Found problems: 6530
2021 Thailand TST, 2
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\]
(A disk is assumed to contain its boundary.)
1986 Kurschak Competition, 2
Let $n>2$ be a positive integer. Find the largest value $h$ and the smallest value $H$ for which
\[h<{a_1\over a_1+a_2}+{a_2\over a_2+a_3}+\cdots+{a_n\over a_n+a_1}<H\]
holds for any positive reals $a_1,\dots,a_n$.
2005 Junior Balkan Team Selection Tests - Moldova, 6
Let $n$ be a nonzero natural number, and $x_1, x_2,..., x_n$ positive real numbers that $ \frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= n$. Find the minimum value of the expression $x_1 +\frac{x_2^2}{2}++\frac{x_3^3}{3}+...++\frac{x_n^n}{n}$.
2006 Costa Rica - Final Round, 2
If $ a$, $ b$, $ c$ are the sidelengths of a triangle, then prove that
$ \frac {3\left(a^4 \plus{} b^4 \plus{} c^4\right)}{\left(a^2 \plus{} b^2 \plus{} c^2\right)^2} \plus{} \frac {bc \plus{} ca \plus{} ab}{a^2 \plus{} b^2 \plus{} c^2}\geq 2$.
2004 Croatia National Olympiad, Problem 2
If $a,b,c$ are the sides and $\alpha,\beta,\gamma$ the corresponding angles of a triangle, prove the inequality
$$\frac{\cos\alpha}{a^3}+\frac{\cos\beta}{b^3}+\frac{\cos\gamma}{c^3}\ge\frac3{2abc}.$$
2025 Malaysian IMO Team Selection Test, 7
Given a real polynomial $P(x)=a_{2024}x^{2024}+\cdots+a_1x+a_0$ with degree $2024$, such that for all positive reals $b_1, b_2,\cdots, b_{2025}$ with product $1$, then; $$P(b_1)+P(b_2)+\cdots +P(b_{2025})\ge 0$$ Suppose there exist positive reals $c_1, c_2, \cdots, c_{2025}$ with product $1$, such that; $$P(c_1)+P(c_2)+ \cdots +P(c_{2025})=0$$ Is it possible that the values $c_1, c_2, \cdots, c_{2025}$ are all distinct?
[i]Proposed by Ivan Chan Kai Chin[/i]
2008 Mathcenter Contest, 3
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]
2009 Brazil National Olympiad, 3
Let $ n > 3$ be a fixed integer and $ x_1,x_2,\ldots, x_n$ be positive real numbers. Find, in terms of $ n$, all possible real values of
\[ {x_1\over x_n\plus{}x_1\plus{}x_2} \plus{} {x_2\over x_1\plus{}x_2\plus{}x_3} \plus{} {x_3\over x_2\plus{}x_3\plus{}x_4} \plus{} \cdots \plus{} {x_{n\minus{}1}\over x_{n\minus{}2}\plus{}x_{n\minus{}1}\plus{}x_n} \plus{} {x_n\over x_{n\minus{}1}\plus{}x_n\plus{}x_1}\]
2014 MMATHS, 4
Determine, with proof, the maximum and minimum among the numbers
$$\sqrt5 - \lfloor \sqrt5 \rfloor, 2\sqrt5 - \lfloor 2\sqrt5 \rfloor, 3\sqrt5 - \lfloor 3
\sqrt5\rfloor, ..., 2013\sqrt5 - \lfloor 2013\sqrt5\rfloor, 2014\sqrt5 - \lfloor 2014\sqrt5\rfloor $$
2003 China National Olympiad, 3
Given a positive integer $n$, find the least $\lambda>0$ such that for any $x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right)$, the condition $\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}}$ implies $\sum_{i=1}^{n}\cos x_i\le\lambda$.
[i]Huang Yumin[/i]
2013 Junior Balkan Team Selection Tests - Romania, 3
Find the minimum and the maximum value of the expression $\sqrt{4 -a^2} +\sqrt{4 -b^2} +\sqrt{4 -c^2}$
where $a,b, c$ are positive real numbers satisfying the condition $a^2 + b^2 + c^2=6$
2001 Grosman Memorial Mathematical Olympiad, 2
If $x_1,x_2,...,x_{2001}$ are real numbers with $0 \le x_n \le 1$ for $n = 1,2,...,2001$, find the maximum value of
$$\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n^2\right)-\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n\right)^2$$
Where is this maximum attained?
2009 German National Olympiad, 4
Let $a$ and $b$ be two fixed positive real numbers. Find all real numbers $x$, such that inequality holds $$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$$
2010 Contests, 3
Let $x_1, x_2, \ldots ,x_n(n\ge 2)$ be real numbers greater than $1$. Suppose that $|x_i-x_{i+1}|<1$ for $i=1, 2,\ldots ,n-1$. Prove that
\[\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots +\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}<2n-1\]
2019 Hong Kong TST, 3
Find the maximal value of
\[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\]
where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$.
[i]Proposed by Evan Chen, Taiwan[/i]
2019 India IMO Training Camp, P2
Let $ABC$ be a triangle with $\angle A=\angle C=30^{\circ}.$ Points $D,E,F$ are chosen on the sides $AB,BC,CA$ respectively so that $\angle BFD=\angle BFE=60^{\circ}.$ Let $p$ and $p_1$ be the perimeters of the triangles $ABC$ and $DEF$, respectively. Prove that $p\le 2p_1.$
1973 Poland - Second Round, 1
Prove that if positive numbers $ x, y, z $ satisfy the inequality
$$
\frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1,$$
then they are the lengths of the sides of a certain triangle.
2004 Vietnam National Olympiad, 2
Let $x$, $y$, $z$ be positive reals satisfying $\left(x+y+z\right)^{3}=32xyz$
Find the minimum and the maximum of $P=\frac{x^{4}+y^{4}+z^{4}}{\left(x+y+z\right)^{4}}$
1982 Vietnam National Olympiad, 2
Let $p$ be a positive integer and $q, z$ be real numbers with $0\le q\le 1$ and $q^{p+1}\le z\le 1$. Prove that
\[\prod_{k=1}^p \left|\frac{z - q^k}{z + q^k}\right| \le\prod_{k=1}^p \left|\frac{1 - q^k}{1 + q^k}\right|.\]
1987 IMO Longlists, 67
If $a, b, c, d$ are real numbers such that $a^2 + b^2 + c^2 + d^2 \leq 1$, find the maximum of the expression
\[(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4.\]
2018 Romania National Olympiad, 3
Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$
$\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$
$\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$
[i]Nicolae Bourbacut[/i]
2007 Turkey Team Selection Test, 3
Let $a, b, c$ be positive reals such that their sum is $1$. Prove that \[\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.\]
2014 Bosnia Herzegovina Team Selection Test, 2
Let $a$ ,$b$ and $c$ be distinct real numbers.
$a)$ Determine value of $ \frac{1+ab }{a-b} \cdot \frac{1+bc }{b-c} + \frac{1+bc }{b-c} \cdot \frac{1+ca }{c-a} + \frac{1+ca }{c-a} \cdot \frac{1+ab}{a-b} $
$b)$ Determine value of $ \frac{1-ab }{a-b} \cdot \frac{1-bc }{b-c} + \frac{1-bc }{b-c} \cdot \frac{1-ca }{c-a} + \frac{1-ca }{c-a} \cdot \frac{1-ab}{a-b} $
$c)$ Prove the following ineqaulity
$ \frac{1+a^2b^2 }{(a-b)^2} + \frac{1+b^2c^2 }{(b-c)^2} + \frac{1+c^2a^2 }{(c-a)^2} \geq \frac{3}{2} $
When does eqaulity holds?
1996 Tuymaada Olympiad, 1
Prove the inequality $x_1y_1+x_2y_2+x_2y_1+2x_2y_2\le 1996$
if $x_1^2+2x_1x_2+2x_2^2\le 998$ and $y_1^2+2y_1y_2+2y_2^2\le 3992$.
1965 AMC 12/AHSME, 23
If we write $ |x^2 \minus{} 4| < N$ for all $ x$ such that $ |x \minus{} 2| < 0.01$, the smallest value we can use for $ N$ is:
$ \textbf{(A)}\ .0301 \qquad \textbf{(B)}\ .0349 \qquad \textbf{(C)}\ .0399 \qquad \textbf{(D)}\ .0401 \qquad \textbf{(E)}\ .0499 \qquad$