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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
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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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My problem that I could not find(NT)
Nuran2010   1
N 28 minutes ago by Nuran2010
Source: Own
While I was thinking on some other geometry problem, a NT problem came to my mind. Despite some tries(which were mostly order), I could not find a way to solve the problem. As I searched, this problem has never been posted before. Here is the problem.

Find all positive integers $a,b$ such that:
$a+b|2^{ab}+1$

Moreover, I wonder if there is a way to solve the question in this variant:

Find all positive integers $a,b,n$ such that:
$a+b|n^{ab}+1$
1 reply
+1 w
Nuran2010
Apr 24, 2025
Nuran2010
28 minutes ago
J
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FE over R+
jasperE3   10
N an hour ago by jasperE3
Source: Slovenia TST 2005 Test 1 Problem 2
Find all functions $f:\mathbb R^+\to\mathbb R^+$ such that for any $x,y>0$,
$$x^2\left(f(x)+f(y)\right)=(x+y)f\left(f(x)y\right).$$
10 replies
jasperE3
Apr 5, 2021
jasperE3
an hour ago
J
H
FE over R
jasperE3   7
N an hour ago by jasperE3
Source: Evan Chen
Find all functions $f:\mathbb R\to\mathbb R$ such that $f(x+f(y))+f(xy)=f(x+1)f(y+1)-1\forall x,y\in\mathbb R$.
7 replies
jasperE3
Apr 27, 2021
jasperE3
an hour ago
J
H
IMO ShortList 2002, geometry problem 7
orl   109
N an hour ago by Ilikeminecraft
Source: IMO ShortList 2002, geometry problem 7
The incircle $ \Omega$ of the acute-angled triangle $ ABC$ is tangent to its side $ BC$ at a point $ K$. Let $ AD$ be an altitude of triangle $ ABC$, and let $ M$ be the midpoint of the segment $ AD$. If $ N$ is the common point of the circle $ \Omega$ and the line $ KM$ (distinct from $ K$), then prove that the incircle $ \Omega$ and the circumcircle of triangle $ BCN$ are tangent to each other at the point $ N$.
109 replies
orl
Sep 28, 2004
Ilikeminecraft
an hour ago
J
No more topics!
H
J
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Inequality on area of quadrilaterals
Goutham   1
N Jun 27, 2011 by mathskid
Source: ToT 2003 SO-4
Each side of $1 \times 1$ square is a hypothenuse of an exterior right triangle. Let $A, B, C, D$ be the vertices of the right angles and $O_1, O_2, O_3, O_4$ be the centers of the incircles of these triangles. Prove that
$a)$ The area of quadrilateral $ABCD$ does not exceed $2$;
$b)$ The area of quadrilateral $O_1O_2O_3O_4$ does not exceed $1$.
1 reply
Goutham
Jun 26, 2011
mathskid
Jun 27, 2011
J
Inequality on area of quadrilaterals
G H J
Reveal topic tags
inequalities
geometry
geometry unsolved
L
G H BBookmark kLocked kLocked NReply
Source: ToT 2003 SO-4
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Goutham
3130 posts
Goutham
#1 h  Jun 26, 2011, 4:01 AM • 2 Y
Y by Adventure10, Mango247
Each side of $1 \times 1$ square is a hypothenuse of an exterior right triangle. Let $A, B, C, D$ be the vertices of the right angles and $O_1, O_2, O_3, O_4$ be the centers of the incircles of these triangles. Prove that
$a)$ The area of quadrilateral $ABCD$ does not exceed $2$;
$b)$ The area of quadrilateral $O_1O_2O_3O_4$ does not exceed $1$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mathskid
13 posts
mathskid
#2 h  Jun 27, 2011, 3:51 AM • 2 Y
Y by Adventure10, Mango247
http://www.math.toronto.edu/oz/turgor/archives/TT2003F_SOsolutions.pdf
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