Found problems: 6530
2012 Irish Math Olympiad, 5
(a) Show that if $x$ and $y$ are positive real numbers, then $$(x+y)^5\ge 12xy(x^3+y^3)$$
(b) Prove that the constant $12$ is the best possible. In other words, prove that for any $K>12$ there exist positive real numbers $x$ and $y$ such that $$(x+y)^5<Kxy(x^3+y^3)$$
2005 VJIMC, Problem 2
Let $(a_{i,j})^n_{i,j=1}$ be a real matrix such that $a_{i,i}=0$ for $i=1,2,\ldots,n$. Prove that there exists a set $\mathcal J\subset\{1,2,\ldots,n\}$ of indices such that
$$\sum_{\begin{smallmatrix}i\in\mathcal J\\j\notin\mathcal J\end{smallmatrix}}a_{i,j}+\sum_{\begin{smallmatrix}i\notin\mathcal J\\j\in\mathcal J\end{smallmatrix}}a_{i,j}\ge\frac12\sum_{i,j=1}^na_{i,j}.$$
2024 Polish Junior MO Finals, 3
Real numbers $a,b,c$ satisfy $a+b \ne 0$, $b+c \ne 0$ and $c+a \ne 0$. Show that
\[\left(\frac{a^2c}{a+b}+\frac{b^2a}{b+c}+\frac{c^2b}{c+a}\right) \cdot \left(\frac{b^2c}{a+b}+\frac{c^2a}{b+c}+\frac{a^2b}{c+a}\right) \ge 0.\]
2003 Estonia National Olympiad, 2
Prove that for all positive real numbers $a, b$, and $c$ , $\sqrt[3]{abc}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 2\sqrt3$.
When does the equality occur?
2022 Sharygin Geometry Olympiad, 6
The incircle and the excircle of triangle $ABC$ touch the side $AC$ at points $P$ and $Q$ respectively. The lines $BP$ and $BQ$ meet the circumcircle of triangle $ABC$ for the second time at points $P'$ and $Q'$ respectively.
Prove that
$$PP' > QQ'$$
2009 Kazakhstan National Olympiad, 4
Let $0<a_1 \leq a_2 \leq \cdots\leq a_n $ ($n \geq 3; n \in \mathbb{N}$) be $n$ real numbers. Prove the inequality
\[\frac{a_1^2}{a_2}+\frac{a_2^3}{a_3^2}+\cdots+\frac{a_n^{n+1}}{a_1^n} \geq a_1+a_2+\cdots+a_n\]
2005 District Olympiad, 1
a) Prove that if $x,y>0$ then
\[ \frac x{y^2} + \frac y{x^2} \geq \frac 1x + \frac 1y. \]
b) Prove that if $a,b,c$ are positive real numbers, then
\[ \frac {a+b}{c^2} + \frac {b+c}{a^2} + \frac {c+a}{b^2} \geq 2 \left( \frac 1a + \frac 1b + \frac 1c \right). \]
2007 Moldova Team Selection Test, 1
Let $ABC$ be a triangle and $M,N,P$ be the midpoints of sides $BC, CA, AB$. The lines $AM, BN, CP$ meet the circumcircle of $ABC$ in the points $A_{1}, B_{1}, C_{1}$. Show that the area of triangle $ABC$ is at most the sum of areas of triangles $BCA_{1}, CAB_{1}, ABC_{1}$.
2008 Iran Team Selection Test, 5
Let $a,b,c > 0$ and $ab+bc+ca = 1$. Prove that:
\[ \sqrt {a^3 + a} + \sqrt {b^3 + b} + \sqrt {c^3 + c}\geq2\sqrt {a + b + c}. \]
2015 European Mathematical Cup, 2
Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$\frac{a+b+c+3}{4}\geqslant \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}.$$
[i]Dimitar Trenevski[/i]
2016 IFYM, Sozopol, 3
Let $x\leq y\leq z$ be real numbers such that $x+y+z=12$, $x^2+y^2+z^2=54$. Prove that:
a) $x\leq 3$ and $z\geq 5$
b) $xy$, $yz$, $zx\in [9,25]$
2010 Putnam, B5
Is there a strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x?$
1998 Greece JBMO TST, 1
If $x,y,z > 0, k>2$ and $a=x+ky+kz, b=kx+y+kz, c=kx+ky+z$, show that $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} \ge \frac{3}{2k+1}$.
2009 Sharygin Geometry Olympiad, 2
Given nonisosceles triangle $ ABC$. Consider three segments passing through different vertices of this triangle and bisecting its perimeter. Are the lengths of these segments certainly different?
1990 IMO Longlists, 6
Let $S, T$ be the circumcenter and centroid of triangle $ABC$, respectively. $M$ is a point in the plane of triangle $ABC$ such that $90^\circ \leq \angle SMT < 180^\circ$. $A_1, B_1, C_1$ are the intersections of $AM, BM, CM$ with the circumcircle of triangle $ABC$ respectively. Prove that $MA_1 + MB_1 + MC_1 \geq MA + MB + MC.$
2007 ITest, 31
Let $x$ be the length of one side of a triangle and let $y$ be the height to that side. If $x+y=418$, find the maximum possible $\textit{integral value}$ of the area of the triangle.
1994 Czech And Slovak Olympiad IIIA, 6
Show that from any four distinct numbers lying in the interval $(0,1)$ one can choose two distinct numbers $a$ and $b$ such that
$$\sqrt{(1-a^2)(1-b^2)} > \frac{a}{2b}+\frac{b}{2a}-ab-\frac{1}{8ab} $$
2009 India National Olympiad, 5
Let $ ABC$ be an acute angled triangle and let $ H$ be its ortho centre. Let $ h_{max}$ denote the largest altitude of the triangle $ ABC$. Prove that:
$AH \plus{} BH \plus{} CH\leq2h_{max}$
2002 District Olympiad, 3
Let $ G $ be the center of mass of a triangle $ ABC, $ and the points $ M,N,P $ on the segments $ AB,BC, $ respectively, $ CA $ (excluding the extremities) such that
$$ \frac{AM}{MB} =\frac{BN}{NC} =\frac{CP}{PA} . $$
$ G_1,G_2,G_3 $ are the centers of mass of the triangles $ AMP, BMN, $ respectively, $ CNP. $ Pove that:
[b]a)[/b] The centers of mas of $ ABC $ and $ G_1G_2G_3 $ are the same.
[b]b)[/b] For any planar point $ D, $ the inequality
$$ 3\cdot DG< DG_1+DG_2+DG_3<DA+DB+DC $$
holds.
2006 JBMO ShortLists, 1
For an acute triangle $ ABC$ prove the inequality:
$ \sum_{cyclic} \frac{m_a^2}{\minus{}a^2\plus{}b^2\plus{}c^2}\ge \frac{9}{4}$ where $ m_a,m_b,m_c$ are lengths of corresponding medians.
2002 Tournament Of Towns, 6
In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.
2003 Alexandru Myller, 3
Let be a nonnegative integer $ n. $ Prove that there exists an increasing and finite sequence of positive real numbers, $
\left( a_k \right)_{0\le k\le n} , $ that satisfy the equality
$$ a_0/0! +a_1/1! +a_2/2! +\cdots +a_n/n! =1/n! , $$
and the inequality
$$ a_0+a_1+a_2+\cdots +a_n<\frac{3}{2^n} . $$
[i]Dorin Andrica[/i]
2018 Kazakhstan National Olympiad, 4
Prove that for all reas $a,b,c,d\in(0,1)$ we have $$\left(ab-cd\right)\left(ac+bd\right)\left(ad-bc\right)+\min{\left(a,b,c,d\right)} < 1.$$
1995 Romania Team Selection Test, 1
Let $a_1, a_2,...., a_n$ be distinct positive integers.
Prove that $(a_1^5 + ...+ a_n^5) + (a_1^7 + ...+ a_n^7) \ge 2(a_1^3 + ...+ a_n^3)^2$ and find the cases of equality.
2017 Swedish Mathematical Competition, 5
Find a costant $C$, such that $$ \frac{S}{ab+bc+ca}\le C$$ where $a,b,c$ are the side lengths of an arbitrary triangle, and $S$ is the area of the triangle.
(The maximal number of points is given for the best possible constant, with proof.)