Infinite sum

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k a May Highlights and 2025 AoPS Online Class Information

jlacosta 0

May 1, 2025

May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies

jlacosta

May 1, 2025

0 replies

J

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k i Adding contests to the Contest Collections

dcouchman 1

N Apr 5, 2023 by v_Enhance

Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add

1 reply

dcouchman

Sep 9, 2019

v_Enhance

Apr 5, 2023

J

H

k i Zero tolerance

ZetaX 49

N May 4, 2019 by NoDealsHere

Source: Use your common sense! (enough is enough)

Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:

To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.

More specifically:

For new threads:

a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"

b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".

c) Good problem statement:
Some recent really bad post was:
[quote] $lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$ [/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.

For answers to already existing threads:

d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$ , do not answer with " $x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like " $x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.

To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!

Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.

49 replies

ZetaX

Feb 27, 2007

NoDealsHere

May 4, 2019

J

H

NT Game in Iran TST

M11100111001Y1R 6

N an hour ago by sami1618

Source: Iran TST 2025 Test 1 Problem 2

Suppose $p$ is a prime number. We have a number of cards, each of which has a number written on it such that each of the numbers $1, \dots, p-1$ appears at most once and $0$ exactly once. To design a game, for each pair of cards $x$ and $y$ , we want to determine which card wins over the other. The following conditions must be satisfied:

$a)$ If card $x$ wins over card $y$ , and card $y$ wins over card $z$ , then card $x$ must also win over card $z$ .

$b)$ If card $x$ does not win over card $y$ , and card $y$ does not win over card $z$ , then for any card $t$ , card $x + z$ must not win over card $y + t$ .

What is the maximum number of cards such that the game can be designed (i.e., one card does not defeat another unless the victory is symmetric or consistent)?

6 replies

M11100111001Y1R

Yesterday at 6:19 AM

sami1618

an hour ago

J

H

Brilliant Problem

M11100111001Y1R 1

N an hour ago by aaravdodhia

Source: Iran TST 2025 Test 3 Problem 3

Find all sequences $(a_n)$ of natural numbers such that for every pair of natural numbers $r$ and $s$ , the following inequality holds:
$\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2$

1 reply

M11100111001Y1R

Yesterday at 7:28 AM

aaravdodhia

an hour ago

J

H

Looking for someone to work with

midacer 3

N 2 hours ago by midacer

I’m looking for a motivated study partner (or small group) to collaborate on college-level competition math problems, particularly from contests like the Putnam, IMO Shortlist, IMC, and similar. My goal is to improve problem-solving skills, explore advanced topics (e.g., combinatorics, NT, analysis), and prepare for upcoming competitions. I’m new to contests but have a strong general math background(CPGE in Morocco). If interested, reply here or DM me to discuss

3 replies

midacer

5 hours ago

midacer

2 hours ago

J

H

Problem 3

blug 2

N 2 hours ago by LeYohan

Source: Polish Junior Math Olympiad Finals 2025

Find all primes $(p, q, r)$ such that
$pq+4=r^4.$

2 replies

blug

Mar 15, 2025

LeYohan

2 hours ago

J

H

Dophantine equation

MENELAUSS 0

2 hours ago

Solve for $x;y \in \mathbb{Z}$ the following equation :
$3^x-8^y =2xy+1$

0 replies

MENELAUSS

2 hours ago

0 replies

J

H

New playlist for Geometry Treasure

Plane_geometry_youtuber 0

2 hours ago

Hi,

I restarted a new playlist which I will introduce about 1500 theorem of plane geometry. This will be a solid foundation for those who want to be familiar and proficient on plane geometry. I put the link below

https://www.youtube.com/watch?v=K7BIBOABuVk&list=PLucWiuOCb2ZrLiPY95kZ6HuywkaNpIEh8

Please share it to the people who need it.

Thank you!

0 replies

Plane_geometry_youtuber

2 hours ago

0 replies

J

H

Inequalities

pcthang 11

N 2 hours ago by flower417477

Prove that $|\cos x|+|\cos 2x|+\ldots+|\cos 2^nx|\geq \frac{n}{3}$

11 replies

pcthang

Dec 21, 2016

flower417477

2 hours ago

J

H

Isogonal Conjugates of Nagel and Gergonne Point

SerdarBozdag 6

N 2 hours ago by ohiorizzler1434

Source: http://math.fau.edu/yiu/Oldwebsites/Geometry2013Fall/Geometry2013Chapter12.pdf

Proposition 12.1.
(a) The isogonal conjugate of the Gergonne point is the insimilicenter of
the circumcircle and the incircle.
(b) The isogonal conjugate of the Nagel point is the exsimilicenter of the circumcircle and
the incircle.
Note: I need synthetic solution.

6 replies

SerdarBozdag

Apr 17, 2021

ohiorizzler1434

2 hours ago

J

H

Looking for someone to work with

midacer 1

N 3 hours ago by wipid98

I’m looking for a motivated study partner (or small group) to collaborate on college-level competition math problems, particularly from contests like the Putnam, IMO Shortlist, IMC, and similar. My goal is to improve problem-solving skills, explore advanced topics (e.g., combinatorics, NT, analysis), and prepare for upcoming competitions. I’m new to contests but have a strong general math background(CPGE in Morocco). If interested, reply here or DM me to discuss

1 reply

midacer

4 hours ago

wipid98

3 hours ago

J

H

USAMO 1983 Problem 2 - Roots of Quintic

Binomial-theorem 33

N 3 hours ago by SomeonecoolLovesMaths

Source: USAMO 1983 Problem 2

Prove that the roots of $x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0$ cannot all be real if $2a^2 < 5b$ .

33 replies

Binomial-theorem

Aug 16, 2011

SomeonecoolLovesMaths

3 hours ago

J

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Compact powers of 2

NO_SQUARES 1

N 4 hours ago by Isolemma

Source: 239 MO 2025 8-9 p3 = 10-11 p2

Let's call a power of two compact if it can be represented as the sum of no more than $10^9$ not necessarily distinct factorials of positive integer numbers. Prove that the set of compact powers of two is finite.

1 reply

NO_SQUARES

May 5, 2025

Isolemma

4 hours ago

J

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Possible values of determinant of 0-1 matrices

mathematics2004 3

N 6 hours ago by Isolemma

Source: 2021 Simon Marais, A3

Let $\mathcal{M}$ be the set of all $2021 \times 2021$ matrices with at most two entries in each row equal to $1$ and all other entries equal to $0$ .
Determine the size of the set $\{ \det A : A \in M \}$ .
Here $\det A$ denotes the determinant of the matrix $A$ .

3 replies

mathematics2004

Nov 2, 2021

Isolemma

6 hours ago

J

H

Infinite Sum

P162008 2

N Yesterday at 5:42 PM by smartvong

Source: Singapore Mathematics Tournament

Let $f(n)$ be the nearest integer to $\sqrt{n}$ .
Find the value of $\sum_{n=1}^{\infty} \frac{(\frac{3}{2})^{f(n)} + (\frac{3}{2})^{-f(n)}}{(\frac{3}{2})^n}.$ Also, generalise your result.