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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
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)!

Be sure to mark your calendars for the following upcoming events:
[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
Jun 2, 2025
0 replies
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My Unsolved Problem
ZeltaQN2008   5
N 9 minutes ago by Funcshun840
Source: IDK
Let \( ABC \) be an acute triangle inscribed in its circumcircle \( (O) \), and let \( (I) \) be its incircle. Let \( K \) be the point where the $A-mixtilinear$ incircle of triangle $ABC$ touches \((O)\). Suppose line \( OI \) intersects segment \( AK \) at \( P \), and intersects line \( BC \) at \( Q \). Let the line through \( I \) perpendicular to \( BC \) intersect line \( KQ \) at \( A' \). Prove that: \[AI \parallel PA'.\]
5 replies
ZeltaQN2008
Yesterday at 1:23 PM
Funcshun840
9 minutes ago
J
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Angle QRP = 90°
orl   13
N 14 minutes ago by Ilikeminecraft
Source: IMO ShortList, Netherlands 1, IMO 1975, Day 1, Problem 3
In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$.

Prove that

a.) $\angle QRP = 90\,^{\circ},$ and

b.) $QR = RP.$
13 replies
orl
Nov 12, 2005
Ilikeminecraft
14 minutes ago
J
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students in a classroom sit in a round table, possible to split into 3 groups
parmenides51   1
N 41 minutes ago by Magnetoninja
Source: Dutch IMO TST 2018 day 2 p4
In the classroom of at least four students the following holds: no matter which four of them take seats around a round table, there is always someone who either knows both of his neighbours, or does not know either of his neighbours. Prove that it is possible to divide the students into two groups such that in one of them, all students know one another, and in the other, none of the students know each other.

(Note: if student A knows student B, then student B knows student A as well.)
1 reply
parmenides51
Aug 30, 2019
Magnetoninja
41 minutes ago
J
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Sharing is a nontrivial task
bjump   11
N an hour ago by HamstPan38825
Source: USA TST 2024 P5
Suppose $a_{1} < a_{2}< \cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.

Ray Li
11 replies
bjump
Jan 15, 2024
HamstPan38825
an hour ago
J
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the finished drawings
pupitrethebean   11
N Jul 25, 2023 by ishipoopman
dont judge
pls
i suck at drawing
so like
just chill
i did ishi's idea and maddie's idea
maddies took so much time like bruh
its fine tho its funny

adios
11 replies
pupitrethebean
Jul 22, 2023
ishipoopman
Jul 25, 2023
J
No more topics!
H
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Inspired by old results
sqing   1
N Apr 16, 2025 by sqing
Source: Own
Let $ a,b \geq 0 $ and $ a^2+b^2+a+b \geq 4 .$ Prove that$$ \frac{1}{a^2+b+1}+\frac{1}{b^2+a+1}+\frac{1}{a+b+1} \leq \frac{7\sqrt{17}-1}{26}$$
1 reply
sqing
Apr 16, 2025
sqing
Apr 16, 2025
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Inspired by old results
G H J
Reveal topic tags
inequalities proposed
algebra
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G H BBookmark kLocked kLocked NReply
Source: Own
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sqing
42601 posts
sqing
#1 h  Apr 16, 2025, 2:44 PM • 1 Y
Y by cubres
Let $ a,b \geq 0 $ and $ a^2+b^2+a+b \geq 4 .$ Prove that$$ \frac{1}{a^2+b+1}+\frac{1}{b^2+a+1}+\frac{1}{a+b+1} \leq \frac{7\sqrt{17}-1}{26}$$
This post has been edited 2 times. Last edited by sqing, Apr 16, 2025, 3:00 PM
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sqing
42601 posts
sqing
#2 h  Apr 16, 2025, 2:58 PM • 1 Y
Y by cubres
Let $ a,b \geq 0 $ and $ a^2+b^2+ab+a+b\geq 5 .$ Prove that$$ \frac{a}{2a+b^2}+\frac{b}{2b+1}+\frac{1}{a^2+2} \leq 1 $$
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