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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Today at 3:57 PM
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
Today at 3:57 PM
0 replies
J
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IMO Genre Predictions
ohiorizzler1434   78
N 3 minutes ago by ohiorizzler1434
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
78 replies
ohiorizzler1434
May 3, 2025
ohiorizzler1434
3 minutes ago
J
H
The Return of Triangle Geometry
peace09   12
N 10 minutes ago by peace09
Source: 2023 ISL A7
Let $N$ be a positive integer. Prove that there exist three permutations $a_1,\dots,a_N$, $b_1,\dots,b_N$, and $c_1,\dots,c_N$ of $1,\dots,N$ such that \[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\]for every $k=1,2,\dots,N$.
12 replies
peace09
Jul 17, 2024
peace09
10 minutes ago
J
H
Find Triples of Integers
termas   40
N 15 minutes ago by monval
Source: IMO 2015 problem 2
Find all positive integers $(a,b,c)$ such that
$$ab-c,\quad bc-a,\quad ca-b$$are all powers of $2$.

Proposed by Serbia
40 replies
termas
Jul 10, 2015
monval
15 minutes ago
J
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AT // BC wanted
parmenides51   106
N 23 minutes ago by happypi31415
Source: IMO 2019 SL G1
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

(Nigeria)
106 replies
parmenides51
Sep 22, 2020
happypi31415
23 minutes ago
J
No more topics!
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Flanders 1
Arne   2
N Sep 7, 2003 by Arne
Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.
2 replies
Arne
Sep 7, 2003
Arne
Sep 7, 2003
J
Flanders 1
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Reveal topic tags
ratio
geometry
vector
geometry solved
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Arne
3660 posts
Arne
#1 h  Sep 7, 2003, 9:41 AM • 2 Y
Y by Adventure10, Mango247
Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.
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grobber
7849 posts
grobber
#2 h  Sep 7, 2003, 10:58 AM • 2 Y
Y by Adventure10, Mango247
The ratio is 3/4; we just construct the triangle having the medians as sides by formung some parallelograms. This is possible because the sum of the vectors AM, BN, CP (where M, N, P are midpts of BC, CA, AB) is 0, so there is a triangle having sides equal to AM, BN, CP and parallel to these.
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Arne
3660 posts
Arne
#3 h  Sep 7, 2003, 11:00 AM • 2 Y
Y by Adventure10, Mango247
Indeed Grobber !
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