Found problems: 6530
2012 India Regional Mathematical Olympiad, 8
Let $x, y, z$ be positive real numbers such that $2(xy + yz + zx) = xyz$.
Prove that $\frac{1}{(x-2)(y-2)(z-2)} + \frac{8}{(x+2)(y+2)(z+2)} \le \frac{1}{32}$
2018 Turkey MO (2nd Round), 4
In a triangle $ABC$, the bisector of the angle $A$ intersects the excircle that is tangential to side $[BC]$ at two points $D$ and $E$ such that $D\in [AE]$. Prove that,
$$
\frac{|AD|}{|AE|}\leq \frac{|BC|^2}{|DE|^2}.
$$
2006 Princeton University Math Competition, 9
Suppose $a,b,c$ are real numbers so that $a+b+c=15$ and $ab+ac+bc=27$. Find the range of values that may be obtained by the expression $abc$.
2011 Croatia Team Selection Test, 1
Let $a,b,c$ be positive reals such that $a+b+c=3$. Prove the inequality
\[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geq \frac{3}{2}.\]
2013 Polish MO Finals, 6
For each positive integer $n$ determine the maximum number of points in space creating the set $A$ which has the following properties:
$1)$ the coordinates of every point from the set $A$ are integers from the range $[0, n]$
$2)$ for each pair of different points $(x_1,x_2,x_3), (y_1,y_2,y_3)$ belonging to the set $A$ it is satisfied at least one of the following inequalities $x_1< y_1, x_2<y_2, x_3<y_3$ and at least one of the following inequalities $x_1>y_1, x_2>y_2,x_3>y_3$.
2013 Korea National Olympiad, 2
Let $ a, b, c>0 $ such that $ ab+bc+ca=3 $. Prove that
\[ \sum_{cyc} { \frac{ (a+b)^{3} }{ {(2(a+b)(a^2 + b^2))}^{\frac{1}{3}}} \ge 12 }\]
1990 Dutch Mathematical Olympiad, 1
Prove that for every integer $ n>1, 1 \cdot 3 \cdot 5 \cdot ... \cdot (2n\minus{}1)<n^n.$
2004 Rioplatense Mathematical Olympiad, Level 3, 3
In a convex hexagon $ABCDEF$, triangles $ACE$ and $BDF$ have the same circumradius $R$. If triangle $ACE$ has inradius $r$, prove that
\[ \text{Area}(ABCDEF)\le\frac{R}{r}\cdot\text{Area}(ACE).\]
2018 IFYM, Sozopol, 2
$x$, $y$, and $z$ are positive real numbers satisfying the equation $x+y+z=\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$.
Prove the following inequality:
$xy + yz + zx \geq 3$.
2019 China National Olympiad, 1
Let $a,b,c,d,e\geq -1$ and $a+b+c+d+e=5.$ Find the maximum and minimum value of $S=(a+b)(b+c)(c+d)(d+e)(e+a).$
2024 Belarus Team Selection Test, 3.1
Triangles $ABC$ and $DEF$, having a common incircle of radius $R$, intersect at points
$X_1, X_2, \ldots , X_6$ and form six triangles (see the figure below). Let $r_1, r_2,\ldots, r_6$ be the radii of the
inscribed circles of these triangles, and let $R_1, R_2, \ldots , R_6$ be the radii of the inscribed circles of the
triangles $AX_1F, FX_2B, BX_3D, DX_4C, CX_5E$ and $EX_6A$ respectively.
[img]https://i.ibb.co/BspgdHB/Image.jpg[/img]
Prove that \[ \sum_{i=1}^{6} \frac{1}{r_i} < \frac{6}{R}+\sum_{i=1}^{6} \frac{1}{R_i} \]
[i]U. Maksimenkau[/i]
2006 Grigore Moisil Urziceni, 3
Let be three positive real numbers $ x,y,z, $ whose product is $ 1. $ Prove that:
$$ \sum_{\text{cyc}} \frac{3}{\sqrt{1+x+xy}} \le \sqrt 3<3\sqrt 3\le \sum_{\text{cyc}} \sqrt{1+x+xy} $$
2012 Putnam, 2
Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P)>0$ with the following property: If a collection of $n$ balls whose volumes sum to $V$ contains the entire surface of $P,$ then $n>c(P)/V^2.$
1999 India Regional Mathematical Olympiad, 5
If $a,b,c$ are sides of a triangle, prove that \[ \frac{a}{c+a-b} + \frac{b}{a+b-c} + \frac{c}{b+c-a} \geq 3. \]
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
\[ \frac{a^2_1 \plus{} a^2_2 \minus{} a^2_3}{a_1 \plus{} a_2 \minus{} a_3} \plus{} \frac{a^2_2 \plus{} a^2_3 \minus{} a^2_4}{a_2 \plus{} a_3 \minus{} a_4} \plus{} \ldots \plus{} \frac{a^2_n \plus{} a^2_1 \minus{} a^2_2}{a_n \plus{} a_1 \minus{} a_2} \leq 2s \minus{} 2n.\]
2017 Austria Beginners' Competition, 1
The nonnegative real numbers $a$ and $b$ satisfy $a + b = 1$. Prove that:
$$\frac{1}{2} \leq \frac{a^3+b^3}{a^2+b^2} \leq 1$$
When do we have equality in the right inequality and when in the left inequality?
[i]Proposed by Walther Janous [/i]
2024 German National Olympiad, 6
Decide whether there exists a largest positive integer $n$ such that the inequality
\[\frac{\frac{a^2}{b}+\frac{b^2}{a}}{2} \ge \sqrt[n]{\frac{a^n+b^n}{2}}\]
holds for all positive real numbers $a$ and $b$. If such a largest positive integer $n$ exists, determine it.
2007 Irish Math Olympiad, 2
Suppose that $ a,b,$ and $ c$ are positive real numbers. Prove that:
$ \frac{a\plus{}b\plus{}c}{3} \le \sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{3}} \le \frac {\frac{ab}{c}\plus{}\frac{bc}{a}\plus{}\frac{ca}{b}}{3}$.
For each of the inequalities, find the conditions on $ a,b,$ and $ c$ such that equality holds.
2021 Vietnam TST, 1
Define the sequence $(a_n)$ as $a_1 = 1$, $a_{2n} = a_n$ and $a_{2n+1} = a_n + 1$ for all $n\geq 1$.
a) Find all positive integers $n$ such that $a_{kn} = a_n$ for all integers $1 \leq k \leq n$.
b) Prove that there exist infinitely many positive integers $m$ such that $a_{km} \geq a_m$ for all positive integers $k$.
2014 USAMTS Problems, 3:
Let $a_1,a_2,a_3,...$ be a sequence of positive real numbers such that:
(i) For all positive integers $m,n$, we have $a_{mn}=a_ma_n$
(ii) There exists a positive real number $B$ such that for all positive integers $m,n$ with $m<n$, we have $a_m < Ba_n$
Find all possible values of $\log_{2015}(a_{2015}) - \log_{2014}(a_{2014})$
1980 Poland - Second Round, 2
Prove that for any real numbers $ x_1, x_2, x_3, \ldots, x_n $ the inequality is true
$$ x_1x_2x_3\ldots x_n \leq \frac{x_1^2}{2} + \frac{x_2^4}{4} + \frac{x_3^8}{8} + \ldots + \frac{x_n^{2^ n}}{2^n} + \frac{1}{2^n}$$
2018 JBMO Shortlist, A3
Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$
1969 All Soviet Union Mathematical Olympiad, 128
Prove that for the arbitrary positive $a_1, a_2, ... , a_n$ the following inequality is held
$$\frac{a_1}{a_2+a_3}+\frac{a_2}{a_3+a_4}+....+\frac{a_{n-1}}{a_n+a_1}+\frac{a_n}{a_1+a_2}>\frac{n}{4}$$
2006 MOP Homework, 7
for real number $a,b,c$ in interval $ (0,1]$ prove that:
$\frac{a}{bc+1}+\frac{b}{ac+1}+\frac{c}{ab+1} \leq 2$
2012 Bogdan Stan, 3
$ \lim_{n\to\infty }\frac{1}{\sqrt[n]{n!}}\left\lfloor \log_5 \sum_{k=2}^{1+5^n} \sqrt[5^n]{k} \right\rfloor $
[i]Taclit Daniela Nadia[/i]