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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

Summer camps are starting this month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have a transformative summer experience. 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 upcoming events:
[list][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
[*]June 9th, Monday, 7:30pm ET, Game Jam: Operation Shuffle!, Come join us to play our second round of Operation Shuffle! If you enjoy number sense, logic, and a healthy dose of luck, this is the game for you. No specific math background is required; all are welcome.[/list]
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0 replies
+1 w
jlacosta
Jun 2, 2025
0 replies
J
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integral
Svyatoslav   0
14 minutes ago
How can we prove that at $\phi\in\Big(-\frac\pi2,\frac\pi2\Big)$
$$ \int_{-\infty}^\infty\frac{\sinh^2(\phi t)}{t\sinh(\pi t)}\,dt=-\ln(\cos\phi)\quad?$$
0 replies
Svyatoslav
14 minutes ago
0 replies
J
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Putnam 1980 B3
sqrtX   3
N 2 hours ago by Ilkin_H
Source: Putnam 1980
For which real numbers $a$ does the sequence $(u_n )$ defined by the initial condition $u_0 =a$ and the recursion $u_{n+1} =2u_n - n^2$ have $u_n >0$ for all $n \geq 0?$
3 replies
sqrtX
Apr 1, 2022
Ilkin_H
2 hours ago
J
H
BA = A^2 + B, then AB=BA?
Cubastic_Boy   8
N 3 hours ago by Ilkin_H
Let $ n \ge 2, A, B \in \mathbb{R} ^{n \times n}$ 2 matrices such that $ BA = A^2 + B$. Prove that$ AB = BA.$
8 replies
Cubastic_Boy
Nov 25, 2024
Ilkin_H
3 hours ago
J
H
Lagrange meets geometric mean
rogue   11
N 4 hours ago by Snoop76
Source: Kyiv Taras Shevchenko University Mechmat Competition 2025
Let $f\in C([a,b])$ and for every $x\in(a,b)$ there exists $f'(x)>0.$ Prove that there exists a point $a<c<b$ such that $f'(c)=\sqrt{\frac{f(c)-f(a)}{c-a}\cdot\frac{f(b)-f(c)}{b-c}}.$ (V.Brayman)
11 replies
rogue
Feb 20, 2025
Snoop76
4 hours ago
J
No more topics!
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Calculus
Bob96   1
N Jun 5, 2025 by Moubinool
Given a sequence \{a_n\} of positive real numbers that decrease to 0, and define f(x)=\sum_{n=1}^\infty a_n^n x^n. If \sum_{n=1}^\infty a_n diverges, prove that \int_1^\infty \frac{\ln f(x)}{x^2}dx diverges
1 reply
Bob96
Jun 5, 2025
Moubinool
Jun 5, 2025
J
Calculus
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Reveal topic tags
Improper integral
infinite series
Sequences
calculus
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Bob96
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Bob96
#1 h  Jun 5, 2025, 4:59 PM
Y by
Given a sequence \{a_n\} of positive real numbers that decrease to 0, and define f(x)=\sum_{n=1}^\infty a_n^n x^n. If \sum_{n=1}^\infty a_n diverges, prove that \int_1^\infty \frac{\ln f(x)}{x^2}dx diverges
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Moubinool
5577 posts
Moubinool
#2 h  Jun 5, 2025, 7:06 PM • 1 Y
Y by Bob96
Bob96 wrote:
Given a sequence $ \{a_n\}$ of positive real numbers that decrease to 0, and define $f(x)=\sum_{n=1}^\infty a_n^n x^n$ , If $\sum_{n=1}^\infty a_n $ diverges, prove that $\int_1^\infty \frac{\ln f(x)}{x^2}dx $diverges
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