Found problems: 787
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 $X$.)
2006 Romania Team Selection Test, 4
The real numbers $a_1,a_2,\dots,a_n$ are given such that $|a_i|\leq 1$
for all $i=1,2,\dots,n$ and
$a_1+a_2+\cdots+a_n=0$.
a) Prove that there exists $k\in\{1,2,\dots,n\}$ such that
\[ |a_1+2a_2+\cdots+ka_k|\leq\frac{2k+1}{4}. \]
b) Prove that for $n > 2$ the bound above is the best possible.
[i]Radu Gologan, Dan Schwarz[/i]
2011 Federal Competition For Advanced Students, Part 1, 2
For a positive integer $k$ and real numbers $x$ and $y$, let
\[f_k(x,y)=(x+y)-\left(x^{2k+1}+y^{2k+1}\right)\mbox{.}\]
If $x^2+y^2=1$, then determine the maximal possible value $c_k$ of $f_k(x,y)$.
2012 China Second Round Olympiad, 3
Let $P_0 ,P_1 ,P_2 , ... ,P_n$ be $n+1$ points in the plane. Let $d$($d>0$) denote the minimal value of all the distances between any two points. Prove that
\[|P_0P_1|\cdot |P_0P_2|\cdot ... \cdot |P_0P_n|>(\frac{d}{3})^n\sqrt{(n+1)!}.\]
2007 India IMO Training Camp, 2
Let $a,b,c$ be non-negative real numbers such that $a+b\leq c+1, b+c\leq a+1$ and $c+a\leq b+1.$ Show that
\[a^2+b^2+c^2\leq 2abc+1.\]
1996 Romania Team Selection Test, 9
Let $ n\geq 3 $ be an integer and let $ x_1,x_2,\ldots,x_{n-1} $ be nonnegative integers such that
\begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n \\ x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*}
Find the minimal value of $ F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k $.
2010 Contests, 2
Let $a,b,c$ be positive reals. Prove that
\[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \]
[i]Calvin Deng.[/i]
2010 ISI B.Stat Entrance Exam, 1
Let $a_1,a_2,\cdots, a_n$ and $b_1,b_2,\cdots, b_n$ be two permutations of the numbers $1,2,\cdots, n$. Show that
\[\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2\]
2023 Indonesia TST, A
Let $a,b,c$ positive real numbers and $a+b+c = 1$. Prove that
\[a^2 + b^2 + c^2 + \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \ge 2(ab + bc + ac)\]
2024 Austrian MO National Competition, 1
Determine the smallest real constant $C$ such that the inequality
\[(X+Y)^2(X^2+Y^2+C)+(1-XY)^2 \ge 0\]
holds for all real numbers $X$ and $Y$. For which values of $X$ and $Y$ does equality hold for this smallest constant $C$?
[i](Walther Janous)[/i]
2024 Mozambique National Olympiad, P2
Prove that if $a+b+c=0$ then $a^3+b^3+c^3=3abc$
2022 Kazakhstan National Olympiad, 5
For positive reals $a,b,c$ with $\sqrt{a}+\sqrt{b}+\sqrt{c}\ge 3$ prove that
$$\frac{a^3}{a^2+b}+\frac{b^3}{b^2+c}+\frac{c^3}{c^2+a}\ge \frac{3}{2}$$
2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 2
Let $f(x,\ y)=\frac{x+y}{(x^2+1)(y^2+1)}.$
(1) Find the maximum value of $f(x,\ y)$ for $0\leq x\leq 1,\ 0\leq y\leq 1.$
(2) Find the maximum value of $f(x,\ y),\ \forall{x,\ y}\in{\mathbb{R}}.$
2007 Romania Team Selection Test, 1
If $a_{1}$, $a_{2}$, $\ldots$, $a_{n}\geq 0$ are such that \[a_{1}^{2}+\cdots+a_{n}^{2}=1,\]
then find the maximum value of the product $(1-a_{1})\cdots (1-a_{n})$.
2014 Contests, 1
Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.
2008 Croatia Team Selection Test, 1
Let $ x$, $ y$, $ z$ be positive numbers. Find the minimum value of:
$ (a)\quad \frac{x^2 \plus{} y^2 \plus{} z^2}{xy \plus{} yz}$
$ (b)\quad \frac{x^2 \plus{} y^2 \plus{} 2z^2}{xy \plus{} yz}$
2014 CentroAmerican, 3
Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal,
\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4\] and $ac=bd$. Find the maximum possible value of
\[\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.\]
2008 China Team Selection Test, 2
For a given integer $ n\geq 2,$ determine the necessary and sufficient conditions that real numbers $ a_{1},a_{2},\cdots, a_{n},$ not all zero satisfy such that there exist integers $ 0<x_{1}<x_{2}<\cdots<x_{n},$ satisfying $ a_{1}x_{1}\plus{}a_{2}x_{2}\plus{}\cdots\plus{}a_{n}x_{n}\geq 0.$
1999 Korea - Final Round, 3
Let $a_1, a_2, ..., a_{1999}$ be nonnegative real numbers satisfying the following conditions:
a. $a_1+a_2+...+a_{1999}=2$
b. $a_1a_2+a_2a_3+...+a_{1999}a_1=1$.
Let $S=a_1^ 2+a_2 ^ 2+...+a_{1999}^2$. Find the maximum and minimum values of $S$.
2005 Germany Team Selection Test, 2
Let n be a positive integer, and let $a_1$, $a_2$, ..., $a_n$, $b_1$, $b_2$, ..., $b_n$ be positive real numbers such that $a_1\geq a_2\geq ...\geq a_n$ and $b_1\geq a_1$, $b_1b_2\geq a_1a_2$, $b_1b_2b_3\geq a_1a_2a_3$, ..., $b_1b_2...b_n\geq a_1a_2...a_n$.
Prove that $b_1+b_2+...+b_n\geq a_1+a_2+...+a_n$.
2009 Bosnia Herzegovina Team Selection Test, 3
$a_{1},a_{2},\dots,a_{100}$ are real numbers such that:\[
a_{1}\geq a_{2}\geq\dots\geq a_{100}\geq0\]
\[
a_{1}^{2}+a_{2}^{2}\geq100\]
\[
a_{3}^{2}+a_{4}^{2}+\dots+a_{100}^{2}\geq100\]
What is the minimum value of sum $a_{1}+a_{2}+\dots+a_{100}.$
Oliforum Contest II 2009, 2
Define $ \phi$ the positive real root of $ x^2 \minus{} x \minus{} 1$ and let $ a,b,c,d$ be positive real numbers such that $ (a \plus{} 2b)^2 \equal{} 4c^2 \plus{} 1$.
Show that $ \displaystyle 2d^2 \plus{} a^2\left(\phi \minus{} \frac {1}{2}\right) \plus{} b^2\left(\frac {1}{\phi \minus{} 1} \plus{} 2\right) \plus{} 2 \ge 4(c \minus{} d) \plus{} 2\sqrt {d^2 \plus{} 2d}$ and find all cases of equality.
[i](A.Naskov)[/i]
2010 Contests, 2
Given the positive real numbers $a_{1},a_{2},\dots,a_{n},$ such that $n>2$ and $a_{1}+a_{2}+\dots+a_{n}=1,$ prove that the inequality
\[
\frac{a_{2}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{1}+n-2}+\frac{a_{1}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{2}+n-2}+\dots+\frac{a_{1}\cdot a_{2}\cdot\dots\cdot a_{n-1}}{a_{n}+n-2}\leq\frac{1}{\left(n-1\right)^{2}}\]
does holds.
2004 China Western Mathematical Olympiad, 3
Find all reals $ k$ such that
\[ a^3 \plus{} b^3 \plus{} c^3 \plus{} d^3 \plus{} 1\geq k(a \plus{} b \plus{} c \plus{} d)
\]
holds for all $ a,b,c,d\geq \minus{} 1$.
[i]Edited by orl.[/i]
2006 Taiwan TST Round 1, 2
Let $a_1<a_2<\cdots<a_n$ be positive integers. Prove that $\displaystyle a_n \ge \sqrt[3]{\frac{(a_1+a_2+\cdots+a_n)^2}{n}}$.