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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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Arbitrary point on BC and its relation with orthocenter
falantrng   31
N 6 minutes ago by NZP_IMOCOMP4
Source: Balkan MO 2025 P2
In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus
31 replies
falantrng
Apr 27, 2025
NZP_IMOCOMP4
6 minutes ago
J
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IMO Genre Predictions
ohiorizzler1434   23
N 12 minutes ago by ohiorizzler1434
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
23 replies
ohiorizzler1434
Yesterday at 6:51 AM
ohiorizzler1434
12 minutes ago
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Number theory
gggzul   0
21 minutes ago
Is the number
$$10^{32}+10^{28}+...+10^4+1$$a perfect square?
0 replies
gggzul
21 minutes ago
0 replies
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3 var inequality
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b,c>0 . $ Prove that
$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{8}{3}\left(1+\frac{a+b}{b+c}+ \frac{b+c}{a+b}\right)$$$$ \left(1 +\frac{a^2}{b^2}\right)\left(1+\frac{b^2}{c^2}\right)\left(1+\frac{c^2}{a^2}\right )\geq \frac{8}{3}\left( 1+\frac{a^2+bc}{b^2+ca}+\frac{b^2+ca }{a^2+bc}\right)$$
1 reply
sqing
May 1, 2025
sqing
an hour ago
J
No more topics!
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Functional Equation Problem
dimi07   2
N Apr 22, 2025 by dimi07
Source: Pang Chung Wu FE Book
Could someone please solve this problem?

Find all functions \( f : \mathbb{Z} \to \mathbb{Z} \) that satisfy \( f(0) = 1 \) and
\[ f(f(n)) = f(f(n+2)+2) = n \]for all integers \( n \).
2 replies
dimi07
Apr 21, 2025
dimi07
Apr 22, 2025
J
Functional Equation Problem
G H J
Reveal topic tags
algebra
functional equation
L
G H BBookmark kLocked kLocked NReply
Source: Pang Chung Wu FE Book
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dimi07
4 posts
dimi07
#1 h  Apr 21, 2025, 12:27 PM
Y by
Could someone please solve this problem?

Find all functions \( f : \mathbb{Z} \to \mathbb{Z} \) that satisfy \( f(0) = 1 \) and
\[ f(f(n)) = f(f(n+2)+2) = n \]for all integers \( n \).
This post has been edited 2 times. Last edited by dimi07, Apr 21, 2025, 12:29 PM
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eevee9406
8 posts
eevee9406
#2 h  Apr 21, 2025, 12:59 PM
Y by
From $f(f(n))=n, f(n)$ is bijective. Then the first equality implies $f(n)=f(n+2)+2$, so using $f(0)=1$ we can show using induction that for even $n$, $f(n)=-n+1$.

From $f(f(n))=n$, clearly $f(f(1))=1=f(0)$; hence by injectivity $f(1)=0$. By similar induction we can show that for odd $n$ the same solution holds; thus $\boxed{f(n)=-n+1}$, which clearly works.
This post has been edited 1 time. Last edited by eevee9406, Apr 21, 2025, 1:05 PM
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dimi07
4 posts
dimi07
#3 h  Apr 22, 2025, 4:13 AM
Y by
Thank you very much good sir!
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