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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

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[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
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0 replies
jlacosta
Mar 2, 2025
0 replies
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a/b + b/a never integer ?
MTA_2024   2
N 6 minutes ago by bogpt
Let $a$ and $b$ be 2 distinct positive integers.
Can $\frac a b +\frac b a $ be in an integer. Prove why ?
2 replies
MTA_2024
Today at 3:08 PM
bogpt
6 minutes ago
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2014 Community AIME / Marathon ... Algebra Medium #1 quartic
parmenides51   5
N 3 hours ago by CubeAlgo15
Let there be a quartic function $f(x)$ with maximums $(4,5)$ and $(5,5)$. If $f(0) = -195$, and $f(10)$ can be expressed as $-n$ where $n$ is a positive integer, find $n$.

proposed by joshualee2000
5 replies
parmenides51
Jan 21, 2024
CubeAlgo15
3 hours ago
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Inequalities
sqing   4
N 4 hours ago by DAVROS
Let $ a, b\geq 0 $ and $ \frac {a^2+a+1}{b^2-b+1}+\frac {b^2+b+1}{a^2-a+1} \geq 3 $. Prove that
$$ a+b+a^2+b^2 \geq 28-6\sqrt{21}$$Let $ a, b\geq 0 $ and $ \frac {a^2+a+1}{b^2-b+1}+\frac {b^2+b+1}{a^2-a+1} \geq 4 $. Prove that
$$ a+b+a^2+b^2 \geq 10-4\sqrt 5$$Let $ a, b\geq 0 $ and $ \frac {a^2+a+1}{b^2-b+1}+\frac {b^2+b+1}{a^2-a+1} \geq 5 $. Prove that
$$ a+b+a^2+b^2 \geq \frac {2(26-5\sqrt{13})}{9} $$Let $ a, b\geq 0 $ and $\frac {a^2+a+1}{b^2-b+1}+\frac {b^2+b+1}{a^2-a+1}\geq 3+ \sqrt {5} $. Prove that
$$ a+b+a^2+b^2 \geq 2$$
4 replies
sqing
Mar 14, 2025
DAVROS
4 hours ago
J
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interesting problem
sausagebun   1
N Today at 4:05 PM by mathprodigy2011
Six points, labeled A, B, C, D, E, and F, are positioned consecutively on a straight line. Let G be a point not located on this line. The following distances are given: AC = 26, BD = 22, CE = 31, DF = 33, AF = 73, CG = 40, and DG = 30. Determine the area of triangle BGE.
I brute forced this with trig, was wondering if theres a more elegant way of doing this
1 reply
sausagebun
Today at 3:21 PM
mathprodigy2011
Today at 4:05 PM
J
No more topics!
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A NIMO Shortlist Problem: Not the Same as #6
djmathman   2
N May 6, 2017 by Ultimate_Drinker
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$, $CD=14$, and $\sin\angle DAC$ $=\sin\angle DBC=\tfrac78$. Denote by $\omega_1$ and $\omega_2$ the incircles of $\triangle DAC$ and $\triangle DBC$ respectively. Suppose that both $\omega_1$ and $\omega_2$ have radii of length $3$. The square of the distance between the centers of the two circles can be written in the form $m\sqrt n - p$, where $m$, $n$, and $p$ are positive integers such that $n$ is not divisible by the square of any prime. What is $m+n+p$?
2 replies
djmathman
Oct 18, 2015
Ultimate_Drinker
May 6, 2017
J
A NIMO Shortlist Problem: Not the Same as #6
G H J
Reveal topic tags
NIMO
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djmathman
7934 posts
djmathman
#1 h  Oct 18, 2015, 5:13 PM • 3 Y
Y by HamstPan38825, Adventure10, Mango247
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$, $CD=14$, and $\sin\angle DAC$ $=\sin\angle DBC=\tfrac78$. Denote by $\omega_1$ and $\omega_2$ the incircles of $\triangle DAC$ and $\triangle DBC$ respectively. Suppose that both $\omega_1$ and $\omega_2$ have radii of length $3$. The square of the distance between the centers of the two circles can be written in the form $m\sqrt n - p$, where $m$, $n$, and $p$ are positive integers such that $n$ is not divisible by the square of any prime. What is $m+n+p$?
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Mathisfun04
752 posts
Mathisfun04
#2 h  May 6, 2017, 1:25 PM • 4 Y
Y by SathWin, suthaparaman, Elephant200, Adventure10
bump. . . cause this looks like a really cool prob
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Ultimate_Drinker
176 posts
Ultimate_Drinker
#3 h  May 6, 2017, 3:05 PM • 4 Y
Y by Math1331Math, tost, Adventure10, Mango247
sol
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