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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.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

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
J
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Hyperbolic tangent
MetaphysicalWukong   1
N a few seconds ago by aidan0626
Source: Kunwei Ding
Consider the function below. Can someone explain to me why that is the correct answer and find the value of a?
1 reply
MetaphysicalWukong
3 minutes ago
aidan0626
a few seconds ago
J
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Inequality with real numbers
JK1603JK   4
N 10 minutes ago by SunnyEvan
Source: unknown
Let a,b,c are real numbers. Prove that (a^3+b^3+c^3+3abc)^4+(a+b+c)^3(a+b-c)^3(-a+b+c)^3(a-b+c)^3>=0
4 replies
JK1603JK
Yesterday at 6:48 AM
SunnyEvan
10 minutes ago
J
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Interesting inequality
sqing   1
N 12 minutes ago by SunnyEvan
Source: Own
Let $ a,b,c\geq 0,(ab+c)(ac+b)\neq 0 $ and $ a+b+c=3 . $ Prove that
$$ \frac{1}{ab+c+k}+\frac{1}{ac+b+k} \geq\frac{2}{k+2} $$Where $ k\geq 0. $
1 reply
sqing
3 hours ago
SunnyEvan
12 minutes ago
J
H
Interesting inequality
sqing   2
N 14 minutes ago by SunnyEvan
Source: Own
Let $ a,b,c\geq 0,(ab+c)(ac+b)\neq 0 $ and $ a+b+c=3 . $ Prove that
$$ \frac{1}{ab+kc}+\frac{1}{ac+kb} \geq\frac{4}{3k} $$Where $ k\geq 3. $
$$ \frac{1}{ab+2c}+\frac{1}{ac+2b} \geq\frac{16}{25} $$$$ \frac{1}{ab+3c}+\frac{1}{ac+3b} \geq\frac{4}{9} $$$$ \frac{1}{ab+4c}+\frac{1}{ac+4b} \geq\frac{1}{3} $$

2 replies
sqing
2 hours ago
SunnyEvan
14 minutes ago
J
No more topics!
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9 Three concurrent chords
v_Enhance   4
N Yesterday at 6:51 AM by cosmicgenius
Three distinct circles $\Omega_1$, $\Omega_2$, $\Omega_3$ cut three common chords concurrent at $X$. Consider two distinct circles $\Gamma_1$, $\Gamma_2$ which are internally tangent to all $\Omega_i$. Determine, with proof, which of the following two statements is true.

(1) $X$ is the insimilicenter of $\Gamma_1$ and $\Gamma_2$
(2) $X$ is the exsimilicenter of $\Gamma_1$ and $\Gamma_2$.
4 replies
v_Enhance
Friday at 8:45 PM
cosmicgenius
Yesterday at 6:51 AM
J
Three concurrent chords
G H J
Reveal topic tags
geometry
poll
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G H BBookmark kLocked kLocked NReply
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v_Enhance
6869 posts
v_Enhance
#1 h  Friday at 8:45 PM • 3 Y
Y by medhimdi, Yiyj1, NaturalSelection
9Poll:
Which statement is true?
81 Votes
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(21)
74%
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Hide Results Show Results
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Three distinct circles $\Omega_1$, $\Omega_2$, $\Omega_3$ cut three common chords concurrent at $X$. Consider two distinct circles $\Gamma_1$, $\Gamma_2$ which are internally tangent to all $\Omega_i$. Determine, with proof, which of the following two statements is true.

(1) $X$ is the insimilicenter of $\Gamma_1$ and $\Gamma_2$
(2) $X$ is the exsimilicenter of $\Gamma_1$ and $\Gamma_2$.
Z K Y
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YaoAOPS
1497 posts
YaoAOPS
#2 h  Friday at 8:51 PM • 3 Y
Y by GrantStar, FunnyKoala17, cj13609517288
I misvoted the answer is exsimilicenter.
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S.Das93
706 posts
S.Das93
#3 h  Friday at 10:44 PM
Y by
it is exsimilcenter by the irrelevant geo knowledge theorem
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ohiorizzler1434
725 posts
ohiorizzler1434
#4 h  Yesterday at 2:20 AM
Y by
Bro! Can we have a diagram for this problem?
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cosmicgenius
1483 posts
cosmicgenius
#5 h  Yesterday at 6:51 AM • 1 Y
Y by MS_asdfgzxcvb
these are probably true:
  1. since the $\Omega_i$ intersect, their radical circle has negative squared radius
  2. this means that a negative inversion at $X$ swaps $\Gamma_1$ and $\Gamma_2$
  3. the bigger of $\Gamma_1$ and $\Gamma_2$ contains the smaller
  4. the center of a negative inversion swapping two circles, the bigger of which contains the smaller, is the two circles' exsimilicenter
This post has been edited 1 time. Last edited by cosmicgenius, Yesterday at 6:52 AM
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