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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
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
Functional equation
socrates   30
N an hour ago by NicoN9
Source: Baltic Way 2014, Problem 4
Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$
30 replies
socrates
Nov 11, 2014
NicoN9
an hour ago
Algebra Functions
pear333   1
N an hour ago by whwlqkd
Let $P(z)=z-1/z$. Prove that there does not exist a pair of rational numbers $x,y$ such that $P(x)+P(y)=4$.
1 reply
pear333
Today at 12:20 AM
whwlqkd
an hour ago
Polynomial equation with rational numbers
Miquel-point   2
N an hour ago by Assassino9931
Source: Romanian TST 1979 day 2 P1
Determine the polynomial $P\in \mathbb{R}[x]$ for which there exists $n\in \mathbb{Z}_{>0}$ such that for all $x\in \mathbb{Q}$ we have: \[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]
Dumitru Bușneag
2 replies
Miquel-point
Apr 15, 2023
Assassino9931
an hour ago
Area problem
MTA_2024   0
an hour ago
Let $\omega$ be a circle inscribed inside a rhombus $ABCD$. Let $P$ and $Q$ be variable points on $AB$ and $AD$ respectively, such as $PQ$ is always the tangent line to $\omega$.
Prove that for any position of $P$ and $Q$ the area of triangle $\triangle CPQ$ is the same.
0 replies
MTA_2024
an hour ago
0 replies
Geometry
srnjbr   0
an hour ago
In triangle ABC, D is the leg of the altitude from A. l is a variable line passing through D. E and F are points on l such that AEB=AFC=90. Find the locus of the midpoint of the line segment EF.
0 replies
srnjbr
an hour ago
0 replies
Geometry
srnjbr   0
2 hours ago
in triangle abc, l is the leg of bisector a, d is the image of c on line al, and e is the image of l on line ab. take f as the intersection of de and bc. show that af is perpendicular to bc
0 replies
srnjbr
2 hours ago
0 replies
<QBC =<PCB if BM = CN, <PMC = <MAB, <QNB = < NAC
parmenides51   1
N 2 hours ago by dotscom26
Source: 2005 Estonia IMO Training Test p2
On the side BC of triangle $ABC$, the points $M$ and $N$ are taken such that the point $M$ lies between the points $B$ and $N$, and $| BM | = | CN |$. On segments $AN$ and $AM$, points $P$ and $Q$ are taken so that $\angle PMC = \angle  MAB$ and $\angle QNB = \angle NAC$. Prove that $\angle QBC = \angle PCB$.
1 reply
parmenides51
Sep 24, 2020
dotscom26
2 hours ago
Bosnia and Herzegovina EGMO TST 2017 Problem 2
gobathegreat   2
N 2 hours ago by anvarbek0813
Source: Bosnia and Herzegovina EGMO Team Selection Test 2017
It is given triangle $ABC$ and points $P$ and $Q$ on sides $AB$ and $AC$, respectively, such that $PQ\mid\mid BC$. Let $X$ and $Y$ be intersection points of lines $BQ$ and $CP$ with circumcircle $k$ of triangle $APQ$, and $D$ and $E$ intersection points of lines $AX$ and $AY$ with side $BC$. If $2\cdot DE=BC$, prove that circle $k$ contains intersection point of angle bisector of $\angle BAC$ with $BC$
2 replies
gobathegreat
Sep 19, 2018
anvarbek0813
2 hours ago
Another NT FE
nukelauncher   58
N 2 hours ago by andrewthenerd
Source: ISL 2019 N4
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
58 replies
nukelauncher
Sep 22, 2020
andrewthenerd
2 hours ago
Easiest Functional Equation
NCbutAN   7
N 2 hours ago by InftyByond
Source: Random book
Find all functions $f: \mathbb R \to \mathbb R$ such that $$f(yf(x)+f(xy))=(x+f(x))f(y)$$Follows for all reals $x,y$.
7 replies
NCbutAN
Mar 2, 2025
InftyByond
2 hours ago
9 Pi or Tau
jkim0656   51
N 3 hours ago by ohiorizzler1434
Hey Aops!
Pi = Circumfrence/Diameter
Tau = Circumfrence/Radius
I have noticed a lot of sites, including Khan Academy, in support of tau over pi...
so what do you think?
https://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/
However i am still in support of the good ol pi :)
(btw this is my first aops poll) :-D

EDIT: 50 votes!!! :play_ball:
EDIT: 100 votes!!! :jump:
EDIT: 150 votes! :trampoline:

If u support pi pls upvote :)
51 replies
jkim0656
Friday at 1:48 PM
ohiorizzler1434
3 hours ago
Is your state listed?
Chatelet1   287
N Today at 3:05 AM by Nioronean
Multiple states have announced their top students who will advance to the 2025 MATHCOUNTS National Competition in May:

• From Alabama: Henry Gladden of Mobile, Austin Lu of Birmingham, Jessie Shi of Vestavia, and Minlu Wang-He of Auburn.

• From Arkansas: Ryan Fan of Fayetteville, Vivek Kalyankar of Fayetteville, Evan Ning of Fayetteville and Charles Yao of Conway.

• From Connecticut: Hayden Hughes of Newtown, Ethan Shi of Riverside, Alex Svoronos of Greenwich and Elaine Zhou of Hamden.

• From the Department of Defense: Narmin Guliyeva of Ankara, Turkey; Taeyul Kim of Manana, Bahrain; Nathan Liang of Wiesbaden, Germany; and Lucas Sze of Okinawa, Japan.

• From Hawaii: Taehwan Jeon, Hilohak Kwak, Isaac Qian and Thien Tran, all from Honolulu.

• From Kansas: Haidan Anderson & Jayden Xue of Overland Park, Christopher Spencer of Manhattan, and Ruby Jiang of Lawrence.

• From Maine: Ana Kanitkar & Connor Kirkham of Falmouth, Anna McClary of Hermon and Poppy Sandin of Bar Harbor.

• From Massachusetts: Eric Huang of Acton, Shlok Mukund & Brandon Ni of Lexington, and Soham Samanta of Medford.

• From Missouri: Lucas Lai of Columbia, Kevin Shi of St. Louis, Charles Yong & Jay Zhou of Chesterfield.

• From Montana: Titus Gilder of Missoula, Otis Heggem of Billings, Kaleb Houtz of Great Falls and Evan Newcomer of Missoula.

• From Nevada: Solomon Dumont of Las Vegas, Aaron Lei of Reno, Leeoz Nebat of Henderson and Maxwell Tsai of Las Vegas.

• From New Mexico: Mark Goldman, Daniel He, Iris Huang and Patrick McArdle, all from Albuquerque.

• From New York: Derrick Chen of Great Neck, Victor Yang of Great Neck, Hanru Zhang of Jericho and Ryan Zhang of Jericho.

• From Rhode Island: Kahlan Anderson of the Wheeler School, Julian Bernhoft & Colin Hegstrom of Providence, and Theodora Watson of Barrington.

• From South Carolina: Yukai Hu of Elgin, Justin Peng of Clemson, Geonhoo Shim of Columbia, and Aaron Wang of Mount Pleasant.

• From South Dakota: Seth Chaplin & Maxwell Wang of Sioux Falls, Laukia Gundewar of Aberdeen, and Cohwen Heimann of Aberdeen.

• From Texas: Shaheem Samsudeen & Ayush Narayan of Plano, Nathan Liu of Richardson, and James Stewart of Southlake.

• From Vermont: Mohid Ali of South Burlington, Vivek Chadive of South Burlington, Joshua Kratze of St. Johnsbury and Albert Zhang of South Burlington.

• From Wisconsin: August Reeder & Lucy Chen of Fitchburg, Junhao Feng of Milwaukee, and Jiyan Singh of River Hills.

===
Updated on 3/15/2025:

• From Colorado: Noah Liu, Christopher Zhu, Neo Luo, and Andrew Zhao.

• From Florida: Arnav Bhatia, Gnaneswar Peddesugari, Edwin Gao, and Rananjay Parmar.

• From Indiana: Roland Li, Hrishabh Bhowmik, Sophia Chen, and Arjun Raman.

• From Kentucky: Sri Shubhaan Vulava, Joyce Liu, Victor Gong, and Brandon Tedja.

• From Maryland: Eric Xie, Angie Zhu, Roger Huang, and Leo Su.

• From Michigan: Arnav Vunnam, Eric Jin, Akshaj Malraj, and Chaithanya Budida.

• From Minnesota: Ahmed Ilyasov, Will Masanz, Anshdeep Singh, and Branden Qiao.

• From New Jersey: Ethan Imanuel, Advait Joshi, Jay Wang, and Easton Wei.

• From North Carolina: Shivank Chintalpati, Steven Wang, Lucas Li, and Leo Hong.

• From Ohio: Henry Lu, Andy Mo, Archishmen Dey, and Caleb Tan.

• From Oregon: Sophia Han, Kevin Cheng, Garud Shah, and Ryan Zhang.
287 replies
Chatelet1
Mar 8, 2025
Nioronean
Today at 3:05 AM
Algebra B Videos Posted
BabaLama   1
N Today at 2:48 AM by jb2015007
I could be really late on this but I just noticed that the Algebra B series videos were finished pretty recently.
1 reply
BabaLama
Today at 2:28 AM
jb2015007
Today at 2:48 AM
quadratics
luciazhu1105   13
N Today at 2:33 AM by Charizard_637
I really need help on quadratics and I don't know why I also kinda need a bit of help on graphing functions and finding the domain and range of them.
13 replies
luciazhu1105
Feb 14, 2025
Charizard_637
Today at 2:33 AM
PLS help me come up with a faster solution to this problem!
ilikemath247365   7
N Mar 13, 2025 by jkim0656
2023 National Sprint Problem 30

Let $M = \frac{9^{40,000}}{9^{200} - 2}$. If $M$ is rounded to the nearest integer and then divided by $100$, what is the remainder?
7 replies
ilikemath247365
Mar 13, 2025
jkim0656
Mar 13, 2025
PLS help me come up with a faster solution to this problem!
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ilikemath247365
219 posts
#1
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2023 National Sprint Problem 30

Let $M = \frac{9^{40,000}}{9^{200} - 2}$. If $M$ is rounded to the nearest integer and then divided by $100$, what is the remainder?
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apex304
519 posts
#2 • 1 Y
Y by ilikemath247365
here is a good solution:

https://www.youtube.com/watch?v=f5VPfewADlM
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ilikemath247365
219 posts
#3
Y by
My solution:
Let $x = 9^{200}$ to make things easier. Now, let us consider the expression $\frac{x^{200}}{x - 2}$. By doing polynomial long division on this expression, I realized that the expression was just:
$x^{199} + 2x^{198} + ..... + 2^{198}x^{2} + 2^{199}x$ plus some random fraction that we get. Since the person is rounding it to the nearest integer, we don't need to worry about that fraction. Now, we realize that $9^{200}$ leaves a remainder of $1$ when divided by $100$(just by simple inspection). Therefore, plugging everything in and taking $mod(100)$, we get the geometric series: $1 + 2 + 4 + 8 + ... + 2^{198} + 2^{199}$. This simply evaluates to $2^{200} - 1$. Now, $2^{200}$ leaves a remainder of $76$ when divided by a $100$ so our final answer should be $\boxed{75}$.

Can someone help me come up with a faster solution under actual time pressure? Thanks!
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aidan0626
1719 posts
#4 • 1 Y
Y by ilikemath247365
imma be fr that's basically how i did it when i solved it rn
um you don't have to expand it into the full polynomial form tho, you can leave it as $\frac{x^{200}-2^{200}}{x-2}$, which might be a bit faster to think about
and to get that $2^{200}\equiv76\pmod{100}$ quickly, you can use euler's and crt to get 0 mod 4 and 1 mod 25 which gives 76 mod 100 (prob how you did it but wtv)
This post has been edited 1 time. Last edited by aidan0626, Mar 13, 2025, 4:36 AM
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ilikemath247365
219 posts
#5
Y by
Explanations to why $9^{200}$ is congruent to $1(mod 100)$ and e.t.c:
By the Euler Totient Function, we have:
$3^{40}$ leaves a remainder of $1$ when divided by $100$. Since $9^{200}$ is equal to $3^{400}$, this is just congruent to $1(mod 100)$.
Now, $2^{10}$ leaves a remainder of $24$ when divided by $100$. Now, we just need to simplify $24^{20}$ modulo $100$. Notice $24^{2}$ leaves a remainder of $76$. So now, we need to figure out $76^{10}$ modulo $100$. Now, $76^{2}$ leaves a remainder of $76$ when divided by a $100$. Therefore, we need to simplify $76^{5}$ modulo $100$. We can simplify this by noting that $76^{4}$ leaves a remainder of $76$ when dividing by a $100$. Lastly, we need to multiply by that last $76$ to get $76^{2}$ modulo $100$ which just simplifies to $76$ modulo $100$ as desired.
So you guys can see how long this problem took me to solve. Under real time pressure, I would take most of the time trying to solve this problem if I were crazy enough to attempt #30. This clearly doesn't help me time manage particularly well. If anyone has a simpler, basic solution that can be used to solve these type of problems really quickly, please let me know. Thanks again!
This post has been edited 1 time. Last edited by ilikemath247365, Mar 13, 2025, 4:39 AM
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aidan0626
1719 posts
#6
Y by
and to get that $2^{200}\equiv76\pmod{100}$ quickly, you can use euler's and crt to get 0 mod 4 and 1 mod 25 which gives 76 mod 100 (prob how you did it but wtv)

this way is probably faster
since 0 mod 4 is very easy to see, and 2^20 is 1 mod 25 by euler's so 2^200 is also 1 mod 25
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ilikemath247365
219 posts
#7
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Oh yes, you're right! Thanks for the faster way to get $2^{200}$ modulo $100$. I did it as shown above which took like 5 minutes! :(
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jkim0656
245 posts
#8 • 1 Y
Y by ilikemath247365
i like the sol :)
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