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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.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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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
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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
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Putnam 1958 November B1
sqrtX   9
N 35 minutes ago by KAME06
Source: Putnam 1958 November
Given
$$b_n = \sum_{k=0}^{n} \binom{n}{k}^{-1}, \;\; n\geq 1,$$prove that
$$b_n = \frac{n+1}{2n} b_{n-1} +1, \;\; n \geq 2.$$Hence, as a corollary, show
$$ \lim_{n \to \infty} b_n =2.$$
9 replies
1 viewing
sqrtX
Jul 19, 2022
KAME06
35 minutes ago
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A Projection Theorem
buratinogigle   2
N an hour ago by wh0nix
Source: VN Math Olympiad For High School Students P1 - 2025
In triangle $ABC$, prove that
\[ a = b\cos C + c\cos B. \]
2 replies
buratinogigle
4 hours ago
wh0nix
an hour ago
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Turbo's en route to visit each cell of the board
Lukaluce   18
N an hour ago by yyhloveu1314
Source: EGMO 2025 P5
Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey
18 replies
Lukaluce
Monday at 11:01 AM
yyhloveu1314
an hour ago
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Perhaps a classic with parameter
mihaig   1
N 2 hours ago by LLriyue
Find the largest positive constant $r$ such that
$$a^2+b^2+c^2+d^2+2\left(abcd\right)^r\geq6$$for all reals $a\geq1\geq b\geq c\geq d\geq0$ satisfying $a+b+c+d=4.$
1 reply
mihaig
Jan 7, 2025
LLriyue
2 hours ago
J
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Distribution of prime numbers
Rainbow1971   6
N 2 hours ago by Rainbow1971
Could anybody possibly prove that the limit of $$(\frac{p_n}{p_n + p_{n-1}})$$is $\tfrac{1}{2}$, maybe even with rather elementary means? As usual, $p_n$ denotes the $n$-th prime number. The problem of that limit came up in my partial solution of this problem: https://artofproblemsolving.com/community/c7h3495516.

Thank you for your efforts.
6 replies
Rainbow1971
Apr 9, 2025
Rainbow1971
2 hours ago
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Picking a College
missionsqhc   1
N 2 hours ago by zkyao
I applied to college as a math major, and my options are Georgetown, UVA, Stony Brook, and Binghamton. I was waitlisted from CMU, Columbia, Northwestern, Berkeley, Williams, UNC, and UMich.

I’ve done competition math throughout middle school and high school and obviously am currently slotted to study math. But I am also very much interested in politics, government, history, etc. I could easily see myself double majoring or even completely switching to something like political science or history. I don’t have a clear-cut vision for a future career. I used to really want to become a mathematician, but now I think it’s more likely that I’ll do something more “practical,” like finance or law. I also have aspirations of working in government, even possibly running for elected office.

If someone has gone to one of the school’s I’ve been accepted by or has experience in one of the careers I’ve mentioned (or possesses some other characteristics that gives insight into my situation), I would greatly appreciate your thoughts. On one hand, I really like Georgetown because of its strong programs in government, international relations, and other social sciences; its DC location; and its stated goal (which I hope is genuine) or educating students for life and not just work. But the hard sciences, and particularly math, are relatively smaller programs and less of the school’s emphasis. I worry that I may end up sticking mainly with math and would have been better off picking something like UVA or even Stony or Bing.

A related question I have regards how the undergraduate math departments compare at different schools. I wouldn't be surprised if the very top-tier places, like MIT, Caltech, CMU, Harvard, Stanford, and Princeton, were significantly stronger than Georgetown. But how does Georgetown compare to places that are good for math but not necessarily hyper-elite, like a Cornell or a UMich?

Also, Georgetown has a 3 + 2 program with Columbia Engineering, in which you study for three years at Georgetown to get a BA/BS in any major in any school (but preferably in math/science) and then study for two years at Columbia to get a BS in their engineering school. This seems like a way to get the best of both worlds between humanities and STEM (and to gain connections in both DC and NYC). If anyone has done this, please do share your experience.
1 reply
missionsqhc
4 hours ago
zkyao
2 hours ago
J
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Connected graph with k edges
orl   26
N 2 hours ago by Maximilian113
Source: IMO 1991, Day 2, Problem 4, IMO ShortList 1991, Problem 10 (USA 5)
Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.

Note: Graph-Definition. A graph consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x = v_{0},v_{1},v_{2},\cdots ,v_{m} = y\,$ such that each pair $ \,v_{i},v_{i + 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.
26 replies
orl
Nov 11, 2005
Maximilian113
2 hours ago
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3 var inquality
sqing   2
N 2 hours ago by sqing
Source: Own
Let $ a,b,c> 0 $ and $ 4(a+b) +3c-ab \geq10$ . Prove that
$$a^2+b^2+c^2+kabc\geq k+3$$Where $0\leq k \leq 1. $
$$a^2+b^2+c^2+abc\geq 4$$
2 replies
sqing
3 hours ago
sqing
2 hours ago
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Pls solve this FE
ItzsleepyXD   2
N 2 hours ago by ItzsleepyXD
Source: My friend
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x^2f(x+y))=f(xyf(x))+xf(x)^2\]for all real numbers $x$ and $y$.
2 replies
ItzsleepyXD
Nov 26, 2023
ItzsleepyXD
2 hours ago
J
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Determinant
Saucepan_man02   0
3 hours ago
Source: Own
Calculate the determinant:
$\begin{vmatrix} 1 & \sin\alpha & \sin2\alpha & \sin3\alpha\\ \sin\alpha & \sin2\alpha & \sin3\alpha & \sin4\alpha\\ \sin2\alpha & \sin3\alpha & \sin4\alpha & \sin5\alpha\\ \sin3\alpha & \sin4\alpha & \sin5\alpha & \sin6\alpha\\ \end{vmatrix}$
0 replies
Saucepan_man02
3 hours ago
0 replies
J
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The old one is gone.
EeEeRUT   3
N 3 hours ago by ItzsleepyXD
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.
3 replies
EeEeRUT
4 hours ago
ItzsleepyXD
3 hours ago
J
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Interesting inequalities
sqing   4
N 3 hours ago by sqing
Source: Own
Let $ a,b $ be reals such that $ a^2+ab+b^2 =1$ . Prove that
$$ \frac{8}{ 5 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq 1$$$$ \frac{9}{ 5 }\geq\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq 1$$$$ \frac{27}{ 14 }\geq \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq 1$$Let $ a,b $ be reals such that $ a^2+ab+b^2 =3$ . Prove that
$$ \frac{13}{ 10 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq \frac{1}{ 2 }$$$$ \frac{6}{ 5 }>\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq \frac{1}{ 5 }$$$$ \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq \frac{1}{ 14 }$$
4 replies
sqing
Yesterday at 8:32 AM
sqing
3 hours ago
J
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Ant wanna come to A
Rohit-2006   3
N 3 hours ago by Rohit-2006
An insect starts from $A$ and in $10$ steps and has to reach $A$ again. But in between one of the s steps and can't go $A$. Find probability. For example $ABCDCDEDEA$ is valid but $ABABCDEABA$ is not valid.

*Too many edits, my brain had gone to a trip
3 replies
Rohit-2006
Yesterday at 6:47 PM
Rohit-2006
3 hours ago
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BMO Shortlist 2021 A5
Lukaluce   16
N 3 hours ago by Sedro
Source: BMO Shortlist 2021
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$f(xf(x + y)) = yf(x) + 1$$holds for all $x, y \in \mathbb{R}^{+}$.

Proposed by Nikola Velov, North Macedonia
16 replies
Lukaluce
May 8, 2022
Sedro
3 hours ago
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J
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Problem on matrix algebra
Fly29   1
N Apr 4, 2025 by alexheinis
Is $\mathrm{GL}(\mathbb R^3)$ isomorphic to $\mathrm{GL}_3(\mathbb R)$?
1 reply
Fly29
Apr 4, 2025
alexheinis
Apr 4, 2025
J
Problem on matrix algebra
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Reveal topic tags
linear algebra
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Fly29
92 posts
Fly29
#1 h  Apr 4, 2025, 7:46 AM
Y by
Is $\mathrm{GL}(\mathbb R^3)$ isomorphic to $\mathrm{GL}_3(\mathbb R)$?
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alexheinis
10543 posts
alexheinis
#2 h  Apr 4, 2025, 8:21 AM
Y by
We fix a basis, for example the canonical basis $e_1,e_2,e_3$ of ${\bf R}^3$.
The group ${\rm GL}(R^3)$ consists of the invertible linear maps from $R^3$ to $R^3$, hence $3\times 3$-matrices over $R$ that are invertible.
The group ${\rm GL}_3(R)$ has the same elements. Since the group action in both actions is given by matrix multiplication, we indeed have ${\rm GL}(R^3)\cong {\rm GL}_3(R)$.
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