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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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Functional Equation with Surjectivity
spherical_charlie   2
N 10 minutes ago by jasperE3
Can someone help me?
Find all surjective functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the following equation for all real numbers \( x, y \):
\[ f\big(f(x - y)\big) = f(x) - f(y). \]
2 replies
spherical_charlie
2 hours ago
jasperE3
10 minutes ago
J
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Functional Equation!
EthanWYX2009   2
N 30 minutes ago by SanFangMath
Source: 2025 TST 24
Find all functions $f:\mathbb Z\to\mathbb Z$ such that $f$ is unbounded and
\[2f(m)f(n)-f(n-m)-1\]is a perfect square for all integer $m,n.$
2 replies
1 viewing
EthanWYX2009
Mar 29, 2025
SanFangMath
30 minutes ago
J
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IMO 2011 Problem 1
Amir Hossein   101
N an hour ago by InterLoop
Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i +a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$.

Proposed by Fernando Campos, Mexico
101 replies
Amir Hossein
Jul 18, 2011
InterLoop
an hour ago
J
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Another NT FE
nukelauncher   61
N an hour ago by InterLoop
Source: ISL 2019 N4
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
61 replies
nukelauncher
Sep 22, 2020
InterLoop
an hour ago
J
No more topics!
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J
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Three vertices consecutives and its circumcircle
Johann Peter Dirichlet   5
N Nov 29, 2013 by GlassBead
Source: Problem 2 - Brazil Math Olympiad 1994
Given any convex polygon, show that there are three consecutive vertices such that the polygon lies inside the circle through them.
5 replies
Johann Peter Dirichlet
Mar 17, 2006
GlassBead
Nov 29, 2013
J
Three vertices consecutives and its circumcircle
G H J
Reveal topic tags
geometry
circumcircle
trigonometry
combinatorial geometry
trig identities
Law of Sines
geometry proposed
L
G H BBookmark kLocked kLocked NReply
Source: Problem 2 - Brazil Math Olympiad 1994
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Johann Peter Dirichlet
376 posts
Johann Peter Dirichlet
#1 h  Mar 17, 2006, 5:00 AM • 2 Y
Y by Adventure10, Mango247
Given any convex polygon, show that there are three consecutive vertices such that the polygon lies inside the circle through them.
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Atlas_ha78
69 posts
Atlas_ha78
#2 h  Nov 19, 2013, 6:08 PM • 2 Y
Y by Adventure10, Mango247
I think we can solve it with extremal.
Any ideas?
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fmasroor
779 posts
fmasroor
#3 h  Nov 27, 2013, 8:04 PM • 2 Y
Y by Adventure10, Mango247
Well look at the convex hull, make a really really really really big circle around it and gradually shrink the circle down. Would that work?
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GlassBead
1583 posts
GlassBead
#4 h  Nov 29, 2013, 7:45 PM • 1 Y
Y by Adventure10
This is a consequence of Bose's theorem for polygons (a somewhat stronger version of the discrete four vertex theorem). The idea is very similar to the idea that every finite set of points has a Delaunay triangulation (i.e. can be triangulated by triangles whose circumcircles do not contain any other vertices).
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infiniteturtle
1131 posts
infiniteturtle
#5 h  Nov 29, 2013, 10:22 PM • 1 Y
Y by Adventure10
Also posted here. However, I believe the last sentence in the proof was never actually resolved...
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GlassBead
1583 posts
GlassBead
#6 h  Nov 29, 2013, 10:58 PM • 1 Y
Y by Adventure10
The last statement (of each solution) seems to be inspired by extended law of sines, though I'm not sure if the arguments are quite correct.
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