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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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Show that this ceil is even
YLG_123   4
N 3 minutes ago by AshAuktober
Source: Brazil EGMO TST1 2023 #2
Let $p$ and $q$ be distinct odd primes. Show that
$$\bigg\lceil \dfrac{p^q+q^p-pq+1}{pq} \bigg\rceil$$is even.
4 replies
YLG_123
Jan 29, 2024
AshAuktober
3 minutes ago
J
H
geo ineq with areas, altitude wrt hypotenuse
parmenides51   2
N 14 minutes ago by AshAuktober
Source: 1998 Germany R2 11-13 p1 https://artofproblemsolving.com/community/c3208025_
As is well known, in every right-angled triangle the two sections into which the hypotenuse is divided by the base point of the height $h$ perpendicular to it are referred to briefly as the two hypotenuse sections. Prove that in every right-angled triangle the sum of the areas of the squares over the two hypotenuse sections is greater than or equal to twice the area of the square above the height $h$.
2 replies
parmenides51
Sep 23, 2024
AshAuktober
14 minutes ago
J
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Old or new
sqing   2
N 16 minutes ago by SunnyEvan
Source: ZDSX 2025 Q845
Let $ a,b,c>0 $ and $ a^2+b^2+c^2+ abc=4 $ . Prove that $$1\leq \frac{1}{2a+bc }+ \frac{1}{2b+ca }+ \frac{1}{2c+ab }\leq \frac{1}{\sqrt{abc} }$$
2 replies
+2 w
sqing
Today at 4:22 AM
SunnyEvan
16 minutes ago
J
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D1010 : How it is possible ?
Dattier   16
N 36 minutes ago by Dattier
Source: les dattes à Dattier
Is it true that$$\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0 $$?

A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975

B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902
16 replies
Dattier
Mar 10, 2025
Dattier
36 minutes ago
J
No more topics!
H
J
H
Geometry
Captainscrubz   2
N Apr 6, 2025 by MrdiuryPeter
Source: Own
Let $D$ be any point on side $BC$ of $\triangle ABC$ .Let $E$ and $F$ be points on $AB$ and $AC$ such that $EB=ED$ and $FD=FC$ respectively. Prove that the locus of circumcenter of $(DEF)$ is a line.
Prove without using moving points :D
2 replies
Captainscrubz
Apr 6, 2025
MrdiuryPeter
Apr 6, 2025
J
Geometry
G H J
Reveal topic tags
Circumcenter
geometry
circumcircle
L
G H BBookmark kLocked kLocked NReply
Source: Own
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Captainscrubz
51 posts
Captainscrubz
#1 h  Apr 6, 2025, 9:07 AM • 1 Y
Y by MrdiuryPeter
Let $D$ be any point on side $BC$ of $\triangle ABC$ .Let $E$ and $F$ be points on $AB$ and $AC$ such that $EB=ED$ and $FD=FC$ respectively. Prove that the locus of circumcenter of $(DEF)$ is a line.
Prove without using moving points :D
This post has been edited 1 time. Last edited by Captainscrubz, Apr 6, 2025, 9:08 AM
Z K Y
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aidenkim119
23 posts
aidenkim119
#2 h  Apr 6, 2025, 10:38 AM
Y by
Trivial by mmp and steiner conic
Z K Y
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MrdiuryPeter
3 posts
MrdiuryPeter
#3 h  Apr 6, 2025, 11:02 AM
Y by
Any synthetic proof?
Z K Y
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