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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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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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Easy P4 combi game with nt flavour
Maths_VC   1
N 3 hours ago by p.lazarov06
Source: Serbia JBMO TST 2025, Problem 4
Two players, Alice and Bob, play the following game, taking turns. In the beginning, the number $1$ is written on the board. A move consists of adding either $1$, $2$ or $3$ to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is $10$, then in a move a player can't choose the number $2$, but he can choose either $1$ or $3$). The player who first writes a perfect square on the board loses. Prove that one of the players has a winning strategy and determine who wins in the game.
1 reply
Maths_VC
May 27, 2025
p.lazarov06
3 hours ago
J
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Central sequences
EeEeRUT   14
N 3 hours ago by HamstPan38825
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.
14 replies
EeEeRUT
Apr 16, 2025
HamstPan38825
3 hours ago
J
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Elementary Problems Compilation
Saucepan_man02   32
N 4 hours ago by atdaotlohbh
Could anyone send some elementary problems, which have tricky and short elegant methods to solve?

For example like this one:
Solve over reals: $$a^2 + b^2 + c^2 + d^2 -ab-bc-cd-d +2/5=0$$
32 replies
Saucepan_man02
May 26, 2025
atdaotlohbh
4 hours ago
J
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Random Points = Problem
kingu   5
N 4 hours ago by happypi31415
Source: Chinese Geometry Handout
Let $ABC$ be a triangle. Let $\omega$ be a circle passing through $B$ intersecting $AB$ at $D$ and $BC$ at $F$. Let $G$ be the intersection of $AF$ and $\omega$. Further, let $M$ and $N$ be the intersections of $FD$ and $DG$ with the tangent to $(ABC)$ at $A$. Now, let $L$ be the second intersection of $MC$ and $(ABC)$. Then, prove that $M$ , $L$ , $D$ , $E$ and $N$ are concyclic.
5 replies
+1 w
kingu
Apr 27, 2024
happypi31415
4 hours ago
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height, circumcenter and arbitrary point
Megus   2
N Apr 22, 2007 by e.lopes
Source: Poland MO 2007
1. In acute triangle $ABC$ point $O$ is circumcenter, segment $CD$ is a height, point $E$ lies on side $AB$ and point $M$ is a midpoint of $CE$. Line through $M$ perpendicular to $OM$ cuts lines $AC$ and $BC$ respectively in $K$, $L$. Prove that $\frac{LM}{MK}=\frac{AD}{DB}$
2 replies
Megus
Apr 22, 2007
e.lopes
Apr 22, 2007
J
height, circumcenter and arbitrary point
G H J
Reveal topic tags
geometry
circumcircle
geometry proposed
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G H BBookmark kLocked kLocked NReply
Source: Poland MO 2007
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Megus
1198 posts
Megus
#1 h  Apr 22, 2007, 1:34 PM • 2 Y
Y by Adventure10, Mango247
1. In acute triangle $ABC$ point $O$ is circumcenter, segment $CD$ is a height, point $E$ lies on side $AB$ and point $M$ is a midpoint of $CE$. Line through $M$ perpendicular to $OM$ cuts lines $AC$ and $BC$ respectively in $K$, $L$. Prove that $\frac{LM}{MK}=\frac{AD}{DB}$
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malinger
89 posts
malinger
#2 h  Apr 22, 2007, 2:06 PM • 2 Y
Y by Adventure10, Mango247
http://www.mathlinks.ro/Forum/viewtopic.php?t=145077 :)
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e.lopes
349 posts
e.lopes
#3 h  Apr 22, 2007, 5:26 PM • 2 Y
Y by Adventure10, Mango247
Let $M_{a}$ and $M_{b}$ the midpoints of CB and CA, respectvely.
Is easy to see that the quadrilaterals $MOM_{a}L$ and $MOKM_{b}$ are cyclics.
So, we easy angle chasing, we get $\angle{LOM}= \angle{B}$, and $\angle{KOM}= \angle{A}$.

now, we have that $\frac{LM}{KM}= \frac{tgB}{tgA}= \frac{AD}{DB}$
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