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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
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
Simple vector geometry existence
AndreiVila   3
N 43 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
43 minutes ago
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
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
No more topics!
A city where the names of people are either A or B and 10 letters long
ZETA_in_olympiad   4
N May 6, 2023 by Banglarmanush
Source: BdMO 2022 Secondary P7
Sabbir noticed one day that everyone in the city of BdMO has a distinct word of length $10$, where each letter is either $A$ or $B$. Sabbir saw that two citizens are friends if one of their words can be altered a few times using a special rule and transformed into the other ones word. The rule is, if somewhere in the word $ABB$ is located consecutively, then these letters can be changed to $BBA$ or if $BBA$ is located somewhere in the word consecutively, then these letters can be changed to $ABB$ (if wanted, the word can be kept as it is, without making this change.) For example $AABBA$ can be transformed into $AAABB$ (the opposite is also possible.) Now Sabbir made a team of $N$ citizens where no one is friends with anyone. What is the highest value of $N.$
4 replies
ZETA_in_olympiad
Apr 12, 2022
Banglarmanush
May 6, 2023
A city where the names of people are either A or B and 10 letters long
G H J
G H BBookmark kLocked kLocked NReply
Source: BdMO 2022 Secondary P7
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ZETA_in_olympiad
2211 posts
#1
Y by
Sabbir noticed one day that everyone in the city of BdMO has a distinct word of length $10$, where each letter is either $A$ or $B$. Sabbir saw that two citizens are friends if one of their words can be altered a few times using a special rule and transformed into the other ones word. The rule is, if somewhere in the word $ABB$ is located consecutively, then these letters can be changed to $BBA$ or if $BBA$ is located somewhere in the word consecutively, then these letters can be changed to $ABB$ (if wanted, the word can be kept as it is, without making this change.) For example $AABBA$ can be transformed into $AAABB$ (the opposite is also possible.) Now Sabbir made a team of $N$ citizens where no one is friends with anyone. What is the highest value of $N.$
Z K Y
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Dr_Misir_Ali
2 posts
#2
Y by
Isn't the answer 9?
Z K Y
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xmt_chn
58 posts
#3
Y by
The sequence and the rest(mod 2) of A won't change,maybe one sequence can show the only man.
Z K Y
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Complete_quadrilateral
144 posts
#4
Y by
If you have 2 same consecutive letters you can "move" them around if you get what i mean. So it suffices to find how many words are there with some number of pairs and some number of isolated letter B (if there are 2k+1 letters in a row then it is as if you have k pairs and one isolated b). Im on my phone right now but that is the main idea pretty much, i think. Also sorry for my English.
Z K Y
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Banglarmanush
32 posts
#5
Y by
Arrange them in all possible ways
AAAAABBBBB
BBBBBAAAAA
ABABABABAB
BABABABABA
AABBAABBAA
BBAABBAABB
AAAABBBBAA
BBBBAAAABB
AAABBBAAAB
BBBAAABBBA
AAAAAABBBB
BBBBBBAAAA
AAAAAAABBB
BBBBBBBAAA
AAAAAAAABB
BBBBBBBBBAA
AAAAAAAAAB
BBBBBBBBBA
THEREFORE,highest value of N is 9
Z K Y
N Quick Reply
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