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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Lower bound on a product
anantmudgal09   11
N an hour ago by AshAuktober
Source: RMO Maharashtra and Goa 2016, P5
Let $x,y,z$ be non-negative real numbers such that $xyz=1$. Prove that $$(x^3+2y)(y^3+2z)(z^3+2x) \ge 27.$$
11 replies
anantmudgal09
Oct 11, 2016
AshAuktober
an hour ago
J
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Product of three positive reals is less than 1/8
kk108   47
N an hour ago by Adywastaken
Source: RMO Hyderabad 2016 , P2 .
Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$Prove that $abc \le \frac{1}{8}$.
47 replies
kk108
Oct 12, 2016
Adywastaken
an hour ago
J
H
Infinite Sum
P162008   1
N an hour ago by cazanova19921
Find $\Omega = \lim_{n \to \infty} \frac{1}{n^2} \left(\sum_{i + j + k + l = n} ijkl\right) \left(\sum_{i + j + k = n} ijk\right)^{-1}.$
1 reply
P162008
4 hours ago
cazanova19921
an hour ago
J
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4 variables with quadrilateral sides
mihaig   0
an hour ago
Source: VL
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$4\left(abc+abd+acd+bcd\right)\geq3\left(a+b+c+d\right)+4.$$
0 replies
mihaig
an hour ago
0 replies
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Incircle and length of segments
_Mirage_   3
N Feb 20, 2024 by AshAuktober
Source: ItaMO 2001 Problem 5
Let $ABC$ be a triangle and $\gamma$ the circle inscribed in $ABC$. The circle $\gamma$ is tangent to side $AB$ at the point $T$. Let $D$ be the point of $\gamma$ diametrically opposite to $T$, and $S$ the intersection point of the line through $C$ and $D$ with side $AB$.
Prove that $AT=SB$.
3 replies
_Mirage_
Feb 17, 2013
AshAuktober
Feb 20, 2024
J
Incircle and length of segments
G H J
Reveal topic tags
geometry
geometric transformation
homothety
incenter
L
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Source: ItaMO 2001 Problem 5
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_Mirage_
15 posts
_Mirage_
#1 h  Feb 17, 2013, 6:13 PM • 2 Y
Y by Adventure10, Mango247
Let $ABC$ be a triangle and $\gamma$ the circle inscribed in $ABC$. The circle $\gamma$ is tangent to side $AB$ at the point $T$. Let $D$ be the point of $\gamma$ diametrically opposite to $T$, and $S$ the intersection point of the line through $C$ and $D$ with side $AB$.
Prove that $AT=SB$.
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fermat007
1006 posts
fermat007
#2 h  Feb 17, 2013, 8:03 PM • 1 Y
Y by Adventure10
Consider the homothety centered at C that takes the incenter to the C-excenter. Note that the tangent at D to the incircle is parallel to AB, hence the homothety takes D to S (since the tangent at S to the excircle is also parallel to AB). It is well known that AT = SB in this case (they both equal s - a, where a is the semi-perimeter).
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jayme
9782 posts
jayme
#3 h  Oct 27, 2014, 11:37 AM • 2 Y
Y by Adventure10, Mango247
Dear Mathlinkers,
you can see
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=504812

Sincerely
Jean-Louis
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AshAuktober
996 posts
AshAuktober
#4 h  Feb 20, 2024, 12:53 PM
Y by
Solution sketch: Homothety taking incircle to excricle, show that $D -> T$, use length properties.
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