3\minus{}x^2\plus{}bx\minus{}1$ are positive real numbers. Prove that: \[ (i)\ 0<3ab\le 1\text{ and }(i)\ b\ge \sqrt{3} \] [19 points out of 100 for the 6 problems]

2007 China Girls Math Olympiad, 3
Tags: induction , inequalities
Let $ n$ be an integer greater than $ 3$, and let $ a_1, a_2, \cdots, a_n$ be non-negative real numbers with $ a_1 \plus{} a_2 \plus{} \cdots \plus{} a_n \equal{} 2$. Determine the minimum value of \[ \frac{a_1}{a_2^2 \plus{} 1}\plus{} \frac{a_2}{a^2_3 \plus{} 1}\plus{} \cdots \plus{} \frac{a_n}{a^2_1 \plus{} 1}.\]

2011 Today's Calculation Of Integral, 746
Tags: calculus , integration , floor function , inequalities , calculus computations
Prove the following inequality. \[n^ne^{-n+1}\leq n!\leq \frac 14(n+1)^{n+1}e^{-n+1}.\]

2011 239 Open Mathematical Olympiad, 7
Tags: inequalities
Prove for positive reals $a,b,c$ that $(ab+bc+ca+1)(a+b)(b+c)(c+a) \ge 2abc(a+b+c+1)^2$

2015 German National Olympiad, 6
Tags: mean , harmonic mean , inequalities
Prove that for all $x,y,z>0$, the inequality \[\frac{x+y+z}{3}+\frac{3}{\frac{1}{x}+\frac{1}{y}+\frac{1}{z}} \ge 5 \sqrt[3]{\frac{xyz}{16}}\] holds. Determine if equality can hold and if so, in which cases it occurs.

2004 Thailand Mathematical Olympiad, 6
Tags: algebra , inequalities
Let $a, b, c > 0$ satisfy $a + b + c \ge \frac{1}{a} +\frac{1}{b} +\frac{1}{c}$. Prove that $a^3 + b^3 + c^3 \ge a + b + c$