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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Tuesday at 2:14 PM
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
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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.

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0 replies
jwelsh
Tuesday at 2:14 PM
0 replies
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Challenge: Make as many positive integers from 2 zeros
Biglion   13
N 4 minutes ago by aidan0626
How many positive integers can you make from at most 2 zeros, any math operation and cocatination?

Starting from 0, 0=0
13 replies
Biglion
Yesterday at 4:48 PM
aidan0626
4 minutes ago
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Inequalities
sqing   4
N 30 minutes ago by DAVROS
Suppose that $x$ and $y$ are nonzero real numbers such that $\left(x + \frac{1}{y} \right) \left(y + \frac{1}{x} \right) = 5$. Prove that
$$\left( x^2-y^2 + \frac{1}{y^2} \right) \left(y^2-x^2 + \frac{1}{x^2} \right)\leq \frac{7+3\sqrt{5}}{2}$$$$\left( x^3 -y^3+ \frac{1}{y^3} \right) \left(y^3 -x^3+ \frac{1}{x^3} \right)\leq 9+4\sqrt{5}$$
4 replies
sqing
Yesterday at 11:54 AM
DAVROS
30 minutes ago
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Counting & Probability
alexaop   2
N an hour ago by HAL9000sk
Six cars pull up to a red light, one at a time. At the light, there are three lanes, one left-turn lane, one straight-going lane, and one right-turn lane. How many ways can the cars stack up so that all three lanes are occupied?

Note that if the first car turns left and the second goes straight, this is considered different from the first car going straight and the second car turning left. In other words, the cars are distinguishable, but pull up to the intersection in a fixed order.

2 replies
alexaop
2 hours ago
HAL9000sk
an hour ago
J
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[PMO25 Areas I.20] A Number and its Divisors
kae_3   2
N 3 hours ago by BinariouslyRandom
Determine the sum of all positive integers $n$ for which $2[\tau(n)]^2=3n$, where $\tau(n)$ is the number of positive divisors of $n$.

Answer Confirmation
2 replies
kae_3
Feb 23, 2025
BinariouslyRandom
3 hours ago
J
No more topics!
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AIME I** 2024/15 misinterpreted
NamelyOrange   2
N Jun 17, 2025 by Yrock
original

Let $\sigma(n)$ be the sum-of-divisors function (as is normally notated, unlike the original question :skull:) How many divisors $n$ of $2024^{2024}$ are there such that $n^{\sigma(n)}\equiv \sigma(n^n) \pmod{128}$?

Is there any good way to do this misinterpreted version?
2 replies
NamelyOrange
Jun 15, 2025
Yrock
Jun 17, 2025
J
AIME I** 2024/15 misinterpreted
G H J
Reveal topic tags
number theory
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NamelyOrange
547 posts
NamelyOrange
#1 h  Jun 15, 2025, 6:01 PM
Y by
original

Let $\sigma(n)$ be the sum-of-divisors function (as is normally notated, unlike the original question :skull:) How many divisors $n$ of $2024^{2024}$ are there such that $n^{\sigma(n)}\equiv \sigma(n^n) \pmod{128}$?

Is there any good way to do this misinterpreted version?
This post has been edited 2 times. Last edited by NamelyOrange, Jun 15, 2025, 6:12 PM
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NamelyOrange
547 posts
NamelyOrange
#2 h  Jun 17, 2025, 3:22 AM
Y by
Bump. Any ideas? Or is this version just hopelessly convoluted or bashy? If so, which is it?
This post has been edited 1 time. Last edited by NamelyOrange, Jun 17, 2025, 3:23 AM
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Yrock
1446 posts
Yrock
#3 h  Jun 17, 2025, 5:19 AM
Y by
NamelyOrange wrote:
Bump. Any ideas? Or is this version just hopelessly convoluted or bashy? If so, which is it?

should be possible with python
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