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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies
jlacosta
Apr 2, 2025
0 replies
2025 ELMOCOUNTS - Mock MATHCOUNTS Nationals
vincentwant   26
N 15 minutes ago by vincentwant
text totally not copied over from wmc (thanks jason <3)
Quick Links:
[list=disc]
[*] National: (Sprint) (Target) (Team) (Sprint + Target Submission) (Team Submission) [/*]
[*] Miscellaneous: (Leaderboard) (Private Discussion Forum) [/*]
[/list]
(Above links will release soon.)
-----
Eddison Chen (KS '22 '24), Aarush Goradia (CO '24), Ethan Imanuel (NJ '24), Benjamin Jiang (FL '23 '24), Rayoon Kim (PA '23 '24), Jason Lee (NC '23 '24), Puranjay Madupu (AZ '23 '24), Andy Mo (OH '23 '24), George Paret (FL '24), Arjun Raman (IN '24), Vincent Wang (TX '24), Channing Yang (TX '23 '24), and Jefferson Zhou (MN '23 '24) present:



[center]IMAGE[/center]

[center]Image credits to Simon Joeng.[/center]

2024 MATHCOUNTS Nationals alumni from all across the nation have come together to administer the first-ever ELMOCOUNTS Competition, a mock written by the 2024 Nationals alumni given to the 2025 Nationals participants. By providing the next generation of mathletes with free, high quality practice, we're here to boast how strong of an alumni community MATHCOUNTS has, as well as foster interest in the beautiful art that is problem writing!

The tests and their corresponding submissions forms will be released here on Monday, April 21, 2025. The deadline is May 10, 2025. Tests can be administered asynchronously at your home or school, and your answers should be submitted to the corresponding submission form. If you include your AoPS username in your submission, you will be granted access to the private discussion forum on AoPS, where you can discuss the tests even before the deadline.
[list=disc]
[*] "How do I know these tests are worth my time?" [/*]
[*] "Who can participate?" [/*]
[*] "How do I sign up?" [/*]
[*] "What if I have multiple students?" [/*]
[*] "What if a problem is ambiguous, incorrect, etc.?" [/*]
[*] "Will there be solutions?" [/*]
[*] "Will there be a Countdown Round administered?" [/*]
[/list]
If you have any other questions, feel free to email us at elmocounts2025@gmail.com!
26 replies
+2 w
vincentwant
3 hours ago
vincentwant
15 minutes ago
2025 USA IMO
john0512   72
N 39 minutes ago by jb2015007
Congratulations to all of you!!!!!!!

Alexander Wang
Hannah Fox
Karn Chutinan
Andrew Lin
Calvin Wang
Tiger Zhang

Good luck in Australia!
72 replies
john0512
Yesterday at 1:40 AM
jb2015007
39 minutes ago
SL Difficulty Level
MajesticCheese   0
an hour ago
Is there a rough difficulty comparison between IMO shortlist questions and USAMO questions? For example,

SL 1, 2, 3 -> USAMO P1
SL 4, 5, 6 -> USAMO P2
SL 7, 8, 9 -> USAMO P3

(This is just my guess; probably not correct)

Also feel free to compare it with other competitions(like the jmo) as well! :-D
0 replies
MajesticCheese
an hour ago
0 replies
VOLUNTEERING OPPORTUNITY OPEN TO HIGH/MIDDLE SCHOOLERS
im_space_cadet   11
N 2 hours ago by Lhaj3
Hi everyone!
Do you specialize in contest math? Do you have a passion for teaching? Do you want to help leverage those college apps? Well, I have something for all of you.

I am im_space_cadet, and during the fall of last year, I opened my non-profit DeltaMathPrep which teaches students preparing for contest math the problem-solving skills they need in order to succeed at these competitions. Currently, we are very much understaffed and would greatly appreciate the help of more tutors on our platform.

Each week on Saturday and Wednesday, we meet once for each competition: Wednesday for AMC 8 and Saturday for AMC 10 and we go over a past year paper for the entire class. On both of these days, we meet at 9PM EST in the night.

This is a great opportunity for anyone who is looking to have a solid activity to add to their college resumes that requires low effort from tutors and is very flexible with regards to time.

This is the link to our non-profit for anyone who would like to view our initiative:
https://www.deltamathprep.org/

If you are interested in this opportunity, please send me a DM on AoPS or respond to this post expressing your interest. I look forward to having you all on the team!

Thanks,
im_space_cadet
11 replies
im_space_cadet
Today at 2:23 PM
Lhaj3
2 hours ago
Find the answer
JetFire008   1
N 3 hours ago by Filipjack
Source: Putnam and Beyond
Find all pairs of real numbers $(a,b)$ such that $ a\lfloor bn \rfloor = b\lfloor an \rfloor$ for all positive integers $n$.
1 reply
JetFire008
Today at 12:31 PM
Filipjack
3 hours ago
Pyramid packing in sphere
smartvong   2
N 5 hours ago by smartvong
Source: own
Let $A_1$ and $B$ be two points that are diametrically opposite to each other on a unit sphere. $n$ right square pyramids are fitted along the line segment $\overline{A_1B}$, such that the apex and altitude of each pyramid $i$, where $1\le i\le n$, are $A_i$ and $\overline{A_iA_{i+1}}$ respectively, and the points $A_1, A_2, \dots, A_n, A_{n+1}, B$ are collinear.

(a) Find the maximum total volume of $n$ pyramids, with altitudes of equal length, that can be fitted in the sphere, in terms of $n$.

(b) Find the maximum total volume of $n$ pyramids that can be fitted in the sphere, in terms of $n$.

(c) Find the maximum total volume of the pyramids that can be fitted in the sphere as $n$ tends to infinity.

Note: The altitudes of the pyramids are not necessarily equal in length for (b) and (c).
2 replies
smartvong
Apr 13, 2025
smartvong
5 hours ago
Converging product
mathkiddus   5
N Today at 2:40 PM by NamelyOrange
Source: mathkiddus
Evaluate the infinite product, $$\prod_{n=1}^{\infty} \frac{7^n - n}{7^n + n}.$$
5 replies
mathkiddus
Apr 18, 2025
NamelyOrange
Today at 2:40 PM
Interesting Limit
Riptide1901   1
N Today at 1:45 PM by Svyatoslav
Find $\displaystyle\lim_{x\to\infty}\left|f(x)-\Gamma^{-1}(x)\right|$ where $\Gamma^{-1}(x)$ is the inverse gamma function, and $f^{-1}$ is the inverse of $f(x)=x^x.$
1 reply
Riptide1901
Apr 18, 2025
Svyatoslav
Today at 1:45 PM
2022 Putnam B1
giginori   25
N Today at 12:13 PM by cursed_tangent1434
Suppose that $P(x)=a_1x+a_2x^2+\ldots+a_nx^n$ is a polynomial with integer coefficients, with $a_1$ odd. Suppose that $e^{P(x)}=b_0+b_1x+b_2x^2+\ldots$ for all $x.$ Prove that $b_k$ is nonzero for all $k \geq 0.$
25 replies
giginori
Dec 4, 2022
cursed_tangent1434
Today at 12:13 PM
Number of A^2=I3
EthanWYX2009   1
N Today at 11:21 AM by loup blanc
Source: 2025 taca-14
Determine the number of $A\in\mathbb F_5^{3\times 3}$, such that $A^2=I_3.$
1 reply
EthanWYX2009
Today at 7:51 AM
loup blanc
Today at 11:21 AM
Prove this recursion!
Entrepreneur   3
N Today at 11:06 AM by quasar_lord
Source: Amit Agarwal
Let $$I_n=\int z^n e^{\frac 1z}dz.$$Prove that $$\color{blue}{I_n=(n+1)!I_0+e^{\frac 1z}\sum_{n=1}^n n! z^{n+1}.}$$
3 replies
Entrepreneur
Jul 31, 2024
quasar_lord
Today at 11:06 AM
Pove or disprove
Butterfly   1
N Today at 10:05 AM by Filipjack

Denote $y_n=\max(x_n,x_{n+1},x_{n+2})$. Prove or disprove that if $\{y_n\}$ converges then so does $\{x_n\}.$
1 reply
Butterfly
Today at 9:34 AM
Filipjack
Today at 10:05 AM
fractional binomial limit sum
Levieee   3
N Today at 9:44 AM by Levieee
this was given to me by a friend

$\lim_{n \to \infty} \sum_{k=1}^{n}{\frac{1}{\binom{n}{k}}}$

a nice solution using sandwich is
$\frac{1}{n}   + \frac{1}{n} + 1 + \frac{n-3}{\binom{n}{2}} \ge \frac{1}{n} +  \sum_{k=2}^{n-2}{\frac{1}{\binom{n}{k}}}+ \frac{1}{n} + 1 \ge \frac{1}{n} +  + \frac{1}{n} + 1$

therefore $\lim_{n \to \infty} \sum_{k=1}^{n}{\frac{1}{\binom{n}{k}}}$ = $1$

ALSO ANOTHER SOLUTION WHICH I WAS THINKING OF WITHOUT SANDWICH BUT I CANT COMPLETE WAS TO USE THE GAMMA FUNCTION

we know

$B(x, y) = \int_0^1 t^{x - 1} (1 - t)^{y - 1} \, dt$

$B(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x + y)}$

and $\Gamma(n) = (n-1)!$ for integers,

$\frac{1}{\binom{n}{k}}$ = $\frac{k! (n-k)!}{n!}$

therefore from the gamma function we get

$ (n+1) \int_{0}^{1}  x^k (1-x)^{n-k} dx$ = $\frac{1}{\binom{n}{k}}$ = $\frac{k! (n-k)!}{n!}$
$\Rightarrow$ $\lim_{n \to \infty} (n+1) \int_{0}^{1} \sum_{k=1}^{n} x^k (1-x)^{n-k} dx$ $=\lim_{n \to \infty} \sum_{k=1}^{n}{\frac{1}{\binom{n}{k}}}$

somehow im supposed to show that

$\lim_{n \to \infty} (n+1) \int_{0}^{1} \sum_{k=1}^{n} x^k (1-x)^{n-k} dx$ $= 1$

all i could observe was if we do L'hopital (which i hate to do as much as you do)

i get $\frac{ \int_{0}^{1} \sum_{k=1}^{n} x^k (1-x)^{n-k} dx}{1/n+1}$

now since $x \in (0,1)$ , as $n \to \infty$ the $(1-x)^{n-k} \to 0$ which gets us the $\frac{0}{0}$ form therefore L'hopital came to my mind , which might be a completely wrong intuition, anyway what should i do to find that limit

:noo: :pilot:
3 replies
Levieee
Yesterday at 9:51 PM
Levieee
Today at 9:44 AM
Integrate exp(x-10cosh(2x))
EthanWYX2009   2
N Today at 8:01 AM by Moubinool
Source: 2024 May taca-14
Determine the value of
\[I=\int\limits_{-\infty}^{\infty}e^{x-10\cosh (2x)}\mathrm dx.\]
2 replies
EthanWYX2009
Apr 18, 2025
Moubinool
Today at 8:01 AM
Question
HopefullyMcNats2025   15
N Apr 9, 2025 by nsking_1209
Is it more difficult to make MOP or make usajmo, usapho, and usabo
15 replies
HopefullyMcNats2025
Apr 7, 2025
nsking_1209
Apr 9, 2025
Question
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HopefullyMcNats2025
45 posts
#1
Y by
Is it more difficult to make MOP or make usajmo, usapho, and usabo
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HopefullyMcNats2025
45 posts
#2
Y by
Btw Usabo and usapho no one cares, I mean usapho and usabo camps
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ConfidentKoala4
597 posts
#3
Y by
all hard, usabo prolly easiest
mop prolly hardest
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HopefullyMcNats2025
45 posts
#4
Y by
Thanks man
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xHypotenuse
772 posts
#5 • 3 Y
Y by happyhippos, aidan0626, Elephant200
No one cares about USABO/USAPhO is wild
This post has been edited 1 time. Last edited by xHypotenuse, Apr 7, 2025, 7:41 PM
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Ruegerbyrd
1061 posts
#6
Y by
MOP, USA(J)MO, usapho, usabo
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MC_ADe
176 posts
#7
Y by
u should talk about camping in these respective things instead cuz MOP is literally camping so u need to like say phys camp or bio camp not jst usapho or usabo
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TetraFish
1088 posts
#8
Y by
I think he means MOP vs making that stage in all 3 of them combined.
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LockingIN2026
5 posts
#9
Y by
TetraFish wrote:
I think he means MOP vs making that stage in all 3 of them combined.

prob the latter
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elasticwealth
317 posts
#10
Y by
MOP >> USABO Camp > USAPhO Camp >>> USAMO Qual > JMO Qual >>> USAPhO qual / USABO Semifinal Qual > AIME
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NoSignOfTheta
1713 posts
#11
Y by
MC_ADe wrote:
u should talk about camping in these respective things instead cuz MOP is literally camping so u need to like say phys camp or bio camp not jst usapho or usabo
HopefullyMcNats2025 wrote:
Btw Usabo and usapho no one cares, I mean usapho and usabo camps
Z K Y
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sadas123
1226 posts
#13
Y by
elasticwealth wrote:
MOP >> USABO Camp > USAPhO Camp >>> USAMO Qual > JMO Qual >>> USAPhO qual / USABO Semifinal Qual > AIME

That is somewhat of the list.
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happyhippos
2197 posts
#14
Y by
HopefullyMcNats2025 wrote:
Btw Usabo and usapho no one cares, I mean usapho and usabo camps

That's an interesting thing to say, considering USAPhO-style physics is much more relevant to actually useful real-life things than USAMO-style math.
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mdk2013
605 posts
#15
Y by
fake123 wrote:
elasticwealth wrote:
MOP >> USABO Camp > USAPhO Camp >>> USAMO Qual > JMO Qual >>> USAPhO qual / USABO Semifinal Qual > AIME

wdym qualifying for AMC 8 is harder

:ewpu: wut
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Jack_w
109 posts
#16
Y by
TetraFish wrote:
I think he means MOP vs making that stage in all 3 of them combined.

MOP harder
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nsking_1209
171 posts
#17
Y by
MOP > usabo camp > usapho camp > usaco camp > usajmo
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