Found problems: 6530
2005 Germany Team Selection Test, 3
Let $ABC$ be a triangle with orthocenter $H$, incenter $I$ and centroid $S$, and let $d$ be the diameter of the circumcircle of triangle $ABC$. Prove the inequality
\[9\cdot HS^2+4\left(AH\cdot AI+BH\cdot BI+CH\cdot CI\right)\geq 3d^2,\]
and determine when equality holds.
2013 Rioplatense Mathematical Olympiad, Level 3, 1
Let $a,b,c,d$ be real positive numbers such that $a^2+b^2+c^2+d^2 = 1$. Prove that $(1-a)(1-b)(1-c)(1-d) \geq abcd$.
2004 Singapore MO Open, 4
If $0 <x_1,x_2,...,x_n\le 1$, where $n \ge 1$, show that
$$\frac{x_1}{1+(n-1)x_1}+\frac{x_2}{1+(n-1)x_2}+...+\frac{x_n}{1+(n-1)x_n}\le 1$$
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
\[a^2+b^2+c^2\leq 2abc+1.\]
2011 Laurențiu Duican, 4
For $a, b, c>0,$ and $k\geq1,$ prove that
\[\frac{a^{k+1}}{b^k+c^k}+\frac{b^{k+1}}{c^k+a^k}+\frac{c^{k+1}}{a^k+b^k}\geq\frac{3}{2}\sqrt{\frac{a^{k+1}+b^{k+1}+c^{k+1}}{{a^{k-1}+b^{k-1}+c^{k-1}}}}\]
Author: MIHALY BENCZE
1986 French Mathematical Olympiad, Problem 3
(a) Prove or find a counter-example: For every two complex numbers $z,w$ the following inequality holds:
$$|z|+|w|\le|z+w|+|z-w|.$$(b) Prove that for all $z_1,z_2,z_3,z_4\in\mathbb C$:
$$\sum_{k=1}^4|z_k|\le\sum_{1\le i<j\le4}|z_i+z_j|.$$
2009 Cuba MO, 7
Let $x_1, x_2, ..., x_n$ be positive reals. Prove that
$$\sum_{k=1}^n \frac{x_k(2x_k - x_{k+1} - x_{k+2})}{x_{k+1} + x_{k+2}} \ge 0$$
In the sum, cyclic indices have been taken, that is, $x_{n+1} = x_1$ and $x_{n+2} = x_2$.
2003 Junior Macedonian Mathematical Olympiad, Problem 4
Let $x$, $y$ and $z$ be positive real numbers such that $x+y+z = 1$. Prove the inequality:
$$\frac{x^2}{1+y}+\frac{y^2}{1+z} +\frac{z^2}{1+x} \leq 1$$
2002 Federal Math Competition of S&M, Problem 2
Points $A_0,A_1,\ldots,A_{2k}$, in this order, divide a circumference into $2k+1$ equal arcs. Point $A_0$ is connected by chords to all the other points. These $2k$ chords divide the interior of the circle into $2k+1$ parts. These parts are alternately painted red and blue so that there are $k+1$ red and $k$ blue parts. Show that the blue area is larger than the red area.
2017 Hanoi Open Mathematics Competitions, 11
Let $ABC$ be an equilateral triangle, and let $P$ stand for an arbitrary point inside the triangle.
Is it true that $| \angle PAB - \angle PAC| \ge | \angle PBC - \angle PCB|$ ?
2015 China Team Selection Test, 1
Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that
\[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]
1997 Singapore Team Selection Test, 3
Suppose the numbers $a_0, a_1, a_2, ... , a_n$ satisfy the following conditions:
$a_0 =\frac12$, $a_{k+1} = a_k +\frac{1}{n}a_k^2$ for $k = 0, 1, ... , n - 1$.
Prove that $1 - \frac{1}{n}< a_n < 1$
2008 Mathcenter Contest, 1
Let $x,y,z$ be a positive real numbers. Prove that $$\frac {x}{\sqrt {x + y}} + \frac {y}{\sqrt {y + z}} + \frac { z}{\sqrt {z + x}}\geq\sqrt [4]{\frac {27(yz + zx + xy)}{4}}$$
[i](dektep)[/i]
2014 Contests, 3
For all $x,y,z\in \mathbb{R}\backslash \{1\}$, such that $xyz=1$, prove that \[ \frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1 \]
1986 USAMO, 1
$(\text{a})$ Do there exist 14 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 11$?
$(\text{b})$ Do there exist 21 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 13$?
1990 IMO Longlists, 42
Find $n$ points $p_1, p_2, \ldots, p_n$ on the circumference of a unit circle, such that $\sum_{1\leq i< j \leq n} p_i p_j$ is maximal.
2008 VJIMC, Problem 4
The numbers of the set $\{1,2,\ldots,n\}$ are colored with $6$ colors. Let
$$S:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have the same color}\}$$and
$$D:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have three different colors}\}.$$Prove that
$$|D|\le2|S|+\frac{n^2}2.$$
2011 Morocco National Olympiad, 2
Let $a,b,c$ be three postive real numbers such that $a+b+c=1$.
Prove that $9abc\leq ab+ac+bc < 1/4 +3abc$.
2006 AMC 12/AHSME, 24
Let $ S$ be the set of all points $ (x,y)$ in the coordinate plane such that $ 0\le x\le \frac \pi2$ and $ 0\le y\le \frac \pi2$. What is the area of the subset of $ S$ for which
\[ \sin^2 x \minus{} \sin x\sin y \plus{} \sin^2 y\le \frac 34?
\]$ \textbf{(A) } \frac {\pi^2}9 \qquad \textbf{(B) } \frac {\pi^2}8 \qquad \textbf{(C) } \frac {\pi^2}6\qquad \textbf{(D) } \frac {3\pi^2}{16} \qquad \textbf{(E) } \frac {2\pi^2}9$
2016 JBMO Shortlist, 2
Let $a,b,c $be positive real numbers.Prove that
$\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}$.
1999 Romania Team Selection Test, 11
Let $a,n$ be integer numbers, $p$ a prime number such that $p>|a|+1$. Prove that the polynomial $f(x)=x^n+ax+p$ cannot be represented as a product of two integer polynomials.
[i]Laurentiu Panaitopol[/i]
1984 Bulgaria National Olympiad, Problem 5
Let $0<x_i<1$ and $x_i+y_i=1$ for $i=1,2,\ldots,n$. Prove that
$$(1-x_1x_2\cdots x_n)^m+(1-y_1^m)(1-y_2^m)\cdots(1-y_n^m)>1$$for any natural numbers $m$ and $n$.
1968 IMO Shortlist, 9
Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality
\[abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.\]
Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.
1986 AMC 12/AHSME, 30
The number of real solutions $(x,y,z,w)$ of the simultaneous equations \[2y = x + \frac{17}{x},\quad 2z = y + \frac{17}{y},\quad 2w = z + \frac{17}{z},\quad 2x = w + \frac{17}{w}\] is
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 16 $
2012 AMC 10, 8
What is the sum of all integer solutions to $1<(x-2)^2<25$?
$ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 25 $