Search found 136 matches
- Wed Jul 15, 2015 1:45 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2015 - Problem 1
- Replies: 1
- Views: 2976
IMO 2015 - Problem 1
We say that a finite set $\mathcal{S}$ of points in the plane is balanced if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is centre-free if for any three different points $A$, $B$ and $C$ in $\mathcal{S...
- Wed May 06, 2015 5:30 pm
- Forum: Geometry
- Topic: Geometric Inequality
- Replies: 1
- Views: 2885
Geometric Inequality
Let $P$ be a point in the interior of $\triangle ABC$ with circumradius $R$. Prove that \[\dfrac{AP}{BC^2}+\dfrac{BP}{CA^2}+\dfrac{CP}{AB^2}\ge \dfrac 1 R.\]
Source: CleverMath homepage.
Those who miss the old Brilliant.org, try CleverMath.
Source: CleverMath homepage.
Those who miss the old Brilliant.org, try CleverMath.
- Mon May 04, 2015 9:43 pm
- Forum: Geometry
- Topic: Inscribed-Quad in an Excribed-Quad
- Replies: 5
- Views: 5259
Re: Inscribed-Quad in an Excribed-Quad
http://s22.postimg.org/e1qxxkkn5/img.png Points $W,X$ are projections of $B$ and $Y,Z$ are projections of $D$ on $IA$ and $IC$ respectively. Multiple configs can occur, so angles are directed mod $\pi$. Since $\measuredangle IWB=\measuredangle IXB=\pi/2$ quad $IWBX$ is cyclic. Similarly quad $IZDY$...
- Mon May 04, 2015 1:28 am
- Forum: Geometry
- Topic: Inscribed-Quad in an Excribed-Quad
- Replies: 5
- Views: 5259
Re: Inscribed-Quad in an Excribed-Quad
You sure about the statement?
- Sun May 03, 2015 1:23 am
- Forum: Geometry
- Topic: triangular inequality [sides and area]
- Replies: 3
- Views: 3817
Re: triangular inequality [sides and area]
Weitzenbock's inequality has like more than ten/eleven different proofs. When I first confronted it, I instead found a stronger one which, surprisingly, is easier to prove. For any $\triangle ABC$ with side lengths $a,b,c$ and area $\Delta$ we have $\displaystyle\sum_{\text{cyc}} a^2-\sum_{\text{cyc...
- Thu Apr 30, 2015 11:26 pm
- Forum: Algebra
- Topic: Inequality (sin, r and s)
- Replies: 2
- Views: 3298
Re: Inequality (sin, r and s)
The inequality simplifies to \[\dfrac{1}{\sqrt{a}}+\dfrac{1}{\sqrt{b}}+\dfrac{1}{\sqrt{c}}\le \dfrac{a+b+c}{\sqrt{abc}}\Rightarrow \sqrt{ab}+\sqrt{bc}+\sqrt{ca}\le a+b+c\] after making the substitutions $2\sin A=\dfrac a R$ etc, $r=\dfrac{abc}{4sR}$, and $2s=a+b+c$. This is a standard AM-GM result w...
- Wed Mar 11, 2015 11:27 pm
- Forum: Geometry
- Topic: Perpendicular from excenter and midpoint of altitude
- Replies: 1
- Views: 2388
Re: Perpendicular from excenter and midpoint of altitude
$1.$ Draw the diameter $GH$ of the incircle perpendicular to $AC$.
$2.$ Homothety.
$2.$ Homothety.
Re: Minimize!
My solution. \[\begin{eqnarray} 4S=\sum_{\text{cyc}}\dfrac{4ab+4}{(a+b)^2}&\ge&\sum_{\text{cyc}}\dfrac{4ab+(a+b+c)^2+(a^2+b^2+c^2)}{(a+b)^2} \\ &=& \sum_{\text{cyc}}\dfrac{2(a+b)^2+2(ab+bc+ca+c^2)}{(a+b)^2} \\ &=& 6+2\sum_{\text{cyc}}\dfrac{(b+c)(c+a)}{(a+b)^2} \\ &\ge & 6+6\sqrt[3]{\prod_{\text{cyc...
Re: Minimize!
It means summing all versions of the inner expression with variables cyclically permuted. I've edited the statement for clarity.
Minimize!
Let $a,b,c>0$ satisfy $\left(a+b+c\right)^2+\left(a^2+b^2+c^2\right)\le 4$. Find the minimum value of \