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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.

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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
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Base 2n of n^k
KevinYang2.71   42
N 41 minutes ago by sansgankrsngupta
Source: USAMO 2025/1, USAJMO 2025/2
Let $k$ and $d$ be positive integers. Prove that there exists a positive integer $N$ such that for every odd integer $n>N$, the digits in the base-$2n$ representation of $n^k$ are all greater than $d$.
42 replies
KevinYang2.71
Mar 20, 2025
sansgankrsngupta
41 minutes ago
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what actually happens after the usamo
bubby617   1
N 3 hours ago by Indpsolver
i keep getting different answers for how the selection process gets down from the usamo winners to the IMO team so can someone set the record straight for me
1 reply
bubby617
Today at 2:47 AM
Indpsolver
3 hours ago
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Inequalities
sqing   4
N 4 hours ago by DAVROS
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that$$a^3b+b^3c+c^3a+\frac{473}{256}abc\le\frac{27}{256}$$Equality holds when $ a=b=c=\frac{1}{3} $ or $ a=0,b=\frac{3}{4},c=\frac{1}{4} $ or $ a=\frac{1}{4} ,b=0,c=\frac{3}{4} $
or $ a=\frac{3}{4} ,b=\frac{1}{4},c=0. $
4 replies
sqing
Yesterday at 3:55 PM
DAVROS
4 hours ago
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Geometry Problem
JetFire008   1
N 4 hours ago by JetFire008
Equilateral $\triangle ADC$ is drawn externally on side $AC$ of $\triangle ABC$. Point $P$ is taken on $BD$. Find $\angle APC$ if $BD=PA+PB+PC$.
1 reply
JetFire008
5 hours ago
JetFire008
4 hours ago
J
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Distributing cupcakes
KevinYang2.71   18
N 5 hours ago by MathLuis
Source: USAMO 2025/6
Let $m$ and $n$ be positive integers with $m\geq n$. There are $m$ cupcakes of different flavors arranged around a circle and $n$ people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person $P$, it is possible to partition the circle of $m$ cupcakes into $n$ groups of consecutive cupcakes so that the sum of $P$'s scores of the cupcakes in each group is at least $1$. Prove that it is possible to distribute the $m$ cupcakes to the $n$ people so that each person $P$ receives cupcakes of total score at least $1$ with respect to $P$.
18 replies
KevinYang2.71
Friday at 12:00 PM
MathLuis
5 hours ago
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usamOOK geometry
KevinYang2.71   73
N 5 hours ago by deduck
Source: USAMO 2025/4, USAJMO 2025/5
Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.
73 replies
KevinYang2.71
Friday at 12:00 PM
deduck
5 hours ago
J
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Gunn Math Competition
the_math_prodigy   15
N 5 hours ago by the_math_prodigy
Gunn Math Circle is excited to host the fourth annual Gunn Math Competition (GMC)! GMC will take place at Gunn High School in Palo Alto, California on Sunday, March 30th. Gather a team of up to four and compete for over $7,500 in prizes! The contest features three rounds: Individual, Guts, and Team. We welcome participants of all skill levels, with separate Beginner and Advanced divisions for all students.

Registration is free and now open at compete.gunnmathcircle.org. The deadline to sign up is March 27th.

Special Guest Speaker: Po-Shen Loh!!!
We are honored to welcome Po-Shen Loh, a world-renowned mathematician, Carnegie Mellon professor, and former coach of the USA International Math Olympiad team. He will deliver a 30-minute talk to both students and parents, offering deep insights into mathematical thinking and problem-solving in the age of AI!

View competition manual, schedule, prize pool at compete.gunnmathcircle.org . For any questions, reach out at ghsmathcircle@gmail.com or ask in Discord.
15 replies
the_math_prodigy
Mar 8, 2025
the_math_prodigy
5 hours ago
J
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k Discord Server
mathprodigy2011   14
N Today at 3:00 AM by KF329
Theres a server where we are all like discussing problems+helping each other practice. Hopefully you guys can join.

https://discord.gg/6hN3w4eK
14 replies
mathprodigy2011
Friday at 11:00 PM
KF329
Today at 3:00 AM
J
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USAMO question
bubby617   2
N Today at 2:44 AM by Andyluo
if i had qualified for the usa(j)mo (i wish), would i have been flown out for free like mathcounts nationals or do you have to plan your own trip for going to the usamo
2 replies
bubby617
Today at 2:32 AM
Andyluo
Today at 2:44 AM
J
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A hard inequality
JK1603JK   2
N Today at 2:25 AM by sqing
Let a,b,c\ge 0: a+b+c=3. Prove \frac{1}{abc}+\frac{12}{a^2b+b^2c+c^2a}\ge 5.
2 replies
JK1603JK
Today at 1:40 AM
sqing
Today at 2:25 AM
J
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Number theory question with many (confusing) variables
urfinalopp   2
N Today at 2:07 AM by urfinalopp
Given m,n,p,q \in \mathbb{N+}, find all solutions to 2^{m}3^{n}+5^{p}=7^{q}$

One of the paths I've found is to boil it down to solving two non-simultaneous equations 2^{m_1}+5^{n_1}=7^{q_1} and
7^{m_1}+5^{n_1}=2^{q_1} but its too hard. Any other approaches/solutions or a continuation of this path?
2 replies
urfinalopp
Yesterday at 4:06 PM
urfinalopp
Today at 2:07 AM
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Number theory national Olympiad
LoRD2022   2
N Today at 12:09 AM by alexheinis
Find all polynomials with integer coefficients such that, $a^2+b^2-c^2|P(a)+P(b)-P(c)$ for all $a,b,c \in mathbb{Z}$.
2 replies
LoRD2022
Yesterday at 8:54 PM
alexheinis
Today at 12:09 AM
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Introduction & Intermediate C&P study guide!
HamstPan38825   25
N Yesterday at 11:47 PM by Andyluo
This took me quite a while to make, but enjoy!

Introduction to C&P (suitable for AMC 8, AMC 10/12)
Chapter 1 - This is like the "introduction", which is pretty easy and is not very important.
Chapter 2 - VERY important! Study this chapter closely, as it contains techniques that will be used again and again in harder problems.
Chapter 3 - Another quite important chapter, though not as important as chapter 2. This chapter covers some of the most confusing parts in C&P and even I can't distinguish that well in that chapter.
Chapter 4 - Interesting but very basic. Not that important, really.
Chapter 5 - Another interesting chapter, which should be studied in greater detail than Chapter 4. The distinguishability section is most important here.
Chapter 6 - Not much, but attempt the problems and read the examples since many of them are very interesting.
Chapter 7 - Pretty important chapter, make sure you read all the sections but not very interesting.
Chapter 8 - Another one of the VERY important sections - make sure read this section closely and do all the problems, since I still compare apples to oranges sometimes.
Chapter 9 - Interesting, but not very important. More important is the concept to "Think About It!"
Chapter 10 - The only topic in the entire C&P series that covers Geometric Probability, this chapter doesn't go into enough detail. Read it closely to get the basics, but I'd recommend doing more practice on Geometric Probability (I'll be making a handout!)
Chapter 11 - This chapter is not really important, reference the section in Intermediate C&P for a deeper understanding of Expected value.
Chapter 12 - Pretty important chapter, study it closely as it gives you the tools to prove combinatorial identities and Pascal's triangle is quite useful.
Chapter 13 - Just get the Hockey Stick Identity - not very useful chapter. Distributions will also be covered in Intermediate C&P.
Chapter 14 - A bit important, but not very - The binomial theorem is easy to master, but if you need more practice read the section in IA.
Chapter 15 - Similar to chapter 6, read all the examples and attempt all the problems here.

AMC 10/12 Chapters: 2, 3, 5, 6, 7, 8, 10, 12, 15

Intermediate C&P Suitable for late AMC 12, AIME + olympiads
Chapter 1 - Review this section thoroughly though there are no exercises here.
Chapter 2 - If you've learned set theory before, this chapter should be a review, but nonetheless skim over this chapter.
Chapter 3 - ANOTHER IMPORTANT CHAPTER! PIE is very important and might be a bit complicated, so study this chapter closely.
Chapter 4 - This chapter is also quite important - Make sure you master both parts of this chapter.
Chapter 5 - A good chapter, but it's a bit too short for my liking. Read extra handouts on the Pigeonhole Principle.
Chapter 6 - Another great chapter - attempt all the problems in this chapter!
Chapter 7 - Yet another very important chapter - distributions tend to pop up all over the place. Attempt all the problems here.
Chapter 8 - This isn't really a chapter - if you've mastered Mathematical Induction, you can just skip this but I recommend doing the problems.
Chapter 9 - This is really just the introduction to Chapter 10, but nonetheless do some of the problems to get a firm recursion basis.
Chapter 10 - Another VERY IMPORTANT CHAPTER! The recursion section is more important than the Catalan Number section unless you're preparing for olympiads.
Chapter 11 - Past this chapter, the concepts start to get quite advanced. This is an interesting chapter and is quite important, so do many of the problems here.
Chapter 12 - A great chapter! This chapter is quite general, but try to learn how to prove combinatorial identities on your own.
Chapter 13 - A quite complex chapter, not that important unless you're preparing for olympiads.
Chapter 14 - A hard but great chapter! GFs are hacks to many common counting problems.
Chapter 15 - Just skip this chapter unless you're doing the Putnam or olympiads, since it's basically nonexistent in the AMC/AIMEs.
Chapter 16 - Many of the problems here are very hard, but do as much as you can here! Try to attempt every single problem though they are very hard.

AMC 12 chapters: 1, 3, 4, 5, 6, 7, 9, 10
AIME chapters: 1, 3, 4, 5, 6, 7, 9, 10, 11
Olympiad chapters: 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15 [basically almost all of them rip]
25 replies
HamstPan38825
Dec 7, 2020
Andyluo
Yesterday at 11:47 PM
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Number theory national Olympiad
LoRD2022   0
Yesterday at 9:09 PM
Find all polynomials with integer coefficients such that, $a^2+b^2-c^2|P(a)+P(b)-P(c)$ for all $a,b,c \in \mathbb{Z}$.
0 replies
LoRD2022
Yesterday at 9:09 PM
0 replies
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USAPhO Exam
happyhippos   1
N 6 hours ago by RYang2
Every other thread on this forum is for USA(J)MO lol.

Anyways, to other USAPhO students, what are you doing to prepare? It seems too close to the test date (April 10) to learn new content, so I am just going through past USAPhO and BPhO exams to practice (untimed for now). How about you? Any predictions for what will be on the test this year? I'm completely cooked if there are any circuitry questions.
1 reply
happyhippos
Yesterday at 3:14 AM
RYang2
6 hours ago
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USAPhO Exam
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happyhippos
2191 posts
happyhippos
#1 h  Yesterday at 3:14 AM
Y by
Every other thread on this forum is for USA(J)MO lol.

Anyways, to other USAPhO students, what are you doing to prepare? It seems too close to the test date (April 10) to learn new content, so I am just going through past USAPhO and BPhO exams to practice (untimed for now). How about you? Any predictions for what will be on the test this year? I'm completely cooked if there are any circuitry questions.
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RYang2
1936 posts
RYang2
#2 h  6 hours ago
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A1: nonlinear circuit element
A2: nonlinear circuit element
A3: nonlinear circuit element
B1: nonlinear circuit element
B2: nonlinear circuit element
B3: easy mech
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