This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 6530

2021 China Second Round Olympiad, Problem 15
Tags: inequalities
Positive real numbers $x, y, z$ satisfy $\sqrt x + \sqrt y + \sqrt z = 1$. Prove that $$\frac{x^4+y^2z^2}{x^{\frac 52}(y+z)} + \frac{y^4+z^2x^2}{y^{\frac 52}(z+x)} + \frac{z^4+y^2x^2}{z^{\frac 52}(y+x)} \geq 1.$$ [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 15)[/i]

2007 China Team Selection Test, 1
Tags: quadratic , function , inequalities , trigonometry
$ u,v,w > 0$,such that $ u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4$ prove that $ \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w$

2010 International Zhautykov Olympiad, 3
Tags: euler , trigonometry , inequalities , geometry , incenter
Let $ABC$ arbitrary triangle ($AB \neq BC \neq AC \neq AB$) And O,I,H it's circum-center, incenter and ortocenter (point of intersection altitudes). Prove, that 1) $\angle OIH > 90^0$(2 points) 2)$\angle OIH >135^0$(7 points) balls for 1) and 2) not additive.

2015 IMC, 6
Tags: inequalities , series
Prove that $$\sum\limits_{n = 1}^{\infty}\frac{1}{\sqrt{n}\left(n+1\right)} < 2.$$ Proposed by Ivan Krijan, University of Zagreb

2011 Morocco National Olympiad, 1
Tags: inequalities , trigonometry
Let $x$, $y$, and $z$ be three real positive numbers such that $x^{2}+y^{2}+z^{2}+2xyz=1$. Prove that $2(x+y+z)\leq 3$.

1952 Moscow Mathematical Olympiad, 219
Tags: inequalities , algebra
Prove that $(1 - x)^n + (1 + x)^n < 2^n$ for an integer $n \ge 2$ and $|x| < 1$.

2017-IMOC, A6
Tags: inequalities , algebra
Show that for all positive reals $a,b,c$ with $a+b+c=3$, $$\sum_{\text{cyc}}\sqrt{a+3b+\frac2c}\ge3\sqrt6.$$

1998 Romania National Olympiad, 2
Tags: cyclic quadrilateral , inequalities , geometry
Let $ABCD$ be a cyclic quadrilateral. Show that $\vert \overline{AC} - \overline{BD} \vert \le \vert \overline{AB}-\overline{CD} \vert$ and determine when does equality hold.

2009 Jozsef Wildt International Math Competition, W. 17
Tags: function , inequalities
If $a$, $b$, $c>0$ and $abc=1$, $\alpha = max\{a,b,c\}$; $f,g : (0, +\infty )\to \mathbb{R}$, where $f(x)=\frac{2(x+1)^2}{x}$ and $g(x)= (x+1)\left (\frac{1}{\sqrt{x}}+1\right )^2$, then $$(a+1)(b+1)(c+1)\geq min\{ \{f(x),g(x) \}\ |\ x\in\{a,b,c\} \backslash \{ \alpha \}\} $$

1991 Swedish Mathematical Competition, 2
Tags: inequalities , algebra
$x, y$ are positive reals such that $x - \sqrt{x} \le y - 1/4 \le x + \sqrt{x}$. Show that $y - \sqrt{y} \le x - 1/4 \le y + \sqrt{y}$.

2015 Junior Balkan Team Selection Tests - Romania, 3
Tags: inequalities
Prove that if $a,b,c>0$ and $a+b+c=1,$ then $$\frac{bc+a+1}{a^2+1}+\frac{ca+b+1}{b^2+1}+\frac{ab+c+1}{c^2+1}\leq \frac{39}{10}$$

2015 Saudi Arabia IMO TST, 3
Tags: min , max , inequalities , algebra
Let $a, b,c$ be positive real numbers satisfying the condition $$(x + y + z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)= 10$$ Find the greatest value and the least value of $$T = (x^2 + y^2 + z^2) \left(\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}\right)$$ Trần Nam Dũng

1978 IMO Longlists, 45
Tags: inequalities , induction
If $r > s >0$ and $a > b > c$, prove that \[a^rb^s + b^rc^s + c^ra^s \ge a^sb^r + b^sc^r + c^sa^r.\]

2008 USA Team Selection Test, 5
Tags: pythagorean theorem , geometry , inequalities , circumcircle , analytic geometry , algebra
Two sequences of integers, $ a_1, a_2, a_3, \ldots$ and $ b_1, b_2, b_3, \ldots$, satisfy the equation \[ (a_n \minus{} a_{n \minus{} 1})(a_n \minus{} a_{n \minus{} 2}) \plus{} (b_n \minus{} b_{n \minus{} 1})(b_n \minus{} b_{n \minus{} 2}) \equal{} 0 \] for each integer $ n$ greater than $ 2$. Prove that there is a positive integer $ k$ such that $ a_k \equal{} a_{k \plus{} 2008}$.

2005 MOP Homework, 5
Tags: geometry , search , inequalities , cyclic quadrilateral
Let $ABCD$ be a cyclic quadrilateral such that $AB \cdot BC=2 \cdot AD \cdot DC$. Prove that its diagonals $AC$ and $BD$ satisfy the inequality $8BD^2 \le 9AC^2$. [color=#FF0000]Moderator says: Use the search before posting contest problems [url]http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=530783[/url][/color]

1999 Vietnam National Olympiad, 1
Tags: inequalities , algebra , trigonometry
Given are three positive real numbers $ a,b,c$ satisfying $ abc \plus{} a \plus{} c \equal{} b$. Find the max value of the expression: \[ P \equal{} \frac {2}{a^2 \plus{} 1} \minus{} \frac {2}{b^2 \plus{} 1} \plus{} \frac {3}{c^2 \plus{} 1}.\]

1998 Austrian-Polish Competition, 1
Tags: inequalities , algebra
Let $x_1, x_2,y _1,y_2$ be real numbers such that $x_1^2 + x_2^2 \le 1$. Prove the inequality $$(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)$$

2005 BAMO, 3
Tags: pigeonhole principle , inequalities , ceiling function , trigonometry
Let $ n\ge12$ be an integer, and let $ P_1,P_2,...P_n, Q$ be distinct points in a plane. Prove that for some $ i$, at least $ \frac{n}{6}\minus{}1$ of the distances $ P_1P_i,P_2P_i,...P_{i\minus{}1}P_i,P_{i\plus{}1}P_i,...P_nP_i$ are less than $ P_iQ$.

2018 Moldova Team Selection Test, 6
Tags: three variable inequality , inequalities , algebra
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Show that $$\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+a^2}\geq \frac{3}{2}.$$

2007 Moldova National Olympiad, 11.5
Tags: inequalities , function , induction
Real numbers $a_{1},a_{2},\dots,a_{n}$ satisfy $a_{i}\geq\frac{1}{i}$, for all $i=\overline{1,n}$. Prove the inequality: \[\left(a_{1}+1\right)\left(a_{2}+\frac{1}{2}\right)\cdot\dots\cdot\left(a_{n}+\frac{1}{n}\right)\geq\frac{2^{n}}{(n+1)!}(1+a_{1}+2a_{2}+\dots+na_{n}).\]

2005 All-Russian Olympiad Regional Round, 11.7
Tags: inequalities , number theory
11.7 Let $N$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 7, and $M$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 8. Compare $M$ and $N$. ([i]A. Golovanov[/i])

2010 Belarus Team Selection Test, 5.2
Tags: geometric inequality , inequalities , geometry , median
Numbers $a, b, c$ are the length of the medians of some triangle. If $ab + bc + ac = 1$ prove that a) $a^2b + b^2c + c^2a > \frac13$ b) $a^2b + b^2c + c^2a > \frac12$ (I. Bliznets)

2008 Spain Mathematical Olympiad, 2
Tags: trigonometry , inequalities
Let $a$ and $b$ be two real numbers, with $0<a,b<1$. Prove that \[\sqrt{ab^2+a^2b}+\sqrt{(1-a)(1-b)^2+(1-a)^2(1-b)}<\sqrt{2}\]

2013 Today's Calculation Of Integral, 884
Tags: integral inequality , calculus , trigonometry , integration , calculus computations , inequalities
Prove that : \[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]

1997 Singapore MO Open, 4
Tags: inequalities , algebra
Let $n \ge 2$ be a positive integer. Suppose that $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$ are 2n numbers such that $\sum_{i=1}^n a_i =\sum_{i=1}^n n_i= 1$ and $a_i\ge 0, 0 \le b_i\le \frac{n-1}{n}, i = 1, 2,..., n$. Show that $$b_1a_2a_3...a_n+a_1b_2a_3...a_n+...+a_1a_2...a_{k-1}b_ka_{k+1}...a_n+ ...+ a_1a_2...a_{n-1}b_n \le \frac{1}{n(n-1)^{n-2}}$$