Found problems: 6530
2003 Turkey MO (2nd round), 2
Let $ABCD$ be a convex quadrilateral and $K,L,M,N$ be points on $[AB],[BC],[CD],[DA]$, respectively. Show that,
\[
\sqrt[3]{s_{1}}+\sqrt[3]{s_{2}}+\sqrt[3]{s_{3}}+\sqrt[3]{s_{4}}\leq 2\sqrt[3]{s}
\]
where $s_1=\text{Area}(AKN)$, $s_2=\text{Area}(BKL)$, $s_3=\text{Area}(CLM)$, $s_4=\text{Area}(DMN)$ and $s=\text{Area}(ABCD)$.
2017 Pan-African Shortlist, I?
Let $x,y$, and $z$ be positive real numbers such that $xy+yz+zx=3xyz$. Prove that $$x^2y+y^2z+z^2x \geq 2(x+y+z)-3.$$
In which cases do we have equality?
2003 Manhattan Mathematical Olympiad, 3
Assume $a,b,c$ are positive numbers, such that
\[ a(1-b) = b(1-c) = c(1-a) = \dfrac14 \]
Prove that $a=b=c$.
2014 Switzerland - Final Round, 6
Let $a,b,c\in \mathbb{R}_{\ge 0}$ satisfy $a+b+c=1$. Prove the inequality :
\[ \frac{3-b}{a+1}+\frac{a+1}{b+1}+\frac{b+1}{c+1}\ge 4 \]
2010 Federal Competition For Advanced Students, P2, 1
Show that $\frac{(x - y)^7 + (y - z)^7 + (z - x)^7 - (x - y)(y - z)(z - x) ((x - y)^4 + (y - z)^4 + (z - x)^4)}
{(x - y)^5 + (y - z)^5 + (z - x)^5} \ge 3$
holds for all triples of distinct integers $x, y, z$. When does equality hold?
1999 German National Olympiad, 2
Determine all real numbers $x$ for which $1+\frac{x}{2} -\frac{x^2}{8} \le \sqrt{1+x} \le 1+\frac{x}{2}$
2014 Hanoi Open Mathematics Competitions, 15
Let $a_1,a_2,...,a_9 \ge - 1$ and $a^3_1+a^3_2+...+a^3_9= 0$.
Determine the maximal value of $M = a_1 + a_2 + ... + a_9$.
2024 Brazil EGMO TST, 1
Decide whether there exists a positive real number \( a < 1 \) such that, for any positive real numbers \( x \) and \( y \), the inequality
\[
\frac{2xy^2}{x^2 + y^2} \leq (1 - a)x + ay
\]
holds true.
2014 Contests, 1
Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that
$ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$
(Proposed by Gerhard Woeginger, Austria)
1995 India National Olympiad, 5
Let $n \geq 2$. Let $a_1 , a_2 , a_3 , \ldots a_n$ be $n$ real numbers all less than $1$ and such that $|a_k - a_{k+1} | < 1$ for $1 \leq k \leq n-1$. Show that \[ \dfrac{a_1}{a_2} + \dfrac{a_2}{a_3} + \dfrac{a_3}{a_4} + \ldots + \dfrac{a_{n-1}}{a_n} + \dfrac{a_n}{a_1} < 2 n - 1 . \]
2010 Postal Coaching, 4
$\triangle ABC$ has semiperimeter $s$ and area $F$ . A square $P QRS$ with side length $x$ is inscribed in $ABC$ with $P$ and $Q$ on $BC$, $R$ on $AC$, and $S$ on $AB$. Similarly, $y$ and $z$ are the sides of squares two vertices of which lie on $AC$ and $AB$, respectively. Prove that
\[\frac 1x +\frac 1y + \frac 1z \le \frac{s(2+\sqrt3)}{2F}\]
2008 Grigore Moisil Intercounty, 3
Let $ f[0,\infty )\longrightarrow\mathbb{R} $ be a convex and differentiable function with $ f(0)=0. $
[b]a)[/b] Prove that $ \int_0^x f(t)dt\le \frac{x^2}{2}f'(x) , $ for any nonnegative $ x. $
[b]b)[/b] Determine $ f $ if the above inequality is actually an equality.
[i]Dorin Andrica[/i] and [i]Mihai Piticari[/i]
1997 Taiwan National Olympiad, 8
Let $O$ be the circumcenter and $R$ be the circumradius of an acute triangle $ABC$. Let $AO$ meet the circumcircle of $OBC$ again at $D$, $BO$ meet the circumcircle of $OCA$ again at $E$, and $CO$ meet the circumcircle of $OAB$ again at $F$. Show that $OD.OE.OF\geq 8R^{3}$.
2011 National Olympiad First Round, 3
How many positive integer $n$ are there satisfying the inequality $1+\sqrt{n^2-9n+20} > \sqrt{n^2-7n+12}$ ?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$
2017 China Team Selection Test, 1
Let $n \geq 4$ be a natural and let $x_1,\ldots,x_n$ be non-negative reals such that $x_1 + \cdots + x_n = 1$. Determine the maximum value of $x_1x_2x_3 + x_2x_3x_4 + \cdots + x_nx_1x_2$.
2010 Peru IMO TST, 7
Let $a, b, c$ be positive real numbers such that $a + b + c = 1.$ Prove that $$ \displaystyle{\frac{1}{a + b}+\frac{1}{b + c}+\frac{1}{c + a}+ 3(ab + bc + ca) \geq \frac{11}{2}.}$$
1966 IMO, 6
Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$.
[i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that
$ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$,
where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.
2024 Germany Team Selection Test, 3
Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that
[list=disc]
[*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and
[*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$.
[/list]
Prove that $\max(a_1,a_{2023})\ge 507$.
1992 China Team Selection Test, 3
For any $n,T \geq 2, n, T \in \mathbb{N}$, find all $a \in \mathbb{N}$ such that $\forall a_i > 0, i = 1, 2, \ldots, n$, we have
\[\sum^n_{k=1} \frac{a \cdot k + \frac{a^2}{4}}{S_k} < T^2 \cdot \sum^n_{k=1} \frac{1}{a_k},\] where $S_k = \sum^k_{i=1} a_i.$
2012 ISI Entrance Examination, 6
[b]i)[/b] Let $0<a<b$.Prove that amongst all triangles having base $a$ and perimeter $a+b$ the triangle having two sides(other than the base) equal to $\frac {b}{2}$ has the maximum area.
[b]ii)[/b]Using $i)$ or otherwise, prove that amongst all quadrilateral having give perimeter the square has the maximum area.
2000 Romania National Olympiad, 1
For the real numbers $a, b, c, d$, the following inequalities hold:
$$a + b + c \le 3d, \,\,\, b + c + d \le 3a, \,\,\,c + d + a \le 3b, \,\,\,d + a + b\le 3c.$$
Compare the numbers $a, b, c, d$.
2016 Bosnia And Herzegovina - Regional Olympiad, 1
Let $a$ and $b$ be real numbers bigger than $1$. Find maximal value of $c \in \mathbb{R}$ such that $$\frac{1}{3+\log _{a} b}+\frac{1}{3+\log _{b} a} \geq c$$
2012 Bosnia and Herzegovina Junior BMO TST, 4
If $a$, $b$ and $c$ are sides of triangle which perimeter equals $1$, prove that:
$a^2+b^2+c^2+4abc<\frac{1}{2}$
2000 Moldova National Olympiad, Problem 7
For any real number $a$, prove the inequality:
$$\left(a^3+a^2+3\right)^2>4a^3(a-1)^2.$$
1998 Baltic Way, 11
If $a,b,c$ be the lengths of the sides of a triangle. Let $R$ denote its circumradius. Prove that
\[ R\ge \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}\]
When does equality hold?