Found problems: 426
2003 Moldova Team Selection Test, 2
The positive reals $ x,y$ and $ z$ are satisfying the relation $ x \plus{} y \plus{} z\geq 1$. Prove that:
$ \frac {x\sqrt {x}}{y \plus{} z} \plus{} \frac {y\sqrt {y}}{z \plus{} x} \plus{} \frac {z\sqrt {z}}{x \plus{} y}\geq \frac {\sqrt {3}}{2}$
[i]Proposer[/i]:[b] Baltag Valeriu[/b]
2013 ELMO Shortlist, 5
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$.
[i]Proposed by Evan Chen[/i]
1994 India National Olympiad, 2
If $x^5 - x ^3 + x = a,$ prove that $x^6 \geq 2a - 1$.
2010 Postal Coaching, 2
Let $M$ be an interior point of a $\triangle ABC$ such that $\angle AM B = 150^{\circ} , \angle BM C = 120^{\circ}$. Let $P, Q, R$ be the circumcentres of the $\triangle AM B, \triangle BM C, \triangle CM A$ respectively. Prove that $[P QR] \ge [ABC]$.
2010 Kyrgyzstan National Olympiad, 1
Given that $a,b,c > 0$ and $a + b + c = 1$. Prove that $\sqrt {\frac{{ab}}{{ab + c}}} + \sqrt {\frac{{bc}}{{bc + a}}} + \sqrt {\frac{{ca}}{{ca + b}}} \leqslant \frac{3}{2}$.
2006 China Western Mathematical Olympiad, 1
Let $n$ be a positive integer with $n \geq 2$, and $0<a_{1}, a_{2},...,a_{n}< 1$. Find the maximum value of the sum
$\sum_{i=1}^{n}(a_{i}(1-a_{i+1}))^{\frac{1}{6}}$
where $a_{n+1}=a_{1}$
2009 Brazil National Olympiad, 3
Let $ n > 3$ be a fixed integer and $ x_1,x_2,\ldots, x_n$ be positive real numbers. Find, in terms of $ n$, all possible real values of
\[ {x_1\over x_n\plus{}x_1\plus{}x_2} \plus{} {x_2\over x_1\plus{}x_2\plus{}x_3} \plus{} {x_3\over x_2\plus{}x_3\plus{}x_4} \plus{} \cdots \plus{} {x_{n\minus{}1}\over x_{n\minus{}2}\plus{}x_{n\minus{}1}\plus{}x_n} \plus{} {x_n\over x_{n\minus{}1}\plus{}x_n\plus{}x_1}\]
2006 MOP Homework, 7
for real number $a,b,c$ in interval $ (0,1]$ prove that:
$\frac{a}{bc+1}+\frac{b}{ac+1}+\frac{c}{ab+1} \leq 2$
1970 IMO Longlists, 19
Let $1<n\in\mathbb{N}$ and $1\le a\in\mathbb{R}$ and there are $n$ number of $x_i, i\in\mathbb{N}, 1\le i\le n$ such that $x_1=1$ and $\frac{x_{i}}{x_{i-1}}=a+\alpha _ i$ for $2\le i\le n$, where $\alpha _i\le \frac{1}{i(i+1)}$. Prove that $\sqrt[n-1]{x_n}< a+\frac{1}{n-1}$.
2010 China Team Selection Test, 1
Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions:
(1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$;
(2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$;
(2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$.
Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.
2002 China Girls Math Olympiad, 3
Find all positive integers $ k$ such that for any positive numbers $ a, b$ and $ c$ satisfying the inequality
\[ k(ab \plus{} bc \plus{} ca) > 5(a^2 \plus{} b^2 \plus{} c^2),\]
there must exist a triangle with $ a, b$ and $ c$ as the length of its three sides respectively.
1994 Taiwan National Olympiad, 2
Let $a,b,c$ are positive real numbers and $\alpha$ be any real number. Denote $f(\alpha)=abc(a^{\alpha}+b^{\alpha}+c^{\alpha}), g(\alpha)=a^{2+\alpha}(b+c-a)+b^{2+\alpha}(-b+c+a)+c^{2+\alpha}(b-c+a)$. Determine $\min{|f(\alpha)-g(\alpha)|}$ and $\max{|f(\alpha)-g(\alpha)|}$, if they are exists.
2013 Polish MO Finals, 5
Let k,m and n be three different positive integers. Prove that \[
\left( k-\frac{1}{k} \right)\left( m-\frac{1}{m} \right)\left( n-\frac{1}{n} \right) \le kmn-(k+m+n). \]
2004 China Team Selection Test, 3
Let $k \geq 2, 1 < n_1 < n_2 < \ldots < n_k$ are positive integers, $a,b \in \mathbb{Z}^+$ satisfy \[ \prod^k_{i=1} \left( 1 - \frac{1}{n_i} \right) \leq \frac{a}{b} < \prod^{k-1}_{i=1} \left( 1 - \frac{1}{n_i} \right) \]
Prove that: \[ \prod^k_{i=1} n_i \geq (4 \cdot a)^{2^k - 1}. \]
2010 Postal Coaching, 4
$\triangle ABC$ has semiperimeter $s$ and area $F$ . A square $P QRS$ with side length $x$ is inscribed in $ABC$ with $P$ and $Q$ on $BC$, $R$ on $AC$, and $S$ on $AB$. Similarly, $y$ and $z$ are the sides of squares two vertices of which lie on $AC$ and $AB$, respectively. Prove that
\[\frac 1x +\frac 1y + \frac 1z \le \frac{s(2+\sqrt3)}{2F}\]
2001 China Western Mathematical Olympiad, 4
Let $ x, y, z$ be real numbers such that $ x \plus{} y \plus{} z \geq xyz$. Find the smallest possible value of $ \frac {x^2 \plus{} y^2 \plus{} z^2}{xyz}$.
1998 China Team Selection Test, 3
For a fixed $\theta \in \lbrack 0, \frac{\pi}{2} \rbrack$, find the smallest $a \in \mathbb{R}^{+}$ which satisfies the following conditions:
[b]I. [/b] $\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} >
1$.
[b]II.[/b] There exists $x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta},
\frac{\sqrt a}{\cos \theta} \rbrack$ such that $\lbrack (1 -
x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} +
\lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta}
\rbrack^{2} \leq a$.
2011 Moldova Team Selection Test, 2
Let $x_1, x_2, \ldots, x_n$ be real positive numbers such that $x_1\cdot x_2\cdots x_n=1$. Prove the inequality
$\frac1{x_1(x_1+1)}+\frac1{x_2(x_2+1)}+\cdots+\frac1{x_n(x_n+1)}\geq\frac n2$
1994 Cono Sur Olympiad, 3
Let $p$ be a positive real number given. Find the minimun vale of $x^3+y^3$, knowing that $x$ and $y$ are positive real numbers such that $xy(x+y)=p$.
1976 IMO Longlists, 24
Let $0 \le x_1 \le x_2\le\cdots\le x_n \le 1$. Prove that for all $A \ge 1$, there exists an interval $I$ of length $2\sqrt[n]{A}$ such that for all $x \in I$,
\[|(x - x_1)(x - x_2) \cdots (x -x_n)| \le A.\]
2012 Junior Balkan Team Selection Tests - Moldova, 1
Let $ 1\leq a,b,c,d,e,f,g,h,k \leq 9 $ and $ a,b,c,d,e,f,g,h,k $ are different integers, find the minimum value of the expression $ E = a*b*c+d*e*f+g*h*k $ and prove that it is minimum.
1989 Austrian-Polish Competition, 1
Show that $(\sum_{i=1}^{n}x_iy_iz_i)^2 \le (\sum_{i=1}^{n}x_i^3) (\sum_{i=1}^{n}y_i^3) (\sum_{i=1}^{n}z_i^3)$ for any positive reals $x_i, y_i, z_i$.
2008 Greece National Olympiad, 4
If $a_1, a_2, \ldots , a_n$ are positive integers and $k = \max\{a_1, \ldots, a_n\}$, $t = \min\{a_1,\ldots, a_n\}$, prove the inequality
\[\left(\frac{a_1^2+a_2^2+\cdots+a_n^2}{a_1+a_2+\cdots+a_n}\right)^{\frac{kn}{t}} \geq a_1a_2\cdots a_n.\]
When does equality hold?
2003 China Team Selection Test, 1
Let $g(x)= \sum_{k=1}^{n} a_k \cos{kx}$, $a_1,a_2, \cdots, a_n, x \in R$. If $g(x) \geq -1$ holds for every $x \in R$, prove that $\sum_{k=1}^{n}a_k \leq n$.
2013 Moldova Team Selection Test, 1
Consider real numbers $x,y,z$ such that $x,y,z>0$. Prove that \[ (xy+yz+xz)\left(\frac{1}{x^2+y^2}+\frac{1}{x^2+z^2}+\frac{1}{y^2+z^2}\right) > \frac{5}{2}. \]