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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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9 Three concurrent chords
v_Enhance   0
3 minutes ago
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 $\Gamma_2$.
0 replies
+2 w
v_Enhance
3 minutes ago
0 replies
J
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Simple vector geometry existence
AndreiVila   3
N 44 minutes ago by Ianis
Source: Romanian District Olympiad 2025 9.1
Let $ABCD$ be a parallelogram of center $O$. Prove that for any point $M\in (AB)$, there exist unique points $N\in (OC)$ and $P\in (OD)$ such that $O$ is the center of mass of $\triangle MNP$.
3 replies
+1 w
AndreiVila
Mar 8, 2025
Ianis
44 minutes ago
J
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Checkerboard
Ecrin_eren   1
N an hour ago by Ecrin_eren
On an 8×8 checkerboard, what is the minimum number of squares that must be marked (including the marked ones) so that every square has exactly one marked neighbor? (We define neighbors as squares that share a common edge, and a square is not considered a neighbor of itself.)
1 reply
1 viewing
Ecrin_eren
Today at 5:20 AM
Ecrin_eren
an hour ago
J
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BD tangent to (MDE) , rhombus ABCD with <DCB=60^o
parmenides51   1
N an hour ago by vanstraelen
Source: 2021 Germany R4 10.6 https://artofproblemsolving.com/community/c3208025_
Let a rhombus $ABCD$ with $|\angle DCB| = 60^o$ be given . On the extension of the segment $\overline{CD}$ beyond $D$, a point $E$ is chosen arbitrarily. Let the line through $E$ and $A$ intersect the line $BC$ at the point $F$. Let $M$ be the intersection of the lines $BE$ and $DF$. Prove that the line $BD$ is tangent to the circumcircle of the triangle $MDE$.
1 reply
parmenides51
Oct 6, 2024
vanstraelen
an hour ago
J
No more topics!
H
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Inspired by Romania JBTST 2014
sqing   5
N Feb 23, 2021 by SMSGodslayer
Source: Own
Let $a,b,c$ be positive real numbers such that $ab+bc+ca+abc=4.$ Prove that
$$(a^n+1)(b^n+1)(c^n+1)\geq8.$$Where $n\in N^+.$
5 replies
sqing
Feb 22, 2021
SMSGodslayer
Feb 23, 2021
J
Inspired by Romania JBTST 2014
G H J
Reveal topic tags
inequalities
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China
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G H BBookmark kLocked kLocked NReply
Source: Own
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sqing
41157 posts
sqing
#1 h  Feb 22, 2021, 1:59 PM • 3 Y
Y by Quantum_fluctuations, SMSGodslayer, son7
Let $a,b,c$ be positive real numbers such that $ab+bc+ca+abc=4.$ Prove that
$$(a^n+1)(b^n+1)(c^n+1)\geq8.$$Where $n\in N^+.$
Z K Y
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xyzz
96 posts
xyzz
#2 h  Feb 22, 2021, 2:02 PM • 1 Y
Y by snakeaid
sqing wrote:
Let $a,b,c$ be positive real numbers such that $ab+bc+ca+abc=4.$ Prove that
$$(a^n+1)(b^n+1)(c^n+1)\geq8.$$Where $n\in N^+.$

as $a+b+c\ge 3$ $$(a+1)(b+1)(c+1)\ge 8$$now using $$a^n+1\ge \frac{{(a+1)}^n}{2^{n-1}}$$we are done!
Z K Y
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sqing
41157 posts
sqing
#3 h  Feb 22, 2021, 2:37 PM
Y by
Thanks.
Let $a,b,c$ be non-negative numbers such that $ab+bc+ca+abc=4.$ Prove that
$$\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}= 1$$$$\frac{1}{2}\leq \frac{1}{a^2+4}+\frac{1}{b^2+4}+\frac{1}{c^2+4}\leq \frac{3}{5}$$
This post has been edited 1 time. Last edited by sqing, Apr 18, 2021, 12:46 AM
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SerdarBozdag
892 posts
SerdarBozdag
#4 h  Feb 22, 2021, 3:40 PM
Y by
@xyzz Can you prove the last one?
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xyzz
96 posts
xyzz
#5 h  Feb 22, 2021, 4:32 PM
Y by
SerdarBozdag wrote:
@xyzz Can you prove the last one?

Its the power mean inequality what is there to prove in it?
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SMSGodslayer
25 posts
SMSGodslayer
#6 h  Feb 23, 2021, 8:48 AM
Y by
SerdarBozdag wrote:
@xyzz Can you prove the last one?
Holder QED
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