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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
6 hours 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)!

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
6 hours ago
0 replies
J
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USAMO 2003 Problem 1
MithsApprentice   71
N an hour ago by cubres
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
71 replies
1 viewing
MithsApprentice
Sep 27, 2005
cubres
an hour ago
J
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IMO Genre Predictions
ohiorizzler1434   76
N an hour ago by aidan0626
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
76 replies
ohiorizzler1434
May 3, 2025
aidan0626
an hour ago
J
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IMO ShortList 2008, Number Theory problem 2
April   40
N an hour ago by ezpotd
Source: IMO ShortList 2008, Number Theory problem 2, German TST 2, P2, 2009
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i + a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.

Proposed by Mohsen Jamaali, Iran
40 replies
April
Jul 9, 2009
ezpotd
an hour ago
J
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A weird problem
jayme   2
N 2 hours ago by lolsamo
Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. I the incenter
4. 1 a circle passing througn B and C
5. X, Y the second points of intersection of 1 wrt BI, CI
6. 2 the circumcircle of the triangle XYI
7. M, N the symetrics of B, C wrt XY.

Question : if 2 is tangent to 0 then, 2 is tangent to MN.

Sincerely
Jean-Louis
2 replies
jayme
Today at 6:52 AM
lolsamo
2 hours ago
J
No more topics!
H
J
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pure geometry(prove)
Pirkuliyev Rovsen   1
N Aug 30, 2014 by sunken rock
Let $ABCD$ be a quadrilateral inscribed in a circle of diameter $BD$ and $M$ be an arbitrary point on the shorter arc $AD$.Let $MN$, $MK$, $MP$, $MT$ be perpendiculars from $M$ onto lines $AB$, $BC$, $CD$, $AD$ respectively.Prove that $S_{MNP}=S_{MKT}$.
1 reply
Pirkuliyev Rovsen
Aug 30, 2014
sunken rock
Aug 30, 2014
J
pure geometry(prove)
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Pirkuliyev Rovsen
5047 posts
Pirkuliyev Rovsen
#1 h  Aug 30, 2014, 5:30 AM • 2 Y
Y by Adventure10, Mango247
Let $ABCD$ be a quadrilateral inscribed in a circle of diameter $BD$ and $M$ be an arbitrary point on the shorter arc $AD$.Let $MN$, $MK$, $MP$, $MT$ be perpendiculars from $M$ onto lines $AB$, $BC$, $CD$, $AD$ respectively.Prove that $S_{MNP}=S_{MKT}$.
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sunken rock
4402 posts
sunken rock
#2 h  Aug 30, 2014, 6:08 AM • 3 Y
Y by Pirkuliyev Rovsen, Adventure10, Mango247
$\angle KMT+\angle NMP=180^\circ\ (\ 1\ )$. $MPDP, MNBK, ABDM$ being cyclic give $\triangle MTP\sim\triangle MNK\implies\frac{MT}{MP}=\frac{MN}{MK}\ (\ 2\ )$. From $(1)$ and $(2)$ we are done.

Best regards,
sunken rock
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