Found problems: 787
2005 Iran Team Selection Test, 1
Suppose that $ a_1$, $ a_2$, ..., $ a_n$ are positive real numbers such that $ a_1 \leq a_2 \leq \dots \leq a_n$. Let
\[ {{a_1 \plus{} a_2 \plus{} \dots \plus{} a_n} \over n} \equal{} m; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{a_1^2 \plus{} a_2^2 \plus{} \dots \plus{} a_n^2} \over n} \equal{} 1.
\]
Suppose that, for some $ i$, we know $ a_i \leq m$. Prove that:
\[ n \minus{} i \geq n \left(m \minus{} a_i\right)^2
\]
2006 Macedonia National Olympiad, 3
Let $a,b,c$ be real numbers distinct from $0$ and $1$, with $a+b+c=1$. Prove that
\[8\left(\frac{1}{2}-ab-bc-ca\right)\left(\frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} \right)\ge 9 \]
2015 Kazakhstan National Olympiad, 1
Prove that $$\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{(n+1)^2} < n \cdot \left(1-\frac{1}{\sqrt[n]{2}}\right).$$
2007 Bulgaria Team Selection Test, 3
Let $n\geq 2$ is positive integer. Find the best constant $C(n)$ such that \[\sum_{i=1}^{n}x_{i}\geq C(n)\sum_{1\leq j<i\leq n}(2x_{i}x_{j}+\sqrt{x_{i}x_{j}})\] is true for all real numbers $x_{i}\in(0,1),i=1,...,n$ for which $(1-x_{i})(1-x_{j})\geq\frac{1}{4},1\leq j<i \leq n.$
2008 Bulgaria Team Selection Test, 2
The point $P$ lies inside, or on the boundary of, the triangle $ABC$. Denote by $d_{a}$, $d_{b}$ and $d_{c}$ the distances between $P$ and $BC$, $CA$, and $AB$, respectively. Prove that $\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}$. When does the equality holds?
2019 Regional Competition For Advanced Students, 1
Let $x,y$ be real numbers such that $(x+1)(y+2)=8.$ Prove that $$(xy-10)^2\ge 64.$$
2007 Iran MO (3rd Round), 3
Find the largest real $ T$ such that for each non-negative real numbers $ a,b,c,d,e$ such that $ a\plus{}b\equal{}c\plus{}d\plus{}e$: \[ \sqrt{a^{2}\plus{}b^{2}\plus{}c^{2}\plus{}d^{2}\plus{}e^{2}}\geq T(\sqrt a\plus{}\sqrt b\plus{}\sqrt c\plus{}\sqrt d\plus{}\sqrt e)^{2}\]
2014 China Team Selection Test, 5
Let $n$ be a given integer which is greater than $1$ . Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that \[\sum_{k\equal{}1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},\] where $z_{n+1}=z_1$.
2015 Turkey MO (2nd round), 2
$x$, $y$ and $z$ are real numbers where the sum of any two among them is not $1$. Show that, \[ \dfrac{(x^2+y)(x+y^2)}{(x+y-1)^2}+\dfrac{(y^2+z)(y+z^2)}{(y+z-1)^2} + \dfrac{(z^2+x)(z+x^2)}{(z+x-1)^2} \ge 2(x+y+z) - \dfrac{3}{4}\]Find all triples $(x,y,z)$ of real numbers satisfying the equality case.
2019 IFYM, Sozopol, 7
Let $a, b, c$ be positive real numbers such that $abc=8$. Prove that
\[
\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}
\]
2005 QEDMO 1st, 6 (U1)
Prove that for any four real numbers $a$, $b$, $c$, $d$, the inequality
\[ \left(a-b\right)\left(b-c\right)\left(c-d\right)\left(d-a\right)+\left(a-c\right)^2\left(b-d\right)^2\geq 0 \]
holds.
[hide="comment"]This is inequality (350) in: Mihai Onucu Drimbe, [i]Inegalitati, idei si metode[/i], Zalau: Gil, 2003.
Posted here only for the sake of completeness; in fact, it is more or less the same as http://www.mathlinks.ro/Forum/viewtopic.php?t=3152 .[/hide]
Darij
2012 Indonesia TST, 1
Let $a,b,c \in \mathbb{C}$ such that $a|bc| + b|ca| + c|ab| = 0$. Prove that $|(a-b)(b-c)(c-a)| \ge 3\sqrt{3}|abc|$.
2010 Kazakhstan National Olympiad, 3
Positive real $A$ is given. Find maximum value of $M$ for which inequality
$ \frac{1}{x}+\frac{1}{y}+\frac{A}{x+y} \geq \frac{M}{\sqrt{xy}} $
holds for all $x, y>0$
2002 Silk Road, 2
I tried to search SRMC problems,but i didn't find them(I found only SRMC 2006).Maybe someone know where on this site i could find SRMC problems?I have all SRMC problems,if someone want i could post them, :wink:
Here is one of them,this is one nice inequality from first SRMC:
Let $ n$ be an integer with $ n>2$ and $ a_{1},a_{2},\dots,a_{n}\in R^{\plus{}}$.Given any positive integers $ t,k,p$ with $ 1<t<n$,set $ m\equal{}k\plus{}p$,prove the following inequalities:
a)
$ \frac{a_{1}^{p}}{a_{2}^{k}\plus{}a_{3}^{k}\plus{}\dots\plus{}a_{t}^{k}}\plus{}\frac{a_{2}^{p}}{a_{3}^{k}\plus{}a_{4}^{k}\plus{}\dots\plus{}a_{t\plus{}1}^{k}}\plus{}\dots\plus{}\frac{a_{n\minus{}1}^{p}}{a_{n}^{k}\plus{}a_{1}^{k}\plus{}\dots\plus{}a_{t\minus{}2}^{k}}\plus{}\frac{a_{n}^{p}}{a_{1}^{k}\plus{}a_{2}^{k}\plus{}\dots\plus{}a_{t\minus{}1}^{k}}\geq\frac{(a_{1}^{p}\plus{}a_{2}^{p}\dots\plus{}a_{n}^{p})^{2}}{(t\minus{}1) ( a_{1}^{m}\plus{}a_{2}^{m}\plus{}\dots\plus{}a_{n}^{m})}$
b)$ \frac{a_{2}^{k}\plus{}a_{3}^{k}\dots\plus{}a_{t}^{k}}{a_{1}^{p}}\plus{}\frac{a_{3}^{k}\plus{}a_{4}^{k}\dots\plus{}a_{t\plus{}1}^{k}}{a_{2}^{p}}\plus{}\dots\plus{}\frac{a_{1}^{k}\plus{}a_{2}^{k}\dots\plus{}a_{t\minus{}1}^{k}}{a_{n}^{p}}\geq\frac{(t\minus{}1)(a_{1}^{k}\plus{}a_{2}^{k}\dots\plus{}a_{n}^{k})^{2}}{( a_{1}^{m}\plus{}a_{2}^{m}\plus{}\dots\plus{}a_{n}^{m})}$ :wink:
2007 Indonesia TST, 1
Let $ a,b,c$ be real numbers. Prove that $ (ab\plus{}bc\plus{}ca\minus{}1)^2 \le (a^2\plus{}1)(b^2\plus{}1)(c^2\plus{}1)$.
2007 Turkey MO (2nd round), 3
If $a,b,c$ are three positive real numbers such that $a+b+c=3$, prove that
$ {\frac{a^{2}+3b^{2}}{ab^{2}(4-ab)}}+{\frac{b^{2}+3c^{2}}{bc^{2}(4-ab)}}+{\frac{c^{2}+3a^{2}}{ca^{2}(4-ca)}}\geq 4 $
2010 Contests, 2
If $ x,y$ are positive real numbers with sum $ 2a$, prove that :
$ x^3y^3(x^2\plus{}y^2)^2 \leq 4a^{10}$
When does equality hold ?
Babis
2010 China Girls Math Olympiad, 4
Let $x_1,x_2,\cdots,x_n$ be real numbers with $x_1^2+x_2^2+\cdots+x_n^2=1$. Prove that
\[\sum_{k=1}^{n}\left(1-\dfrac{k}{{\displaystyle \sum_{i=1}^{n} ix_i^2}}\right)^2 \cdot \dfrac{x_k^2}{k} \leq \left(\dfrac{n-1}{n+1}\right)^2 \sum_{k=1}^{n} \dfrac{x_k^2}{k}\]
Determine when does the equality hold?
2007 QEDMO 4th, 7
For any three nonnegative reals $a$, $b$, $c$, prove that
$\left|ca-ab\right|+\left|ab-bc\right|+\left|bc-ca\right|\leq\left|b^{2}-c^{2}\right|+\left|c^{2}-a^{2}\right|+\left|a^{2}-b^{2}\right|$.
[i]Generalization.[/i] For any $n$ nonnegative reals $a_{1}$, $a_{2}$, ..., $a_{n}$, prove that
$\sum_{i=1}^{n}\left|a_{i-1}a_{i}-a_{i}a_{i+1}\right|\leq\sum_{i=1}^{n}\left|a_{i}^{2}-a_{i+1}^{2}\right|$.
Here, the indices are cyclic modulo $n$; this means that we set $a_{0}=a_{n}$ and $a_{n+1}=a_{1}$.
darij
2023 All-Russian Olympiad Regional Round, 11.9
If $a, b, c$ are non-zero reals, prove that $|\frac{b} {a}-\frac{b} {c}|+|\frac{c} {a}-\frac{c}{b}|+|bc+1|>1$.
2005 Morocco TST, 2
Let $a,b,c$ be positive real numbers. Prove the inequality
\[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.\]
When does equality occur?
2013 India National Olympiad, 6
Let $a,b,c,x,y,z$ be six positive real numbers satisfying $x+y+z=a+b+c$ and $xyz=abc.$ Further, suppose that $a\leq x<y<z\leq c$ and $a<b<c.$ Prove that $a=x,b=y$ and $c=z.$
2011 ISI B.Math Entrance Exam, 3
For $n\in\mathbb{N}$ prove that
\[\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\leq\frac{1}{\sqrt{2n+1}}.\]
2020 German National Olympiad, 5
Let $a_1,a_2,\dots,a_{22}$ be positive integers with sum $59$.
Prove the inequality
\[\frac{a_1}{a_1+1}+\frac{a_2}{a_2+1}+\dots+\frac{a_{22}}{a_{22}+1}<16.\]
2006 India National Olympiad, 6
(a) Prove that if $n$ is a integer such that $n \geq 4011^2$ then there exists an integer $l$ such that \[ n < l^2 < (1 + \frac{1}{{2005}})n . \]
(b) Find the smallest positive integer $M$ for which whenever an integer $n$ is such that $n \geq M$
then there exists an integer $l$ such that \[ n < l^2 < (1 + \frac{1}{{2005}})n . \]