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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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0 replies
jlacosta
Apr 2, 2025
0 replies
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Problem 6
blug   1
N an hour ago by atdaotlohbh
Source: Polish Math Olympiad 2025 Finals P6
A strictly decreasing function $f:(0, \infty)\Rightarrow (0, \infty)$ attaining all positive values and positive numbers $a_1\ne b_1$ are given. Numbers $a_2, b_2, a_3, b_3, ...$ satisfy
$$a_{n+1}=a_n+f(b_n),\;\;\;\;\;\;\;b_{n+1}=b_n+f(a_n)$$for every $n\geq 1$. Prove that there exists a positive integer $n$ satisfying $|a_n-b_n| >2025$.
1 reply
blug
Yesterday at 12:17 PM
atdaotlohbh
an hour ago
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Regarding Maaths olympiad prepration
omega2007   13
N an hour ago by omega2007
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compiled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your perspective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
13 replies
omega2007
Yesterday at 3:13 PM
omega2007
an hour ago
J
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square root problem that involves geometry
kjhgyuio   5
N an hour ago by kjhgyuio
If x is a nonnegative real number , find the minimum value of √x^2+4 + √x^2 -24x +153

5 replies
kjhgyuio
Today at 3:56 AM
kjhgyuio
an hour ago
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PoP+Parallel
Solilin   3
N an hour ago by ND_
Source: Titu Andreescu, Lemmas in Olympiad Geometry
Let ABC be a triangle and let D, E, F be the feet of the altitudes, with D on BC, E on CA, and F on AB. Let the parallel through D to EF meet AB at X and AC at Y. Let T be the intersection of EF with BC and let M be the midpoint of side BC. Prove that the points T, M, X, Y are concyclic.
3 replies
Solilin
5 hours ago
ND_
an hour ago
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No more topics!
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A,P,M,N are concyclic
MRF2017   3
N Nov 23, 2016 by hectorraul
Source: Bulgaria 2016,P 5
Let $ABC$ be an isosceles triangle with $AC=BC$.Take a point $D$ on the extention of $AC$ from $C$ such that $AC> CD$.The angle bisector of $\angle BCD$ intersect with $BD$ at $N$ and $M$ is midpoint of $BD$.The tangent from $M$ to circumcircle of triangle $AMD$ intersect with $BC$ at $P$.Prove that $A,P,M,N$ are concyclic.
3 replies
MRF2017
Sep 25, 2016
hectorraul
Nov 23, 2016
J
A,P,M,N are concyclic
G H J
Reveal topic tags
geometry
angle bisector
circumcircle
cyclic quadrilateral
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Source: Bulgaria 2016,P 5
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MRF2017
237 posts
MRF2017
#1 h  Sep 25, 2016, 8:10 AM • 1 Y
Y by Adventure10
Let $ABC$ be an isosceles triangle with $AC=BC$.Take a point $D$ on the extention of $AC$ from $C$ such that $AC> CD$.The angle bisector of $\angle BCD$ intersect with $BD$ at $N$ and $M$ is midpoint of $BD$.The tangent from $M$ to circumcircle of triangle $AMD$ intersect with $BC$ at $P$.Prove that $A,P,M,N$ are concyclic.
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Stranger8
238 posts
Stranger8
#2 h  Nov 22, 2016, 7:36 AM • 2 Y
Y by Adventure10, Mango247
any idea please :)
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lminsl
544 posts
lminsl
#3 h  Nov 23, 2016, 7:44 AM • 2 Y
Y by Adventure10, Mango247
Bumping this :)
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hectorraul
361 posts
hectorraul
#4 h  Nov 23, 2016, 11:23 AM • 5 Y
Y by SHARKYBOY, qweDota, onlygeo, thczarif, Adventure10
cool problem, I will sketch my solution:

Let $L$ be the midpoint of $AB$, $O$ the intersection of $BD$ and the perpendicular bisector of $AB$ and $E$ the intersection of $OA$ and $ML$,

1- $AC\parallel ME, \Rightarrow \angle AEM=\angle OAC=\angle OBC$, by angle chasing $\angle AME=\angle PMB$ and we get $\triangle MAE\sim \triangle MPB$
2- spiral similarity centered at $M$ shows that $\triangle MAP \sim \triangle MEB\Rightarrow \angle MAP = \angle MEB $
3- $O$ is the center of the homotecy sending $(A,C,N)$ to $(E,L,B)$, then $\angle CAN= \angle LEB$, then $\angle CAN= \angle MAP$
4-this later fact implies $\angle NAP= \angle DAM =\angle AML= \angle PMB$, and we are done.
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