Found problems: 112
2000 All-Russian Olympiad, 2
Let $-1 < x_1 < x_2 , \cdots < x_n < 1$ and $x_1^{13} + x_2^{13} + \cdots + x_n^{13} = x_1 + x_2 + \cdots + x_n$. Prove that if $y_1 < y_2 < \cdots < y_n$, then \[ x_1^{13}y_1 + \cdots + x_n^{13}y_n < x_1y_1 + x_2y_2 + \cdots + x_ny_n. \]
2012 IMO Shortlist, A3
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
2019 Canada National Olympiad, 4
Prove that for $n>1$ and real numbers $a_0,a_1,\dots, a_n,k$ with $a_1=a_{n-1}=0$,
\[|a_0|-|a_n|\leq \sum_{i=0}^{n-2}|a_i-ka_{i+1}-a_{i+2}|.\]
2005 Korea Junior Math Olympiad, 7
If positive reals $ x_1,x_2,\cdots,x_n $ satisfy $\sum_{i=1}^{n}x_i=1.$ Prove that$$\sum_{i=1}^{n}\frac{1}{1+\sum_{j=1}^{i}x_j}<\sqrt{\frac{2}{3}\sum_{i=1}^{n}\frac{1}{x_i}}
$$
2016 Croatia Team Selection Test, Problem 1
Let $n \ge 1$ and $x_1, \ldots, x_n \ge 0$. Prove that
$$ (x_1 + \frac{x_2}{2} + \ldots + \frac{x_n}{n}) (x_1 + 2x_2 + \ldots + nx_n) \le \frac{(n+1)^2}{4n} (x_1 + x_2 + \ldots + x_n)^2 .$$
2004 China National Olympiad, 2
For a given positive integer $n\ge 2$, suppose positive integers $a_i$ where $1\le i\le n$ satisfy $a_1<a_2<\ldots <a_n$ and $\sum_{i=1}^n \frac{1}{a_i}\le 1$. Prove that, for any real number $x$, the following inequality holds
\[\left(\sum_{i=1}^n\frac{1}{a_i^2+x^2}\right)^2\le\frac{1}{2}\cdot\frac{1}{a_1(a_1-1)+x^2} \]
[i]Li Shenghong[/i]
2014 Taiwan TST Round 2, 1
Let $a_i > 0$ for $i=1,2,\dots,n$ and suppose $a_1 + a_2 + \dots + a_n = 1$. Prove that for any positive integer $k$,
\[ \left( a_1^k + \frac{1}{a_1^k} \right) \left( a_2^k + \frac{1}{a_2^k} \right) \dots \left( a_n^k + \frac{1}{a_n^k} \right) \ge \left( n^k + \frac{1}{n^k} \right)^n. \]
2021 Canada National Olympiad, 2
Let $n\geq 2$ be some fixed positive integer and suppose that $a_1, a_2,\dots,a_n$ are positive real numbers satisfying $a_1+a_2+\cdots+a_n=2^n-1$.
Find the minimum possible value of $$\frac{a_1}{1}+\frac{a_2}{1+a_1}+\frac{a_3}{1+a_1+a_2}+\cdots+\frac{a_n}{1+a_1+a_2+\cdots+a_{n-1}}$$
1998 IMO Shortlist, 2
Let $r_{1},r_{2},\ldots ,r_{n}$ be real numbers greater than or equal to 1. Prove that
\[ \frac{1}{r_{1} + 1} + \frac{1}{r_{2} + 1} + \cdots +\frac{1}{r_{n}+1} \geq \frac{n}{ \sqrt[n]{r_{1}r_{2} \cdots r_{n}}+1}. \]
2014 Mediterranean Mathematics Olympiad, 1
Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that
$ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$
(Proposed by Gerhard Woeginger, Austria)
2004 IMO Shortlist, 7
Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[
n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality.
[i]Proposed by Finbarr Holland, Ireland[/i]
2012 IMO, 2
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
1997 IMO Shortlist, 19
Let $ a_1\geq \cdots \geq a_n \geq a_{n \plus{} 1} \equal{} 0$ be real numbers. Show that
\[ \sqrt {\sum_{k \equal{} 1}^n a_k} \leq \sum_{k \equal{} 1}^n \sqrt k (\sqrt {a_k} \minus{} \sqrt {a_{k \plus{} 1}}).
\]
[i]Proposed by Romania[/i]
1995 Poland - Second Round, 4
Positive real numbers $x_1,x_2,...,x_n$ satisfy the condition $\sum_{i=1}^n x_i \le \sum_{i=1}^n x_i ^2$ .
Prove the inequality $\sum_{i=1}^n x_i^t \le \sum_{i=1}^n x_i ^{t+1}$ for all real numbers $t > 1$.
1995 IMO Shortlist, 3
Let $ n$ be an integer, $ n \geq 3.$ Let $ a_1, a_2, \ldots, a_n$ be real numbers such that $ 2 \leq a_i \leq 3$ for $ i \equal{} 1, 2, \ldots, n.$ If $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n,$ prove that
\[ \frac{a^2_1 \plus{} a^2_2 \minus{} a^2_3}{a_1 \plus{} a_2 \minus{} a_3} \plus{} \frac{a^2_2 \plus{} a^2_3 \minus{} a^2_4}{a_2 \plus{} a_3 \minus{} a_4} \plus{} \ldots \plus{} \frac{a^2_n \plus{} a^2_1 \minus{} a^2_2}{a_n \plus{} a_1 \minus{} a_2} \leq 2s \minus{} 2n.\]
2016 Iran Team Selection Test, 4
Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.
2011 ISI B.Stat Entrance Exam, 1
Let $x_1, x_2, \cdots , x_n$ be positive reals with $x_1+x_2+\cdots+x_n=1$. Then show that
\[\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}\]
2010 Mediterranean Mathematics Olympiad, 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.
1998 IMO Shortlist, 1
Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$. Prove that
\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]
1966 IMO Longlists, 13
Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove the inequality
\[\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2\]
2011 Morocco TST, 2
Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$
[i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]
Russian TST 2021, P3
Let $n$ and $k$ be positive integers. Prove that for $a_1, \dots, a_n \in [1,2^k]$ one has
\[ \sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}. \]
2017 China Girls Math Olympiad, 5
Let $0=x_0<x_1<\cdots<x_n=1$ .Find the largest real number$ C$ such that for any positive integer $ n $ , we have $$\sum_{k=1}^n x^2_k (x_k - x_{k-1})>C$$
2016 Taiwan TST Round 1, 2
Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.
2015 Baltic Way, 2
Let $n$ be a positive integer and let $a_1,\cdots ,a_n$ be real numbers satisfying $0\le a_i\le 1$ for $i=1,\cdots ,n.$ Prove the inequality \[(1-{a_i}^n)(1-{a_2}^n)\cdots (1-{a_n}^n)\le (1-a_1a_2\cdots a_n)^n.\]