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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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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Log Rationality
Magdalo   0
a few seconds ago
Let $x+y=36$ be positive integers. How many pairs of $x,y$ are there such that $\log_xy$ is rational?
0 replies
Magdalo
a few seconds ago
0 replies
J
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Original Problem U1
NeoAzure   1
N 4 minutes ago by NeoAzure
How many three-digit positive integers are divisible by 4 and have exactly two even digits?

Solution

Answer
1 reply
NeoAzure
5 minutes ago
NeoAzure
4 minutes ago
J
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[PMO22 Qualifying] I.11
Magdalo   2
N 16 minutes ago by Shan3t
Let $x$ and $y$ be positive real numbers such that
\[\log_x64+\log_{y^2}16=\dfrac{5}{3}\text{ and }\log_y64+\log_{x^2}16=1 \]Find $\log_2(xy)$
2 replies
Magdalo
39 minutes ago
Shan3t
16 minutes ago
J
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[Sipnayan 2017 JHS] Semifinals A, Difficult, P2
NeoAzure   1
N 22 minutes ago by NeoAzure
Akira fills an urn with 10 chips such that 1 chip is labeled “1”, 2 chips are labeled “2”, 3
chips are labeled “3”, and 4 chips are labeled “4”. She draws 4 chips from the box without
replacement. What is the probability the sum of the numbers labeled on the 4 chips is divisible
by 3

Solution

Answer
1 reply
NeoAzure
23 minutes ago
NeoAzure
22 minutes ago
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No more topics!
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J
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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
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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
599 posts
lbh_qys
#4 h  Apr 21, 2025, 3:04 AM
Y by
Hint
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lbh_qys
599 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
599 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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