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Summer is a great time to explore cool problems to keep your skills sharp!  Schedule a class today!
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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
H
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!
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circle tangent to 3 circles on the same hemisphere
parmenides51   0
Oct 18, 2021
Source: 1999 Argentina IberoAmerican Training List p4 https://artofproblemsolving.com/community/c2434876_argentina_training_lists_ibero
In a sphere of radius $ 1$, consider a maximum circle $H$. On one of the hemispheres determined by $H$ consider three circles of equal radii, tangent to each other, in pairs, and tangent to $H$. Determine the radius of another circle, located on the same hemisphere and that is tangent to the three circles.
0 replies
parmenides51
Oct 18, 2021
0 replies
J
circle tangent to 3 circles on the same hemisphere
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Reveal topic tags
3D geometry
tangent
geometry
Spheres
tangent circles
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Source: 1999 Argentina IberoAmerican Training List p4 https://artofproblemsolving.com/community/c2434876_argentina_training_lists_ibero
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parmenides51
30653 posts
parmenides51
#1 h  Oct 18, 2021, 6:50 PM
Y by
In a sphere of radius $ 1$, consider a maximum circle $H$. On one of the hemispheres determined by $H$ consider three circles of equal radii, tangent to each other, in pairs, and tangent to $H$. Determine the radius of another circle, located on the same hemisphere and that is tangent to the three circles.
This post has been edited 1 time. Last edited by parmenides51, Oct 18, 2021, 6:53 PM
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