Plan ahead for the next school year. Schedule your class today!

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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
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0 replies
jwelsh
Jul 1, 2025
0 replies
Putnam 2018 A1
62861   32
N Yesterday at 8:40 PM by mudkip42
Find all ordered pairs $(a, b)$ of positive integers for which
\[\frac{1}{a} + \frac{1}{b} = \frac{3}{2018}.\]
32 replies
62861
Dec 2, 2018
mudkip42
Yesterday at 8:40 PM
Matrix equation
Natrium   4
N Yesterday at 8:05 PM by GreenKeeper
If $A$ is a complex matrix with $AA^*A=A^3,$ prove that $A$ is self-adjoint, i.e., that $A^*=A.$
4 replies
Natrium
Jul 12, 2025
GreenKeeper
Yesterday at 8:05 PM
NT By Probabilistic Method
EthanWYX2009   1
N Yesterday at 7:41 PM by Manivs
Source: 2024 March 谜之竞赛-6
Given a positive integer \( k \) and a positive real number \( \varepsilon \), prove that there exist infinitely many positive integers \( n \) for which we can find pairwise coprime integers \( n_1, n_2, \cdots, n_k \) less than \( n \) satisfying
\[\text{gcd}(\varphi(n_1), \varphi(n_2), \cdots, \varphi(n_k)) \geq n^{1-\varepsilon}.\]Proposed by Cheng Jiang from Tsinghua University
1 reply
EthanWYX2009
Yesterday at 6:57 AM
Manivs
Yesterday at 7:41 PM
Probability that x/y is odd
Stear14   2
N Yesterday at 4:12 PM by Stear14
Part A. $\ $ Let $\ x,y\in U[0,1]$. $\ $ Find the probability that the nearest integer to the ratio $\ x/y\ $ is odd.

Part B. $\ $ Let $\ x,y\in N(0,1)$. $\ $ Find the probability that the nearest integer to the ratio $\ x/y\ $ is odd.

In both cases, give the answers not as infinite series, but in terms of elementary functions and known constants.
2 replies
Stear14
Jul 13, 2025
Stear14
Yesterday at 4:12 PM
Go on a Journey
eashang1   1
N Sep 1, 2020 by fields123
After postponing it for well over a year, I finally watched Your Lie in April this week. The vibrant art and beautiful music complemented the bittersweet story, so chock-full of joy, sorrow, hope, and above all, passion.

When I read or watch something that really touches me, I often get a lingering sense of existential sadness. I definitely felt this with Your Lie in April. Maybe it's because I've been quarantined for months. Or that I've had a subtle sense of lacking direction in life since leaving high school. Or that, as a violinist and math contest enthusiast myself, I could identify with the music and themes of struggle and competition. Regardless, YLiA reminded me how lucky I am to have my life, and got me thinking about what I want to do with it.

When I was an underclassman in high school, I'd hear people talk about me as a tryhard who took part in all those math/QB contests despite being mediocre at best. Back then I told myself I wouldn't let others' opinions stop me from pursuing something I enjoyed. I was fueled purely by passion. These days I feel like I do my coursework more out of obligation. In life, I want to do something I have that true passion for. Something I love through which I can impact others. In YLiA, Kaori, the violinist, says that through his work, Mozart is asking us to "go on a journey." I suppose right now I'm at the point in my journey where I'm finding that something.

Problem 1
Solution
Remark
1 reply
eashang1
Aug 29, 2020
fields123
Sep 1, 2020
Investing
eashang1   0
Aug 1, 2020
My average day still pretty much consists of the same activities I talked about in my last few posts, but it's been almost two months since I last updated this blog, so I thought I'd write something here. Quarantine time really flies by...

Out of the blue, a friend of mine sent me a sign-up link to get a free stock. It looked kind off sketch so I ignored it, but when talking to another friend a couple of weeks later I learned that link was for the commission-free investing service Robinhood. As I haven't been doing much over quarantine, I thought I'd give it a try, and I set up a small account with $50.

At first, I invested in a bunch of well-known companies that took a big hit from quarantine and were slowly recovering. By the next day, I realized that it would take an eternity to make a significant amount from my non-volatile fractional stocks. I researched ways to make more significant gains in shorter periods of time, and there seemed to be a few answers. The first is to trade options. But with $50, options weren't an option (because). The second way is to day trade. Unfortunately, there is some regulation that prevents you from making more than three-day trades in a week unless your account is worth at least $25,000. The last, and arguably riskiest, way is to invest in penny stocks. These are highly volatile stocks worth under $5. The benefit of this is that with a small amount of money you can buy many cheap stocks. With more stocks of a company, a small percent increase in the value of the stock will have many times the effect on your portfolio. Since the other options weren't possible for me, I went with the penny stock strategy.

So I had $50. After the first few days of investing in safer companies and getting my free stock, my account was worth $56. One of the first penny stocks I invested heavily in was NOVN, a biopharmaceutical company (biopharma companies are generally very, very volatile). A few hours later their CEO left and the stock went down 20% :| . Until then most of my plays came from the wallstreetbets subreddit, since people on there sounded like they knew what they were doing. When choosing a penny stock to invest in, one of the companies I was looking at was Kodak. The stock price was $2 a few of days ago... and, after a $750M deal with the U.S. government, it reached a high of $60. I'm not really that sad I missed it since I was looking at KODK for all the wrong reasons (I should've looked at them since they had much higher than average volume with a relatively constant price, hinting at some sort of announcement). After NOVN and missing KODK, I started doing more research into what I invested in, and am currently making an analysis tool in Python, which I hope to talk about in more detail in my next post. The investing experience so far has been surprisingly fun, though I guess it's mostly because manyof my friends are also into it and I can't really lose that much money with $50.

Here's an interesting problem from a recent Codeforces contest:
Problem
Solution
Remark
0 replies
eashang1
Aug 1, 2020
0 replies
No more topics!
3xn matrice with combinatorical property
Sebaj71Tobias   1
N Jun 5, 2025 by c00lb0y
Let"s have a 3xn matrice with the following properties:
The firs row of the matrice is 1,2,3,... ,n in this order.
The second and the third rows are permutations of the first.
Very important, that in each column thera are different entries.
How many matrices with thees properties are there?

The answer for 2xn matrices is well-known, but what is the answer for 3xn, or for kxn ( k<=n) ?
1 reply
Sebaj71Tobias
Jun 1, 2025
c00lb0y
Jun 5, 2025
3xn matrice with combinatorical property
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Sebaj71Tobias
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#1
Y by
Let"s have a 3xn matrice with the following properties:
The firs row of the matrice is 1,2,3,... ,n in this order.
The second and the third rows are permutations of the first.
Very important, that in each column thera are different entries.
How many matrices with thees properties are there?

The answer for 2xn matrices is well-known, but what is the answer for 3xn, or for kxn ( k<=n) ?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
c00lb0y
22 posts
#2
Y by
my laTex is very bad, so i will write how i can. we just do combinatorics here, whenever u choose a_21 there are n-1 choices, after for a_22 u have n-2 choices, continue this process until 2nd row ends, then u will have (n-1)! , then start choosing 3rd row with same way, a_31 have n-3 choices, ..., after all 3rd row has (n-2)! choices, then our answer is (n-1)!(n-2)!
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