Found problems: 6530
1991 Austrian-Polish Competition, 5
If $x,y, z$ are arbitrary positive numbers with $xyz = 1$, prove the inequality
$$x^2+y^2+z^2 + xy+yz + zx \ge 2(\sqrt{x} +\sqrt{y}+ \sqrt{z})$$.
2003 China Team Selection Test, 3
Let $a_{1},a_{2},...,a_{n}$ be positive real number $(n \geq 2)$,not all equal,such that $\sum_{k=1}^n a_{k}^{-2n}=1$,prove that:
$\sum_{k=1}^n a_{k}^{2n}-n^2.\sum_{1 \leq i<j \leq n}(\frac{a_{i}}{a_{j}}-\frac{a_{j}}{a_{i}})^2 >n^2$
2010 Rioplatense Mathematical Olympiad, Level 3, 2
Find the minimum and maximum values of $ S=\frac{a}{b}+\frac{c}{d} $ where $a$, $b$, $c$, $d$ are positive integers satisfying $a + c = 20202$ and $b + d = 20200$.
2011 USA TSTST, 6
Let $a, b, c$ be positive real numbers in the interval $[0, 1]$ with $a+b, b+c, c+a \ge 1$. Prove that
\[
1 \le (1-a)^2 + (1-b)^2 + (1-c)^2 +
\frac{2\sqrt{2} abc}{\sqrt{a^2+b^2+c^2}}.
\]
2025 Bulgarian Spring Mathematical Competition, 12.1
In terms of the real numbers $a$ and $b$ determine the minimum value of $$ \sqrt{(x+a)^2+1}+\sqrt{(x+1-a)^2+1}+\sqrt{(x+b)^2+1}+\sqrt{(x+1-b)^2+1}$$ as well as all values of $x$ which attain it.
1997 Pre-Preparation Course Examination, 5
Let $ABC$ be an acute angled triangle, $O$ be the circumcenter of $ABC$, and $R$ be the cicumradius. $AO$ meets the circumcircle of $BOC$ at $A'$, $BO$ meets the circumcircle of $COA$, and $CO$ meets the circumcircle of $AOB$ at $C'$. Prove that
\[OA' \cdot OB' \cdot OC' \geq 8R^3.\]
When does inequality occur?
2005 France Pre-TST, 4
Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2 = 25.$
Find the minimum of $\frac {xy} z + \frac {yz} x + \frac {zx} y .$
Pierre.
2020 South East Mathematical Olympiad, 1
Let $f(x)=a(3a+2c)x^2-2b(2a+c)x+b^2+(c+a)^2$ $(a,b,c\in R, a(3a+2c)\neq 0).$ If $$f(x)\leq 1$$for any real $x$, find the maximum of $|ab|.$
1984 IMO Longlists, 19
Let $ABC$ be an isosceles triangle with right angle at point $A$. Find the minimum of the function $F$ given by
\[F(M) = BM +CM-\sqrt{3}AM\]
1968 All Soviet Union Mathematical Olympiad, 109
Two finite sequences $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ are just rearranged sequence $1, 1/2, ... , 1/n$ with $$a_1+b_1\ge a_2+b_2\ge...\ge a_n+b_n.$$ Prove that $a_m+a_n\ge 4/m$ for every $m$ ($1\le m\le n$) .
2018 Hanoi Open Mathematics Competitions, 14
Let $a,b, c$ denote the real numbers such that $1 \le a, b, c\le 2$.
Consider $T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}$.
Determine the largest possible value of $T$.
2013 Turkmenistan National Math Olympiad, 1
Find the product $ \cos a \cdot \cos 2a\cdot \cos 3a \cdots \cos 1006a$ where $a=\frac{2\pi}{2013}$.
2015 Dutch Mathematical Olympiad, 5
Given are (not necessarily positive) real numbers $a, b$, and $c$ for which $|a - b| \ge |c| , |b - c| \ge |a|$ and $|c - a| \ge |b|$ . Prove that one of the numbers $a, b$, and $c$ is the sum of the other two.
2010 Albania Team Selection Test, 4
With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?
2010 Iran MO (2nd Round), 4
Let $P(x)=ax^3+bx^2+cx+d$ be a polynomial with real coefficients such that \[\min\{d,b+d\}> \max\{|{c}|,|{a+c}|\}\]
Prove that $P(x)$ do not have a real root in $[-1,1]$.
2000 Stanford Mathematics Tournament, 25
How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart?
2020 Iran MO (3rd Round), 2
let $a_1,a_2,...,a_n$,$b_1,b_2,...,b_n$,$c_1,c_2,...,c_n$ be real numbers. prove that
$$ \sum_{cyc}{ \sqrt{\sum_{i \in \{1,...,n\} }{ (3a_i-b_i-c_i)^2}}} \ge \sum_{cyc}{\sqrt{\sum_{i \in \{1,2,...,n\}}{a_i^2}}}$$
2017 Saudi Arabia JBMO TST, 1
Let $a,b,c>0$ and $a^2+b^2+c^2=3$ . Prove that $$ \frac{a(a-b^2)}{a+b^2}+\frac{b(b-c^2)}{b+c^2}+\frac{c(c-a^2)}{c+a^2}\ge 0.$$
2023 Regional Competition For Advanced Students, 1
Let $a$, $b$ and $c$ be real numbers with $0 \le a, b, c \le 2$. Prove that
$$(a - b)(b - c)(a- c) \le 2.$$
When does equality hold?
[i](Karl Czakler)[/i]
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]
2014 Indonesia MO Shortlist, A5
Determine the largest natural number $m$ such that for each non negative real numbers $a_1 \ge a_2 \ge ... \ge a_{2014} \ge 0$ , it is true that $$\frac{a_1+a_2+...+a_m}{m}\ge \sqrt{\frac{a_1^2+a_2^2+...+a_{2014}^2}{2014}}$$
2007 India National Olympiad, 1
In a triangle $ ABC$ right-angled at $ C$ , the median through $ B$ bisects the angle between $ BA$ and the bisector of $ \angle B$. Prove that
\[ \frac{5}{2} < \frac{AB}{BC} < 3\]
2006 All-Russian Olympiad, 1
Prove that $\sin\sqrt{x}<\sqrt{\sin x}$ for every real $x$ such that $0<x<\frac{\pi}{2}$.
1995 China National Olympiad, 3
Find the minimun value of $\sum_{i=1}^{10} \sum_{j=1}^{10} \sum_{k=1}^{10}|k(x+y-10i)(3x-6y-36j)(19x+95y-95k)|$ , where $x,y$ are integers.
2012 India IMO Training Camp, 2
Let $0<x<y<z<p$ be integers where $p$ is a prime. Prove that the following statements are equivalent:
$(a) x^3\equiv y^3\pmod p\text{ and }x^3\equiv z^3\pmod p$
$(b) y^2\equiv zx\pmod p\text{ and }z^2\equiv xy\pmod p$