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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)!

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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
Jun 2, 2025
0 replies
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2-var inequality
sqing   4
N 2 minutes ago by Rohit-2006
Source: Own
Let $ a,b\geq 0 $ and $\frac{1}{a^2+3} + \frac{1}{b^2+3} -ab\leq \frac{1}{2}.$ Prove that
$$ a^2+ab+b^2 \geq \frac{3(\sqrt{57}-7)}{4}$$Let $ a,b\geq 0 $ and $\frac{a}{b^2+3} + \frac{b}{a^2+3} +ab\leq \frac{1}{2}.$ Prove that
$$ a^2+ab+b^2 \leq \frac{9}{4}$$Let $ a,b\geq 0 $ and $ \frac{a}{b^3+3}+\frac{b}{a^3+3}-ab\leq \frac{1}{2}.$ Prove that
$$ a^2+ab+b^2 \geq \frac{9}{4}$$
4 replies
sqing
Yesterday at 12:55 PM
Rohit-2006
2 minutes ago
J
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Easy Diff NT
xToiletG   3
N 4 minutes ago by elizhang101412
Prove that for every $n \geq 2$ there exists positive integers $x, y, z$ such that
$$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$
3 replies
xToiletG
Today at 7:35 AM
elizhang101412
4 minutes ago
J
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Number Theory
Resolut1on07   1
N 5 minutes ago by Resolut1on07
Given a positive integer \( n \). Prove that always exists a positive integer \( k \) such that \( a^2 + b^2 + n \) is not divisible by \( k \), for all positive integers \( a \) and \( b \).
1 reply
Resolut1on07
Yesterday at 11:49 AM
Resolut1on07
5 minutes ago
J
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A series of transfers
Fermat -Euler   12
N 16 minutes ago by kotmhn
Source: IMO Shortlist 1994, C3
Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. Prove that Peter can always transfer all his money into two accounts. Can Peter always transfer all his money into one account?
12 replies
+1 w
Fermat -Euler
Oct 22, 2005
kotmhn
16 minutes ago
J
No more topics!
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equilateral wanted and given by bisector, median, altitude of diff. vertices
parmenides51   0
Sep 5, 2020
Source: Champions Tournament (Ukraine) - Турнір чемпіонів - 2006 Juniors p3
The bisector $AK$, the median $BL$ and the altitude $CM$ are drawn in the triangle $ABC$. The triangle $KML$ is equilateral. Prove that the triangle $ABC$ is equilateral.
0 replies
parmenides51
Sep 5, 2020
0 replies
J
equilateral wanted and given by bisector, median, altitude of diff. vertices
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Reveal topic tags
geometry
Equilateral
Champions Tournament
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G H BBookmark kLocked kLocked NReply
Source: Champions Tournament (Ukraine) - Турнір чемпіонів - 2006 Juniors p3
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parmenides51
30653 posts
parmenides51
#1 h  Sep 5, 2020, 4:39 PM
Y by
The bisector $AK$, the median $BL$ and the altitude $CM$ are drawn in the triangle $ABC$. The triangle $KML$ is equilateral. Prove that the triangle $ABC$ is equilateral.
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