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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.

Be sure to mark your calendars for the following events:
[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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Funny system of equations in three variables
Tintarn   10
N 36 minutes ago by Marcus_Zhang
Source: Baltic Way 2020, Problem 5
Find all real numbers $x,y,z$ so that
\begin{align*} x^2 y + y^2 z + z^2 &= 0 \\ z^3 + z^2 y + z y^3 + x^2 y &= \frac{1}{4}(x^4 + y^4). \end{align*}
10 replies
Tintarn
Nov 14, 2020
Marcus_Zhang
36 minutes ago
J
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a^{2m}+a^{n}+1 is perfect square
kmh1   1
N an hour ago by kmh1
Source: own
Find all positive integer triplets $(a,m,n)$ such that $2m>n$ and $a^{2m}+a^{n}+1$ is a perfect square.
1 reply
kmh1
Mar 20, 2025
kmh1
an hour ago
J
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Interesting problem
deraxenrovalo   0
an hour ago
Given $\triangle$$ABC$ with circumcenter $O$$.\;$Let $P$ be an arbitrary point on $(BOC)$ such that $P$ is outside $(ABC)$$.\;$Let $Q$ be an arbitrary point on $(ABC)$$.\;$$AB$ cuts $(ACP)$ again at $E$ and $AC$ cuts $(ABP)$ again at $F$$.\;$The intersection of $BF$ and $CE$ is $R$$.\;$Let $X$ and $Y$ be the intersection of $EF$ with $(PQC)$ and $(PQR)$ respectively such that $X$, $Y$, $P$ are pairwise distinct.
Show that : $(APX)$, $(BPY)$, $(QPE)$ are coaxial circles

hint
0 replies
deraxenrovalo
an hour ago
0 replies
J
H
Vieta Jumping Unsolved(Reposted)
Eagle116   0
an hour ago
Source: MONT, Vieta Jumping part
The question is:
Let $x_1$, $x_2$, $\dots$, $x_n$ be $n$ integers. If $k>n$ is an integer, prove that the only solution to
$$x_1^2 + x_2^2 + \dots + x_n^2 = kx_1x_2\dots x_n $$is is $x_1 = x_2 = \dots = x_n = 0$.
0 replies
Eagle116
an hour ago
0 replies
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No more topics!
H
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inequalities
Aritra12   5
N May 15, 2020 by JosefSvejk
prove this, this is a nice problem you can try :)

\[ \frac{a}{b+2 c+d}+\frac{b}{c+2 d+a}+\frac{c}{d+2 a+b}+\frac{d}{a+2 b+c} \geq 1 \]
5 replies
Aritra12
May 15, 2020
JosefSvejk
May 15, 2020
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inequalities
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Reveal topic tags
inequalities proposed
inequalities
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Aritra12
1026 posts
Aritra12
#1 h  May 15, 2020, 11:54 AM • 2 Y
Y by abhradeep12, SD_360MathLover
prove this, this is a nice problem you can try :)

\[ \frac{a}{b+2 c+d}+\frac{b}{c+2 d+a}+\frac{c}{d+2 a+b}+\frac{d}{a+2 b+c} \geq 1 \]
This post has been edited 1 time. Last edited by Aritra12, Aug 1, 2021, 3:53 PM
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Aritra12
1026 posts
Aritra12
#2 h  May 15, 2020, 11:54 AM • 2 Y
Y by abhradeep12, SD_360MathLover
this was proposed problem
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Aritra12
1026 posts
Aritra12
#3 h  May 15, 2020, 11:56 AM • 2 Y
Y by abhradeep12, SD_360MathLover
of romania 2006 trian tamain
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Aritra12
1026 posts
Aritra12
#4 h  May 15, 2020, 11:56 AM • 2 Y
Y by abhradeep12, SD_360MathLover
one mor conditions are a,b,c are>0
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arqady
30152 posts
arqady
#5 h  May 15, 2020, 2:14 PM • 4 Y
Y by abhradeep12, Mango247, Mango247, Mango247
If also $d>0$ so use C-S.
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JosefSvejk
111 posts
JosefSvejk
#6 h  May 15, 2020, 2:41 PM • 1 Y
Y by abhradeep12
$\frac{a}{b+2 c+d}+\frac{b}{c+2 d+a}+\frac{c}{d+2 a+b}+\frac{d}{a+2 b+c} =\sum_{cyc} \frac{a^2}{ab+2ca+da} \ge \frac{(a+b+c+d)^2}{2(ab+bc+2ac+cd+2bd+da)} \ge 1$, so
$(a+b+c+d)^2 \ge 2(ab+bc+2ac+cd+2bd+da) \implies \sum_{cyc}a^2 \ge 2bd + 2ac$ that directly follows from AM-GM
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