Found problems: 426
2010 German National Olympiad, 2
Let $a,b,c$ be pairwise distinct real numbers. Show that
\[
(\frac{2a-b}{a-b})^2+(\frac{2b-c}{b-c})^2+(\frac{2c-a}{c-a})^2 \ge 5.
\]
2010 China Team Selection Test, 1
Given integer $n\geq 2$ and positive real number $a$, find the smallest real number $M=M(n,a)$, such that for any positive real numbers $x_1,x_2,\cdots,x_n$ with $x_1 x_2\cdots x_n=1$, the following inequality holds:
\[\sum_{i=1}^n \frac {1}{a+S-x_i}\leq M\]
where $S=\sum_{i=1}^n x_i$.
2009 Mexico National Olympiad, 3
Let $a$, $b$, and $c$ be positive numbers satisfying $abc=1$. Show that
\[\frac{a^3}{a^3+2}+\frac{b^3}{b^3+2}+\frac{c^3}{c^3+2}\ge1\text{ and }\frac1{a^3+2}+\frac1{b^3+2}+\frac1{c^3+2}\le1\]
2006 IberoAmerican, 2
For n real numbers $a_{1},\, a_{2},\, \ldots\, , a_{n},$ let $d$ denote the difference between the greatest and smallest of them and $S = \sum_{i<j}\left |a_i-a_j \right|.$ Prove that \[(n-1)d\le S\le\frac{n^{2}}{4}d\] and find when each equality holds.
2005 China Team Selection Test, 1
Find all positive integers $m$ and $n$ such that the inequality:
\[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \]
is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.
2012 Federal Competition For Advanced Students, Part 2, 1
Determine the maximum value of $m$, such that the inequality
\[ (a^2+4(b^2+c^2))(b^2+4(a^2+c^2))(c^2+4(a^2+b^2)) \ge m \]
holds for every $a,b,c \in \mathbb{R} \setminus \{0\}$ with $\left|\frac{1}{a}\right|+\left|\frac{1}{b}\right|+\left|\frac{1}{c}\right|\le 3$.
When does equality occur?
1984 IMO Longlists, 19
Let $ABC$ be an isosceles triangle with right angle at point $A$. Find the minimum of the function $F$ given by
\[F(M) = BM +CM-\sqrt{3}AM\]
1989 IMO Longlists, 92
Prove that $ a < b$ implies that $ a^3 \minus{} 3a \leq b^3 \minus{} 3b \plus{} 4.$ When does equality occur?
2010 South East Mathematical Olympiad, 3
Let $n$ be a positive integer. The real numbers $a_1,a_2,\cdots,a_n$ and $r_1,r_2,\cdots,r_n$ are such that $a_1\leq a_2\leq \cdots \leq a_n$ and $0\leq r_1\leq r_2\leq \cdots \leq r_n$.
Prove that $\sum_{i=1}^n\sum_{j=1}^n a_i a_j \min (r_i,r_j)\geq 0$
Russian TST 2018, P3
Let $a,b,c>0.$ Prove that $\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \ge \frac{1}{\sqrt{2a^2+2bc}}+\frac{1}{\sqrt{2b^2+2ca}}+\frac{1}{\sqrt{2c^2+2ab}}$
1970 Regional Competition For Advanced Students, 1
Let $x,y,z$ be positive real numbers such that $x+y+z=1$ Prove that always $\left( 1+\frac1x\right)\times\left(1+\frac1y\right)\times\left(1 +\frac1z\right)\ge 64$
When does equality hold?
2006 China National Olympiad, 5
Let $\{a_n\}$ be a sequence such that: $a_1 = \frac{1}{2}$, $a_{k+1}=-a_k+\frac{1}{2-a_k}$ for all $k = 1, 2,\ldots$. Prove that
\[ \left(\frac{n}{2(a_1+a_2+\cdots+a_n)}-1\right)^n \leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n\left(\frac{1}{a_1}-1\right)\left(\frac{1}{a_2}-1\right)\cdots \left(\frac{1}{a_n}-1\right). \]
2011 Macedonia National Olympiad, 1
Let $~$ $ a,\,b,\,c,\,d\, >\, 0$ $~$ and $~$ $a+b+c+d\, =\, 1\, .$ $~$ Prove the inequality
\[ \frac{1}{4a+3b+c}+\frac{1}{3a+b+4d}+\frac{1}{a+4c+3d}+\frac{1}{4b+3c+d}\; \ge\; 2\, . \]
2010 Contests, 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.
2014 Iran Team Selection Test, 5
$n$ is a natural number. for every positive real numbers $x_{1},x_{2},...,x_{n+1}$ such that $x_{1}x_{2}...x_{n+1}=1$ prove that:
$\sqrt[x_{1}]{n}+...+\sqrt[x_{n+1}]{n} \geq n^{\sqrt[n]{x_{1}}}+...+n^{\sqrt[n]{x_{n+1}}}$
1999 Mediterranean Mathematics Olympiad, 3
Let $a,b,c\not= 0$ and $x,y,z\in\mathbb{R}^+$ such that $x+y+z=3$. Prove that \[\frac{3}{2}\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}\geq\frac{x}{1+a^2}+\frac{y}{1+b^2}+\frac{z}{1+c^2}\]
[color=#FF0000]Mod: before the edit, it was [/color]
\[\frac{3}{2}\left (\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right )\geq\frac{x}{1+a^2}+\frac{y}{1+b^2}+\frac{z}{1+c^2}\]
2009 China Girls Math Olympiad, 5
Let $ x,y,z$ be real numbers greater than or equal to $ 1.$ Prove that
\[ \prod(x^{2} \minus{} 2x \plus{} 2)\le (xyz)^{2} \minus{} 2xyz \plus{} 2.\]
1990 IMO Longlists, 4
Find the minimal value of the function
\[\begin{array}{c}\ f(x) =\sqrt{15 - 12 \cos x} + \sqrt{4 -2 \sqrt 3 \sin x}+\sqrt{7-4\sqrt 3 \sin x} +\sqrt{10-4 \sqrt 3 \sin x - 6 \cos x}\end{array}\]
2009 Indonesia MO, 2
Find the lowest possible values from the function
\[ f(x) \equal{} x^{2008} \minus{} 2x^{2007} \plus{} 3x^{2006} \minus{} 4x^{2005} \plus{} 5x^{2004} \minus{} \cdots \minus{} 2006x^3 \plus{} 2007x^2 \minus{} 2008x \plus{} 2009\]
for any real numbers $ x$.
1989 Polish MO Finals, 3
Show that for positive reals $a, b, c, d$ we have
\[ \left(\dfrac{ab + ac + ad + bc + bd + cd}{6} \right)^3 \geq \left(\dfrac{abc + abd + acd + bcd}{4}\right)^2 \]
1988 China National Olympiad, 4
(1) Let $a,b,c$ be positive real numbers satisfying $(a^2+b^2+c^2)^2>2(a^4+b^4+c^4)$. Prove that $a,b,c$ can be the lengths of three sides of a triangle respectively.
(2) Let $a_1,a_2,\dots ,a_n$ be $n$ ($n>3$) positive real numbers satisfying $(a_1^2+a_2^2+\dots +a_n^2)^2>(n-1)(a_1^4+ a_2^4+\dots +a_n^4)$. Prove that any three of $a_1,a_2,\dots ,a_n$ can be the lengths of three sides of a triangle respectively.
2005 MOP Homework, 1
Let $a$ and $b$ be nonnegative real numbers. Prove that \[\sqrt{2}\left(\sqrt{a(a+b)^3}+b\sqrt{a^2+b^2}\right) \le 3(a^2+b^2).\]
2008 Vietnam National Olympiad, 6
Let $ x, y, z$ be distinct non-negative real numbers. Prove that
\[ \frac{1}{(x\minus{}y)^2} \plus{} \frac{1}{(y\minus{}z)^2} \plus{} \frac{1}{(z\minus{}x)^2} \geq \frac{4}{xy \plus{} yz \plus{} zx}.\]
When does the equality hold?
2005 China Girls Math Olympiad, 5
Let $ x$ and $ y$ be positive real numbers with $ x^3 \plus{} y^3 \equal{} x \minus{} y.$ Prove that \[ x^2 \plus{} 4y^2 < 1.\]
1987 Vietnam National Olympiad, 1
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be positive real numbers ($ n \ge 2$) whose sum is $ S$. Show
that \[ \sum_{i\equal{}1}^n\frac{a_i^{2^{k}}}{\left(S\minus{}a_i\right)^{2^t\minus{}1}}\ge\frac{S^{1\plus{}2^k\minus{}2^t}}{\left(n\minus{}1\right)^{2^t\minus{}1}n^{2^k\minus{}2^t}}\] for any nonnegative integers $ k$, $ t$ with $ k \ge t$. When does equality occur?