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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
an hour ago
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

Summer camps are starting this 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 a transformative 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][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
[*]June 9th, Monday, 7:30pm ET, Game Jam: Operation Shuffle!, Come join us to play our second round of Operation Shuffle! If you enjoy number sense, logic, and a healthy dose of luck, this is the game for you. No specific math background is required; all are welcome.[/list]
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0 replies
jlacosta
an hour ago
0 replies
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Orthocenter lies on circumcircle
whatshisbucket   90
N 7 minutes ago by bjump
Source: 2017 ELMO #2
Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$

Proposed by Michael Ren
90 replies
whatshisbucket
Jun 26, 2017
bjump
7 minutes ago
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Polish MO Finals 2014, Problem 4
j___d   3
N 25 minutes ago by ariopro1387
Source: Polish MO Finals 2014
Denote the set of positive rational numbers by $\mathbb{Q}_{+}$. Find all functions $f: \mathbb{Q}_{+}\rightarrow \mathbb{Q}_{+}$ that satisfy
$$\underbrace{f(f(f(\dots f(f}_{n}(q))\dots )))=f(nq)$$for all integers $n\ge 1$ and rational numbers $q>0$.
3 replies
+1 w
j___d
Jul 27, 2016
ariopro1387
25 minutes ago
J
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S(an) greater than S(n)
ilovemath0402   1
N 38 minutes ago by ilovemath0402
Source: Inspired by an old result
Find all positive integer $n$ such that $S(an)\ge S(n) \quad \forall a \in \mathbb{Z}^{+}$ ($S(n)$ is sum of digit of $n$ in base 10)
P/s: Original problem
1 reply
1 viewing
ilovemath0402
an hour ago
ilovemath0402
38 minutes ago
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Hagge-like circles, Jerabek hyperbola, Lemoine cubic
kosmonauten3114   0
40 minutes ago
Source: My own
Let $\triangle{ABC}$ be a scalene oblique triangle with circumcenter $O$ and orthocenter $H$, and $P$ ($\neq \text{X(3), X(4)}$, $\notin \odot(ABC)$) a point in the plane.
Let $\triangle{A_1B_1C_1}$, $\triangle{A_2B_2C_2}$ be the circumcevian triangles of $O$, $P$, respectively.
Let $\triangle{P_AP_BP_C}$ be the pedal triangle of $P$ with respect to $\triangle{ABC}$.
Let $A_1'$ be the reflection in $P_A$ of $A_1$. Define $B_1'$, $C_1'$ cyclically.
Let $A_2'$ be the reflection in $P_A$ of $A_2$. Define $B_2'$, $C_2'$ cyclically.
Let $O_1$, $O_2$ be the circumcenters of $\triangle{A_1'B_1'C_1'}$, $\triangle{A_2'B_2'C_2'}$, respectively.

Prove that:
1) $P$, $O_1$, $O_2$ are collinear if and only if $P$ lies on the Jerabek hyperbola of $\triangle{ABC}$.
2) $H$, $O_1$, $O_2$ are collinear if and only if $P$ lies on the Lemoine cubic (= $\text{K009}$) of $\triangle{ABC}$.
0 replies
kosmonauten3114
40 minutes ago
0 replies
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No more topics!
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Mobius thingy
Hip1zzzil   1
N Mar 30, 2025 by seoneo
Source: FKMO 2025
For all natural numbers $n$, sequence $a_{n}$ satisfies the equation:
$\sum_{k=1}^{n}\frac{1}{2}(1-(-1)^{[\frac{n}{k}]})a_{k}=1$
When $m=1001\times 2^{2025}$, find the value of $a_{m}$.
1 reply
Hip1zzzil
Mar 29, 2025
seoneo
Mar 30, 2025
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Mobius thingy
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Reveal topic tags
number theory
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G H BBookmark kLocked kLocked NReply
Source: FKMO 2025
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Hip1zzzil
16 posts
Hip1zzzil
#1 h  Mar 29, 2025, 10:03 AM
Y by
For all natural numbers $n$, sequence $a_{n}$ satisfies the equation:
$\sum_{k=1}^{n}\frac{1}{2}(1-(-1)^{[\frac{n}{k}]})a_{k}=1$
When $m=1001\times 2^{2025}$, find the value of $a_{m}$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
seoneo
339 posts
seoneo
#2 h  Mar 30, 2025, 7:27 AM
Y by
Please visit
https://artofproblemsolving.com/community/q1h3537647p34401505[
This post has been edited 1 time. Last edited by seoneo, Mar 30, 2025, 7:27 AM
Reason: To fix errata
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