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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.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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0 replies
jlacosta
Mar 2, 2025
0 replies
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Prime for square numbers
giangtruong13   1
N 26 minutes ago by shanelin-sigma
Source: City’s Specialized Math Examination
Given that $a,b$ are natural numbers satisfy that: $\frac{a^3}{a+b}$ and $\frac{b^3}{a+b}$ are prime numbers. Prove that $$a^2+3ab+3a+b+1$$is a perfect squared number
1 reply
1 viewing
giangtruong13
an hour ago
shanelin-sigma
26 minutes ago
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Inspired by hunghd8
sqing   0
26 minutes ago
Source: Own
Let $ a,b,c\geq 0 $ and $ a+b+c\geq 2+abc . $ Prove that
$$a^2+b^2+c^2-\frac{1}{2}a^2b^2c^2\geq 2$$$$a^2+b^2+c^2-abc-\frac{1}{2}a^2b^2c^2\geq \frac{3}{2}$$$$a^2+b^2+c^2- \frac{19}{10}abc-\frac{1}{2}a^2b^2c^2\geq -\frac{12}{25}$$$$a^2+b^2+c^2- \frac{3}{2}abc-\frac{1}{2}a^2b^2c^2\geq \frac{17\sqrt{17}-71}{16}$$
0 replies
2 viewing
sqing
26 minutes ago
0 replies
J
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Find min
hunghd8   3
N 41 minutes ago by sqing
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$
3 replies
hunghd8
4 hours ago
sqing
41 minutes ago
J
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Interesting inequality
sqing   5
N an hour ago by sqing
Source: Own
Let $ a,b >0. $ Prove that
$$ \frac{1}{\frac{a}{a+b}+\frac{a}{2b}} +\frac{1}{\frac{b}{a+b}+\frac{1}{2}} +\frac{a}{2b} \geq \frac{5}{2} $$
5 replies
sqing
Feb 26, 2025
sqing
an hour ago
J
No more topics!
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AK is perpendicular to OH
TienLQD   2
N Aug 10, 2020 by jayme
Let the triangle ABC have H, O be the orthocenter and the circumcenter of ABC respectively. The line through H parallel to OA intersects AB at D . The perpendicular bisector of BC intersects AB, AC at E, F. Let K be the circumcenter of the triangle DEF. Prove that AK is perpendicular to OH
2 replies
TienLQD
Aug 8, 2020
jayme
Aug 10, 2020
J
AK is perpendicular to OH
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TienLQD
82 posts
TienLQD
#1 h  Aug 8, 2020, 9:59 AM • 1 Y
Y by Mango247
Let the triangle ABC have H, O be the orthocenter and the circumcenter of ABC respectively. The line through H parallel to OA intersects AB at D . The perpendicular bisector of BC intersects AB, AC at E, F. Let K be the circumcenter of the triangle DEF. Prove that AK is perpendicular to OH
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dagezjm
88 posts
dagezjm
#2 h  Aug 8, 2020, 4:56 PM • 3 Y
Y by AlastorMoody, like123, Mango247
Let $HD\cap AC=U$. By $\measuredangle AUD=\measuredangle CAO=\measuredangle HAB=\measuredangle FEA$ we get $D,E,F,U$ are concyclic. Now suppose $XY$ be the line parallel to $OH$ passing through $A$, and $XY\cap EF=X,XY\cap DU=Y$. By $AHOX$ and $AOHY$ are parallelograms we have $AX=OH=AY$. So the butterfly theorem implies that $AK\bot XY$ and $AK\bot OH$.
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jayme
9767 posts
jayme
#3 h  Aug 10, 2020, 9:00 AM
Y by
Dear mathlinkers,

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=564805

Sincerely
Jean-Louis
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