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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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geometry
JetFire008   2
N 14 minutes ago by sunken rock
Given four concyclic points. For each subset of three points take the incenter. Show that the four incentres form a rectangle.
2 replies
JetFire008
Yesterday at 4:14 PM
sunken rock
14 minutes ago
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volume 9f a pentagonal base pyramid circumscribed around a right circular cone
FOL   1
N 23 minutes ago by Mathzeus1024
A pentagonal base pyramid is circumscribed around a right circular cone, whose height is equal to the radius of the base. The total surface area of the pyramid is d times greater than that of the cone. Find the volume of the pyramid if the lateral surface area of the cone is equal to $\pi\sqrt{2}$.
1 reply
1 viewing
FOL
Jul 22, 2023
Mathzeus1024
23 minutes ago
J
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old problem from an open contest
Darealzolt   1
N 4 hours ago by alexheinis
Given that $a, b \in \mathbb{R}$ satisfy
\[ a + \frac{1}{a + 2015} = b - 4030 + \frac{1}{b - 2015} \]and $|a - b| > 5000$. Determine the value of
\[ \frac{ab}{2015} - a + b. \]
1 reply
Darealzolt
Yesterday at 1:41 AM
alexheinis
4 hours ago
J
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A Collection of Good Problems from my end
SomeonecoolLovesMaths   12
N 5 hours ago by SomeonecoolLovesMaths
This is a collection of good problems and my respective attempts to solve them. I would like to encourage everyone to post their solutions to these problems, if any. This will not only help others verify theirs but also perhaps bring forward a different approach to the problem. I will constantly try to update the pool of questions.

The difficulty level of these questions vary from AMC 10 to AIME. (Although the main pool of questions were prepared as a mock test for IOQM over the years)

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5
12 replies
SomeonecoolLovesMaths
May 4, 2025
SomeonecoolLovesMaths
5 hours ago
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No more topics!
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Three variables inequality
Headhunter   6
N Apr 30, 2025 by lbh_qys
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
Prove that at least one of $(a-b)^{2}$, $(b-c)^{2}$, $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$.

I assume that all are greater than it, but can't go more.
6 replies
Headhunter
Apr 20, 2025
lbh_qys
Apr 30, 2025
J
Three variables inequality
G H J
Reveal topic tags
inequalities
algebra
proof-writing
L
G H BBookmark kLocked kLocked NReply
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Headhunter
1963 posts
Headhunter
#1 h  Apr 20, 2025, 6:58 AM
Y by
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
Prove that at least one of $(a-b)^{2}$, $(b-c)^{2}$, $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$.

I assume that all are greater than it, but can't go more.
Z K Y
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programjames1
3046 posts
programjames1
#2 h  Apr 20, 2025, 7:20 AM
Y by
Can you give the reference? I think this is from a USAJMO contest in the 2010s.

EDIT: I was thinking of USA(J)MO 2018 #1 (#2) which can be rearranged to a similar inequality.
This post has been edited 2 times. Last edited by programjames1, Apr 20, 2025, 7:23 AM
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Headhunter
1963 posts
Headhunter
#3 h  Apr 20, 2025, 9:01 AM
Y by
I guess that this problem is from chinese materials at 1990~2004. but I'm not sure. Thanks.
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lbh_qys
558 posts
lbh_qys
#4 h  Apr 21, 2025, 3:04 AM
Y by
Hint
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lbh_qys
558 posts
lbh_qys
#5 h  Apr 21, 2025, 3:18 AM • 2 Y
Y by programjames1, spy27
another solution
This post has been edited 1 time. Last edited by lbh_qys, Apr 21, 2025, 3:18 AM
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spy27
8 posts
spy27
#6 h  Apr 30, 2025, 5:38 AM
Y by
lbh_qys wrote:
another solution

Can you explain the \( a + b + c \neq 0 \) case in some detail?
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lbh_qys
558 posts
lbh_qys
#7 h  Apr 30, 2025, 6:08 AM
Y by
spy27 wrote:
lbh_qys wrote:
another solution

Can you explain the \( a + b + c \neq 0 \) case in some detail?

$f(x) = (a+x)^2 + (b+x)^2 + (c+x)^2 $ get minimum at $x=0$ iff $a+b+c=0$.
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