Found problems: 426
2011 Morocco National Olympiad, 1
Let $a$ and $b$ be two positive real numbers such that $a+b=ab$.
Prove that $\frac{a}{b^{2}+4}+\frac{b}{a^{2}+4}\geq \frac{1}{2}$.
2005 Federal Competition For Advanced Students, Part 2, 2
Prove that for all positive reals $a,b,c,d$, we have $\frac{a+b+c+d}{abcd}\leq \frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}+\frac{1}{d^{3}}$
1992 China Team Selection Test, 3
For any $n,T \geq 2, n, T \in \mathbb{N}$, find all $a \in \mathbb{N}$ such that $\forall a_i > 0, i = 1, 2, \ldots, n$, we have
\[\sum^n_{k=1} \frac{a \cdot k + \frac{a^2}{4}}{S_k} < T^2 \cdot \sum^n_{k=1} \frac{1}{a_k},\] where $S_k = \sum^k_{i=1} a_i.$
2003 USA Team Selection Test, 5
Let $A, B, C$ be real numbers in the interval $\left(0,\frac{\pi}{2}\right)$. Let \begin{align*} X &= \frac{\sin A\sin (A-B)\sin (A-C)}{\sin (B+C)} \\ Y &= \frac{\sin B\sin(B-C)\sin (B-A)}{\sin (C+A)} \\ Z &= \frac{\sin C\sin (C-A)\sin (C-B)}{\sin (A+B)} . \end{align*} Prove that $X+Y+Z \geq 0$.
2011 Junior Balkan MO, 1
Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that:
$\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$
2007 CHKMO, 4
Let a_1, a_2, a_3,... be a sequence of positive numbers. If there exists a positive number M such that for n = 1,2,3,...,
$a^{2}_{1}+a^{2}_{2}+...+a^{2}_{n}< Ma^{2}_{n+1}$
then prove that there exist a positive number M' such that for every n = 1,2,3,...,
$a_{1}+a_{2}+...+a_{n}< M'a_{n+1}$
1997 Hungary-Israel Binational, 2
Find all the real numbers $ \alpha$ satisfy the following property: for any positive integer $ n$ there exists an integer $ m$ such that $ \left |\alpha\minus{}\frac{m}{n}\right|<\frac{1}{3n}$.
1995 China National Olympiad, 1
Let $a_1,a_2,\cdots ,a_n; b_1,b_2,\cdots ,b_n (n\ge 3)$ be real numbers satisfying the following conditions:
(1) $a_1+a_2+\cdots +a_n= b_1+b_2+\cdots +b_n $;
(2) $0<a_1=a_2, a_i+a_{i+1}=a_{i+2}$ ($i=1,2,\cdots ,n-2$);
(3) $0<b_1\le b_2, b_i+b_{i+1}\le b_{i+2}$ ($i=1,2,\cdots ,n-2$).
Prove that $a_{n-1}+a_n\le b_{n-1}+b_n$.
1999 Belarusian National Olympiad, 5
Determine the maximal value of $ k $, such that for positive reals $ a,b $ and $ c $ from inequality $ kabc >a^3+b^3+c^3 $ it follows that $ a,b $ and $ c $ are sides of a triangle.
2007 Regional Competition For Advanced Students, 1
Let $ 0<x_0,x_1, \dots , x_{669}<1$ be pairwise distinct real numbers. Show that there exists a pair $ (x_i,x_j)$ with
$ 0<x_ix_j(x_j\minus{}x_i)<\frac{1}{2007}$
2000 Greece National Olympiad, 3
Find the maximum value of $k$ such that \[\frac{xy}{\sqrt{(x^2 + y^2)(3x^2 + y^2)}}\leq \frac{1}{k}\]
holds for all positive numbers $x$ and $y.$
2007 Mexico National Olympiad, 3
Given $a$, $b$, and $c$ be positive real numbers with $a+b+c=1$, prove that
\[\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}\le2\]
2005 South East Mathematical Olympiad, 8
Let $0 < \alpha, \beta, \gamma < \frac{\pi}{2}$ and $\sin^{3} \alpha + \sin^{3} \beta + \sin^3 \gamma = 1$. Prove that
\[ \tan^{2} \alpha + \tan^{2} \beta + \tan^{2} \gamma \geq \frac{3 \sqrt{3}}{2} . \]
2011 Mongolia Team Selection Test, 1
Let $t,k,m$ be positive integers and $t>\sqrt{km}$. Prove that
$\dbinom{2m}{0}+\dbinom{2m}{1}+\cdots+\dbinom{2m}{m-t-1}<\dfrac{2^{2m}}{2k}$
(proposed by B. Amarsanaa, folklore)
2013 Turkmenistan National Math Olympiad, 3
If a,b,c positive numbers and such that $a+\sqrt{b+\sqrt{c}}=c+\sqrt{b+\sqrt{a}}$. Prove that if $a\neq c$ then $40ac<1$.
2005 Moldova National Olympiad, 10.4
Real numbers $ x_{1},x_{2},..,x_{n}$ are positive. Prove the inequality:
$ \frac{x_{1}}{x_{2}\plus{}x_{3}}\plus{}\frac{x_{2}}{x_{3}\plus{}x_{4}}\plus{}...\plus{} \frac{x_{n\minus{}1}}{x_{n}\plus{}x_{1}}\plus{}\frac{x_{n}}{x_{1}\plus{}x_{2}}>(\sqrt{2}\minus{}1)n$
2007 Korea National Olympiad, 1
For all positive reals $ a$, $ b$, and $ c$, what is the value of positive constant $ k$ satisfies the following inequality?
$ \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007}$ .
2014 South East Mathematical Olympiad, 5
Let $\triangle ABC $ and $\triangle A'B'C'$ are acute triangles.Prove that\[Max\{cotA'(cotB+cotC),cotB'(cotC+cotA),cotC'(cotA+cotB)\}\ge \frac{2}{3}.\]
2005 Korea National Olympiad, 6
Real numbers $x_1, x_2, x_3, \cdots , x_n$ satisfy $x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2 = 1$. Show that \[ \frac{x_1}{1+x_1^2}+\frac{x_2}{1+x_1^2+x_2^2}+\cdots+\frac{x_n}{1+ x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2} < \sqrt{\frac n2} . \]
2011 China Western Mathematical Olympiad, 2
Let $a,b,c > 0$, prove that
\[\frac{(a-b)^2}{(c+a)(c+b)} + \frac{(b-c)^2}{(a+b)(a+c)} + \frac{(c-a)^2}{(b+c)(b+a)} \geq \frac{(a-b)^2}{a^2+b^2+c^2}\]
1985 IMO Longlists, 24
Let $d \geq 1$ be an integer that is not the square of an integer. Prove that for every integer $n \geq 1,$
\[(n \sqrt d +1) \cdot | \sin(n \pi \sqrt d )| \geq 1\]
1991 Vietnam National Olympiad, 3
Prove that:
$ \frac {x^{2}y}{z} \plus{} \frac {y^{2}z}{x} \plus{} \frac {z^{2}x}{y}\geq x^{2} \plus{} y^{2} \plus{} z^{2}$
where $ x;y;z$ are real numbers saisfying $ x \geq y \geq z \geq 0$
1995 India National Olympiad, 5
Let $n \geq 2$. Let $a_1 , a_2 , a_3 , \ldots a_n$ be $n$ real numbers all less than $1$ and such that $|a_k - a_{k+1} | < 1$ for $1 \leq k \leq n-1$. Show that \[ \dfrac{a_1}{a_2} + \dfrac{a_2}{a_3} + \dfrac{a_3}{a_4} + \ldots + \dfrac{a_{n-1}}{a_n} + \dfrac{a_n}{a_1} < 2 n - 1 . \]
2010 Contests, 3
Given $a_1\ge 1$ and $a_{k+1}\ge a_k+1$ for all $k\ge 1,2,\dots,n$, show that $a_1^3+a_2^3+\dots+a_n^3\ge (a_1+a_2+\dots+a_n)^2$
2014 Contests, 2
Let $a$ ,$b$ and $c$ be distinct real numbers.
$a)$ Determine value of $ \frac{1+ab }{a-b} \cdot \frac{1+bc }{b-c} + \frac{1+bc }{b-c} \cdot \frac{1+ca }{c-a} + \frac{1+ca }{c-a} \cdot \frac{1+ab}{a-b} $
$b)$ Determine value of $ \frac{1-ab }{a-b} \cdot \frac{1-bc }{b-c} + \frac{1-bc }{b-c} \cdot \frac{1-ca }{c-a} + \frac{1-ca }{c-a} \cdot \frac{1-ab}{a-b} $
$c)$ Prove the following ineqaulity
$ \frac{1+a^2b^2 }{(a-b)^2} + \frac{1+b^2c^2 }{(b-c)^2} + \frac{1+c^2a^2 }{(c-a)^2} \geq \frac{3}{2} $
When does eqaulity holds?