Found problems: 6530
2018 Cyprus IMO TST, 4
Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$.
(a) Prove that
$$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$
(b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence:
$$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$
2001 VJIMC, Problem 3
Let $n\ge2$ be a natural number. Prove that
$$\prod_{k=2}^n\ln k<\frac{\sqrt{n!}}n.$$
2007 Hanoi Open Mathematics Competitions, 8
Let a; b; c be positive integers. Prove that
$$ \frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac{3}{5}$$
1980 Swedish Mathematical Competition, 4
The functions $f$ and $g$ are positive and continuous. $f$ is increasing and $g$ is decreasing. Show that
\[
\int\limits_0^1 f(x)g(x) dx \leq \int\limits_0^1 f(x)g(1-x) dx
\]
2003 Polish MO Finals, 2
Let $0 < a < 1$ be a real number. Prove that for all finite, strictly increasing sequences $k_1, k_2, \ldots , k_n$ of non-negative integers we have the inequality
\[\biggl( \sum_{i=1}^n a^{k_i} \biggr)^2 < \frac{1+a}{1-a} \sum_{i=1}^n a^{2k_i}.\]
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 \[(1-{a_i}^n)(1-{a_2}^n)\cdots (1-{a_n}^n)\le (1-a_1a_2\cdots a_n)^n.\]
2008 Iran Team Selection Test, 4
Let $ P_1,P_2,P_3,P_4$ be points on the unit sphere. Prove that $ \sum_{i\neq j}\frac1{|P_i\minus{}P_j|}$ takes its minimum value if and only if these four points are vertices of a regular pyramid.
2003 Greece National Olympiad, 1
If $a, b, c, d$ are positive numbers satisfying $a^3 + b^3 +3ab = c + d = 1,$ prove that \[\left(a+\frac{1}{a}\right)^3+\left(b+\frac{1}{b}\right)^3+\left(c+\frac{1}{c}\right)^3+\left(d+\frac{1}{d}\right)^3\geq 40.\]
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 $z_{n+1}=z_1$.
2019 Bundeswettbewerb Mathematik, 2
Determine the smallest possible value of the sum $S (a, b, c) = \frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}$ where $a, b, c$ are three positive real numbers with $a^2 + b^2 + c^2 = 1$
2009 Kyiv Mathematical Festival, 2
Let $x,y,z$ be positive numebrs such that $x+y+z\le x^3+y^3+z^3$. Is it true that
a) $x^2+y^2+z^2 \le x^3+y^3+z^3$ ?
b) $x+y+z\le x^2+y^2+z^2$ ?
2016 Indonesia TST, 2
Given a convex polygon with $n$ sides and perimeter $S$, which has an incircle $\omega$ with radius $R$. A regular polygon with $n$ sides, whose vertices lie on $\omega$, has a perimeter $s$. Determine whether the following inequality holds:
\[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]
PEN P Problems, 11
For each positive integer $n$, let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4)=4$, because the number $4$ can be represented in the following four ways: \[4, 2+2, 2+1+1, 1+1+1+1.\] Prove that, for any integer $n \geq 3$, \[2^{n^{2}/4}< f(2^{n}) < 2^{n^{2}/2}.\]
2019 Balkan MO Shortlist, A4
Let $a_{ij}, i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ be positive real numbers. Prove that
\[ \sum_{i = 1}^m \left( \sum_{j = 1}^n \frac{1}{a_{ij}} \right)^{-1} \le \left( \sum_{j = 1}^n \left( \sum_{i = 1}^m a_{ij} \right)^{-1} \right)^{-1} \]
2011 Today's Calculation Of Integral, 747
Prove that $\int_0^4 \left(1-\cos \frac{x}{2}\right)e^{\sqrt{x}}dx\leq -2e^2+30.$
2015 Indonesia MO, 7
Let $a,b,c$ be positive real numbers. Prove that
$\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$
2011 Estonia Team Selection Test, 4
Let $a,b,c$ be positive real numbers such that $2a^2 +b^2=9c^2$.Prove that $\displaystyle \frac{2c}{a}+\frac cb \ge\sqrt 3$.
2017 Junior Balkan Team Selection Tests - Romania, 2
Given $x_1,x_2,...,x_n$ real numbers, prove that there exists a real number $y$, such that,
$$\{y-x_1\}+\{y-x_2\}+...+\{y-x_n\} \leq \frac{n-1}{2}$$
2015 JBMO Shortlist, A5
The positive real $x, y, z$ are such that $x^2+y^2+z^2 = 3$. Prove that$$\frac{x^2+yz}{x^2+yz +1}+\frac{y^2+zx}{y^2+zx+1}+\frac{z^2+xy}{z^2+xy+1}\leq 2$$
1989 Vietnam National Olympiad, 2
The sequence of polynomials $ \left\{P_n(x)\right\}_{n\equal{}0}^{\plus{}\infty}$ is defined inductively by $ P_0(x) \equal{} 0$ and $ P_{n\plus{}1}(x) \equal{} P_n(x)\plus{}\frac{x \minus{} P_n^2(x)}{2}$. Prove that for any $ x \in [0, 1]$ and any natural number $ n$ it holds that $ 0\le\sqrt x\minus{} P_n(x)\le\frac{2}{n \plus{} 1}$.
2001 Federal Math Competition of S&M, Problem 3
Let $p_{1}, p_{2},...,p_{n}$, where $n>2$, be the first $n$ prime numbers. Prove that
$\frac{1}{p_{1}^2}+\frac{1}{p_{2}^2}+...+\frac{1}{p_{n}^2}+\frac{1}{p_{1}p_{2}...p_{n}}<\frac{1}{2}$
1987 IMO Longlists, 75
Let $a_k$ be positive numbers such that $a_1 \geq 1$ and $a_{k+1} -a_k \geq 1 \ (k = 1, 2, . . . )$. Prove that for every $n \in \mathbb N,$
\[\sum_{k=1}^{1987}\frac{1}{a_{k+1} \sqrt[1987]{a_k}} <1987\]
1994 Polish MO Finals, 3
The distinct reals $x_1, x_2, ... , x_n$ ,($n > 3$) satisfy $\sum_{i=1}^n x_i = 0$, $\sum_{i=1}^n x_i ^2 = 1$. Show that four of the numbers $a, b, c, d$ must satisfy:
\[ a + b + c + nabc \leq \sum_{i=1}^n x_i ^3 \leq a + b + d + nabd \].
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)$.
2005 IMC, 3
3) $f$ cont diff, $R\rightarrow ]0,+\infty[$, prove $|\int_{0}^{1}f^{3}-{f(0)}^{2}\int_{0}^{1}f| \leq \max_{[0,1]} |f'|(\int_{0}^{1}f)^{2}$