Found problems: 6530
2005 Turkey MO (2nd round), 1
For all positive real numbers $a,b,c,d$ prove the inequality
\[\sqrt{a^4+c^4}+\sqrt{a^4+d^4}+\sqrt{b^4+c^4}+\sqrt{b^4+d^4} \ge 2\sqrt{2}(ad+bc)\]
2009 Serbia Team Selection Test, 2
Let $ x,y,z$ be positive real numbers such that $ xy \plus{} yz \plus{} zx \equal{} x \plus{} y \plus{} z$. Prove the inequality
$ \frac1{x^2 \plus{} y \plus{} 1} \plus{} \frac1{y^2 \plus{} z \plus{} 1} \plus{} \frac1{z^2 \plus{} x \plus{} 1}\le1$
When does the equality hold?
2008 Abels Math Contest (Norwegian MO) Final, 3
a) Let $x$ and $y$ be positive numbers such that $x + y = 2$.
Show that $\frac{1}{x}+\frac{1}{y} \le \frac{1}{x^2}+\frac{1}{y^2}$
b) Let $x,y$ and $z$ be positive numbers such that $x + y +z= 2$.
Show that $\frac{1}{x}+\frac{1}{y} +\frac{1}{z} +\frac{9}{4} \le \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$
.
1990 ITAMO, 4
Let $a,b,c$ be side lengths of a triangle with $a+b+c = 1$. Prove that $a^2 +b^2 +c^2 +4abc \le \frac12$ .
2006 Switzerland Team Selection Test, 1
Let $a,b,c \in \mathbb{R^+}$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show $\sqrt{ab+c} + \sqrt{bc+a} + \sqrt{ca+b} \ge \sqrt{abc} + \sqrt{a} + \sqrt{b} + \sqrt{c}$. :D
2010 Romanian Master of Mathematics, 2
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]
[i]Marko Radovanović, Serbia[/i]
2004 All-Russian Olympiad, 4
Let $n > 3$ be a natural number, and let $x_1$, $x_2$, ..., $x_n$ be $n$ positive real numbers whose product is $1$. Prove the inequality \[ \frac {1}{1 + x_1 + x_1\cdot x_2} + \frac {1}{1 + x_2 + x_2\cdot x_3} + ... + \frac {1}{1 + x_n + x_n\cdot x_1} > 1. \]
2004 Baltic Way, 4
Let $x_1$, $x_2$, ..., $x_n$ be real numbers with arithmetic mean $X$. Prove that there is a positive integer $K$ such that for any integer $i$ satisfying $0\leq i<K$, we have $\frac{1}{K-i}\sum_{j=i+1}^{K} x_j \leq X$. (In other words, prove that there is a positive integer $K$ such that the arithmetic mean of each of the lists $\left\{x_1,x_2,...,x_K\right\}$, $\left\{x_2,x_3,...,x_K\right\}$, $\left\{x_3,...,x_K\right\}$, ..., $\left\{x_{K-1},x_K\right\}$, $\left\{x_K\right\}$ is not greater than $X$.)
2013 Indonesia MO, 3
Determine all positive real $M$ such that for any positive reals $a,b,c$, at least one of $a + \dfrac{M}{ab}, b + \dfrac{M}{bc}, c + \dfrac{M}{ca}$ is greater than or equal to $1+M$.
2015 Czech-Polish-Slovak Junior Match, 3
Real numbers $x, y$ satisfy the inequality $x^2 + y^2 \le 2$. Orove that $xy + 3 \ge 2x + 2y$
2013 ELMO Shortlist, 6
Let $a, b, c$ be positive reals such that $a+b+c=3$. Prove that \[18\sum_{\text{cyc}}\frac{1}{(3-c)(4-c)}+2(ab+bc+ca)\ge 15. \][i]Proposed by David Stoner[/i]
2002 VJIMC, Problem 3
Let $E$ be the set of all continuous functions $u:[0,1]\to\mathbb R$ satisfying
$$u^2(t)\le1+4\int^t_0s|u(s)|\text ds,\qquad\forall t\in[0,1].$$Let $\varphi:E\to\mathbb R$ be defined by
$$\varphi(u)=\int^1_0\left(u^2(x)-u(x)\right)\text dx.$$Prove that $\varphi$ has a maximum value and find it.
2002 Croatia National Olympiad, Problem 2
Prove that for any positive number $a,b,c$ and any nonnegative integer $p$
$$a^{p+2}+b^{p+2}+c^{p+2}\ge a^pbc+b^pca+c^pab.$$
Russian TST 2016, P1
The positive numbers $a, b, c$ are such that $a^2<16bc, b^2<16ca$ and $c^2<16ab$. Prove that \[a^2+b^2+c^2<2(ab+bc+ca).\]
2023 Irish Math Olympiad, P8
Suppose that $a, b, c$ are positive real numbers and $a + b + c = 3$. Prove that
$$\frac{a+b}{c+2} + \frac{b+c}{a+2} + \frac{c+a}{b+2} \geq 2$$
and determine when equality holds.
2011 Junior Balkan Team Selection Tests - Romania, 3
a) Find the largest possible value of the number $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n$, if $x_1, x_2, ... , x_n$ ($n \ge 2$) are non-negative integers and their sum is $2011$.
b) Find the numbers $x_1, x_2, ... , x_n$ for which the maximum value determined at a) is obtained
2007 Irish Math Olympiad, 3
Let $ ABC$ be a triangle the lengths of whose sides $ BC,CA,AB,$ respectively, are denoted by $ a,b,$ and $ c$. Let the internal bisectors of the angles $ \angle BAC, \angle ABC, \angle BCA,$ respectively, meet the sides $ BC,CA,$ and $ AB$ at $ D,E,$ and $ F$. Denote the lengths of the line segments $ AD,BE,CF$ by $ d,e,$ and $ f$, respectively. Prove that:
$ def\equal{}\frac{4abc(a\plus{}b\plus{}c) \Delta}{(a\plus{}b)(b\plus{}c)(c\plus{}a)}$ where $ \Delta$ stands for the area of the triangle $ ABC$.
1975 USAMO, 2
Let $ A,B,C,$ and $ D$ denote four points in space and $ AB$ the distance between $ A$ and $ B$, and so on. Show that \[ AC^2\plus{}BD^2\plus{}AD^2\plus{}BC^2 \ge AB^2\plus{}CD^2.\]
2019 Polish MO Finals, 5
The sequence $a_1, a_2, \ldots, a_n$ of positive real numbers satisfies the following conditions:
\begin{align*}
\sum_{i=1}^n \frac{1}{a_i} \le 1 \ \ \ \ \hbox{and} \ \ \ \ a_i \le a_{i-1}+1
\end{align*}
for all $i\in \lbrace 1, 2, \ldots, n \rbrace$, where $a_0$ is an integer. Prove that
\begin{align*}
n \le 4a_0 \cdot \sum_{i=1}^n \frac{1}{a_i}
\end{align*}
1970 IMO Longlists, 5
Prove that $\sqrt[n]{\sum_{i=1}^{n}{\frac{i}{n+1}}}\ge 1$ for $2 \le n \in \mathbb{N}$.
1988 IMO Longlists, 70
$ABC$ is a triangle, with inradius $r$ and circumradius $R.$ Show that: \[ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{B}{2} \right) + \sin \left( \frac{B}{2} \right) \cdot \sin \left( \frac{C}{2} \right) + \sin \left( \frac{C}{2} \right) \cdot \sin \left( \frac{A}{2} \right) \leq \frac{5}{8} + \frac{r}{4 \cdot R}. \]
1977 Bulgaria National Olympiad, Problem 1
For natural number $n$ and real numbers $\alpha$ and $x$ satisfy the inequalities $\alpha^{n+1}\le x\le1$ and $0<\alpha<1$. Prove that
$$\prod_{k=1}^n\left|\frac{x-\alpha^k}{x+\alpha^k}\right|\le\prod_{k=1}^n\left|\frac{1-\alpha^k}{1+\alpha^k}\right|.$$
[i]Borislav Boyanov[/i]
2022 Junior Balkan Team Selection Tests - Romania, P1
Let $a\geq b\geq c\geq d$ be real numbers such that $(a-b)(b-c)(c-d)(d-a)=-3.$
[list=a]
[*]If $a+b+c+d=6,$ prove that $d<0,36.$
[*]If $a^2+b^2+c^2+d^2=14,$ prove that $(a+c)(b+d)\leq 8.$ When does equality hold?
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1962 AMC 12/AHSME, 33
The set of $ x$-values satisfying the inequality $ 2 \leq |x\minus{}1| \leq 5$ is:
$ \textbf{(A)}\ \minus{}4 \leq x \leq \minus{}1 \text{ or } 3 \leq x \leq 6 \qquad
\textbf{(B)}\ 3 \leq x \leq 6 \text{ or } \minus{}6 \leq x \leq \minus{}3 \qquad
\textbf{(C)}\ x \leq \minus{}1 \text{ or } x \geq 3 \qquad
\textbf{(D)}\ \minus{}1 \leq x \leq 3 \qquad
\textbf{(E)}\ \minus{}4 \leq x \leq 6$
2010 Korea Junior Math Olympiad, 5
If reals $x, y, z $ satises $tan x + tan y + tan z = 2$ and $0 < x, y,z < \frac{\pi}{2}.$ Prove that
$$sin^2 x + sin^2 y + sin^2 z < 1.$$