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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.

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0 replies
jlacosta
May 1, 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
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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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ISI UGB 2025 P1
SomeonecoolLovesMaths   7
N 8 minutes ago by SatisfiedMagma
Source: ISI UGB 2025 P1
Suppose $f \colon \mathbb{R} \longrightarrow \mathbb{R}$ is differentiable and $| f'(x)| < \frac{1}{2}$ for all $x \in \mathbb{R}$. Show that for some $x_0 \in \mathbb{R}$, $f \left( x_0 \right) = x_0$.
7 replies
+1 w
SomeonecoolLovesMaths
May 11, 2025
SatisfiedMagma
8 minutes ago
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hard inequality omg
tokitaohma   5
N 16 minutes ago by math90
1. Given $a, b, c > 0$ and $abc=1$
Prove that: $ \sqrt{a^2+1} + \sqrt{b^2+1} + \sqrt{c^2+1} \leq \sqrt{2}(a+b+c) $

2. Given $a, b, c > 0$ and $a+b+c=1 $
Prove that: $ \dfrac{\sqrt{a^2+2ab}}{\sqrt{b^2+2c^2}} + \dfrac{\sqrt{b^2+2bc}}{\sqrt{c^2+2a^2}} + \dfrac{\sqrt{c^2+2ca}}{\sqrt{a^2+2b^2}} \geq \dfrac{1}{a^2+b^2+c^2} $
5 replies
+2 w
tokitaohma
May 11, 2025
math90
16 minutes ago
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Non-decelarating sequence is convergence-inducing
Miquel-point   0
17 minutes ago
Source: KoMaL A. 905
We say that a strictly increasing sequence of positive integers $n_1, n_2,\ldots$ is non-decelerating if $n_{k+1}-n_k\le n_{k+2}-n_{k+1}$ holds for all positive integers $k$. We say that a strictly increasing sequence $n_1, n_2, \ldots$ is convergence-inducing, if the following statement is true for all real sequences $a_1, a_2, \ldots$: if subsequence $a_{m+n_1}, a_{m+n_2}, \ldots$ is convergent and tends to $0$ for all positive integers $m$, then sequence $a_1, a_2, \ldots$ is also convergent and tends to $0$. Prove that a non-decelerating sequence $n_1, n_2,\ldots$ is convergence-inducing if and only if sequence $n_2-n_1$, $n_3-n_2$, $\ldots$ is bounded from above.

Proposed by András Imolay
0 replies
Miquel-point
17 minutes ago
0 replies
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Changing the states of light bulbs
Lukaluce   1
N 19 minutes ago by sarjinius
Source: 2025 Macedonian Balkan Math Olympiad TST Problem 1
A set of $n \ge 2$ light bulbs are arranged around a circle, and are consecutively numbered with
$1, 2, . . . , n$. Each bulb can be in one of two states: either it is on or off. In the initial configuration,
at least one bulb is turned on. On each one of $n$ days we change the current on/off configuration as
follows: for $1 \le k \le n$, on the $k$-th day we start from the $k$-th bulb and moving in clockwise direction
along the circle, we change the state of every traversed bulb until we switch on a bulb which was
previously off.
Prove that the final configuration, reached on the $n$-th day, coincides with the initial one.
1 reply
Lukaluce
Apr 14, 2025
sarjinius
19 minutes ago
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Strange domain
Besh00   1
N 20 minutes ago by Mathzeus1024
Find the $dom f$ of
$$f(x)=\sqrt{x^2 +\sin^2(x) +x\arctan(e^x)}$$
1 reply
Besh00
Jan 22, 2018
Mathzeus1024
20 minutes ago
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Proving radical axis through orthocenter
azzam2912   0
34 minutes ago
In acute triangle $ABC$ let $D, E$ and $F$ denote the feet of the altitudes from $A, B$ and $C$, respectively. Let line $DE$ intersect circumcircle $ABC$ at points $G, H$. Similarly, let line $DF$ intersect circumcircle $ABC$ at points $I, J$. Prove that the radical axis of circles $EIJ$ and $FGH$ passes through the orthocenter of triangle $ABC$
0 replies
azzam2912
34 minutes ago
0 replies
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Ez induction to start it off
alexanderhamilton124   22
N 40 minutes ago by Adywastaken
Source: Inmo 2025 p1
Consider the sequence defined by \(a_1 = 2\), \(a_2 = 3\), and
\[ a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1}, \]for all integers \(k \geq 1\). Determine all positive integers \(n\) such that
\[ \frac{a_n}{n} \]is an integer.

Proposed by Niranjan Balachandran, SS Krishnan, and Prithwijit De.
22 replies
alexanderhamilton124
Jan 19, 2025
Adywastaken
40 minutes ago
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Weird Algebra?
JARP091   0
42 minutes ago
Source: Art and Craft of Problem Solving 2.2.16
For each positive integer \( n \), find positive integer solutions \( x_1, x_2, \ldots, x_n \) to the equation

\[ \frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n} + \frac{1}{x_1 x_2 \cdots x_n} = 1 \]
0 replies
JARP091
42 minutes ago
0 replies
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Parallel lines in incircle configuration
GeorgeRP   2
N an hour ago by bin_sherlo
Source: Bulgaria IMO TST 2025 P1
Let $I$ be the incenter of triangle $\triangle ABC$. Let $H_A$, $H_B$, and $H_C$ be the orthocenters of triangles $\triangle BCI$, $\triangle ACI$, and $\triangle ABI$, respectively. Prove that the lines through $H_A$, $H_B$, and $H_C$, parallel to $AI$, $BI$, and $CI$, respectively, are concurrent.
2 replies
GeorgeRP
5 hours ago
bin_sherlo
an hour ago
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Transposition?
EeEeRUT   1
N an hour ago by ItzsleepyXD
Source: Thailand MO 2025 P8
For each integer sequence $a_1, a_2, a_3, \dots, a_n$, a single parity swapping is to choose $2$ terms in this sequence, say $a_i$ and $a_j$, such that $a_i + a_j$ is odd, then switch their placement, while the other terms stay in place. This creates a new sequence.

Find the minimal number of single parity swapping to transform the sequence $1,2,3, \dots, 2025$ to $2025, \dots, 3, 2, 1$, using only single parity swapping.
1 reply
EeEeRUT
6 hours ago
ItzsleepyXD
an hour ago
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Zero-Player Card Game
pieater314159   15
N an hour ago by N3bula
Source: ELMO 2019 Problem 3, 2019 ELMO Shortlist C4
Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card.

Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$.

Proposed by Carl Schildkraut and Colin Tang
15 replies
pieater314159
Jun 19, 2019
N3bula
an hour ago
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Replace a,b by a+b/2
mathscrazy   16
N an hour ago by Adywastaken
Source: INMO 2025/2
Let $n\ge 2$ be a positive integer. The integers $1,2,\cdots,n$ are written on a board. In a move, Alice can pick two integers written on the board $a\neq b$ such that $a+b$ is an even number, erase both $a$ and $b$ from the board and write the number $\frac{a+b}{2}$ on the board instead. Find all $n$ for which Alice can make a sequence of moves so that she ends up with only one number remaining on the board.
Note. When $n=3$, Alice changes $(1,2,3)$ to $(2,2)$ and can't make any further moves.

Proposed by Rohan Goyal
16 replies
mathscrazy
Jan 19, 2025
Adywastaken
an hour ago
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UMich Math
missionsqhc   1
N 3 hours ago by Mathzeus1024
I was recently accepted into the University of Michigan as a math major. If anyone studies math at UMich or knows anything about the program, could you share your experience? How would you rate the program? I know UMich is well-regarded for math (among many other things) but from my understanding, it is not quite at the level of an MIT or CalTech. What math programs is it comparable to? How does the rigor of the curricula compare to other top math programs? What are the other students like—is there a thriving contest math community? How accessible are research opportunities and graduate-level classes? Are most students looking to get into pure math and become research mathematicians or are most people focused on applied fields?

Also, aside from the math program, how is UMich overall? What were the advantages and disadvantages from being at such a large school? I was admitted to the Residential College (RC) within the College of Literature, Science, and the Arts. This is supposed to emulate a liberal arts college (while still allowing me access to the resources of a major research university). Could anyone speak on the RC?

How academically-inclined are UMich students? I’ve heard the school is big on sports and school spirit. I am just concerned that there may be a lot of subpar in-state students. How is the climate of Ann Arbor and how is the city in general?

Finally, how is UMich generally regarded? I’m also considering Georgetown. Am I right in viewing the latter as more well-regarded for humanities and the former better-known for STEM?
1 reply
missionsqhc
Yesterday at 4:31 PM
Mathzeus1024
3 hours ago
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Integral and Derivative Equation
ahaanomegas   6
N 4 hours ago by Sagnik123Biswas
Source: Putnam 1990 B1
Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, \[ \left( f(x) \right)^2 = \displaystyle\int_0^x \left[ \left( f(t) \right)^2 + \left( f'(t) \right)^2 \right] \, \mathrm{d}t + 1990. \]
6 replies
ahaanomegas
Jul 12, 2013
Sagnik123Biswas
4 hours ago
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A very simple question about calculus for middle school students
Craftybutterfly   19
N Apr 15, 2025 by Craftybutterfly
Source: own
$\lim_{x \to 8} \frac{2x^2+13x+6}{x^2+14x+48}=$ ? (there is an easy workaround)
(I know this is very easy- a little child can solve this in 1 second kinda problem so don't argue or mock me please)
19 replies
Craftybutterfly
Apr 9, 2025
Craftybutterfly
Apr 15, 2025
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A very simple question about calculus for middle school students
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Reveal topic tags
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Craftybutterfly
498 posts
Craftybutterfly
#1 h  Apr 9, 2025, 7:41 AM
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$\lim_{x \to 8} \frac{2x^2+13x+6}{x^2+14x+48}=$ ? (there is an easy workaround)
(I know this is very easy- a little child can solve this in 1 second kinda problem so don't argue or mock me please)
This post has been edited 2 times. Last edited by Craftybutterfly, Apr 10, 2025, 9:26 PM
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Craftybutterfly
498 posts
Craftybutterfly
#2 h  Apr 9, 2025, 7:44 AM
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sol
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Yiyj1
1266 posts
Yiyj1
#3 h  Apr 9, 2025, 7:49 AM
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GentlePanda24
662 posts
GentlePanda24
#4 h  Apr 9, 2025, 2:33 PM
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solution
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KAME06
159 posts
KAME06
#5 h  Apr 9, 2025, 4:10 PM
Y by
We can plug because the function is continuous in $8$.
Pretty well known that polynomials are continuous, and $2x^2+13x+6$ and $x^2+14x+48$ are continuous on $8$ and different from $0$ when you evaluate them on $8$.
That implies that $\frac{2x^2+13x+6}{x^2+14x+48}$ is continuous on $8$, so $\lim_{x \to 8} \frac{2x^2+13x+6}{x^2+14x+48}=\frac{2(8)^2+13(8)+6}{(8)^2+14(8)+48}$ (factorize if u want to do less work).
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HacheB2031
396 posts
HacheB2031
#6 h  Apr 10, 2025, 5:22 AM
Y by
Craftybutterfly wrote:
Find $\lim_{x \to -6} \frac{2x^2+13x+6}{x^2+14x+48}.$

FTFY
This post has been edited 1 time. Last edited by HacheB2031, Apr 10, 2025, 2:37 PM
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Craftybutterfly
498 posts
Craftybutterfly
#7 h  Apr 10, 2025, 5:24 AM
Y by
HacheB2031 wrote:
Craftybutterfly wrote:
Find $\lim_{x \to -8} \frac{2x^2+13x+6}{x^2+14x+48}.$

FTFY
?
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Craftybutterfly
498 posts
Craftybutterfly
#9 h  Apr 10, 2025, 6:02 AM
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@sp0rtman00000Click to reveal hidden text
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HacheB2031
396 posts
HacheB2031
#10 h  Apr 10, 2025, 2:37 PM • 1 Y
Y by LawofCosine
Craftybutterfly wrote:
HacheB2031 wrote:
Craftybutterfly wrote:
Find $\lim_{x \to -6} \frac{2x^2+13x+6}{x^2+14x+48}.$

FTFY
?
I'm sorry, I realized it should probably be as $x\to-6.$ Let me explain why.
When you make limits, you often want them to tend to a value where a function is not defined. Take the limit \[\lim_{x\to0}\frac{\sin x}x.\]Clearly, direct evaluation gives \[\frac{\sin0}0=\frac00,\]an indeterminate form, but (using methods such as the squeeze theorem) you can show that \[\lim_{x\to0}\frac{\sin x}x=1.\]Despite the function not being defined there, it has a well-defined limit. In my [edited] post, notice that as $x\to-6,$ you also have $x^2+14x+48\to0$ and $2x^2+13x+6\to0,$ making this limit not evaluatable with direct evaluation. (The limit may exist, but in the old post where $x\to-8,$ it blows up.) If you have the limit of a rational function, a function where the numerator and denominator are both polynomials, you can cancel common factors to redefine the function where it wasn't defined. Also, note that you can't always use direct evaluation. Let \[f(x)=\begin{cases}x^2&x\ne0\\1&x=0\end{cases}.\]Try to evaluate the limit \[\lim_{x\to0}f(x).\]If we directly evaluate it, we get that \[\lim_{x\to0}f(x)=f(0)=1.\]However, $f(x)$ has a problem: it isn't continuous, or, if you drew its graph, you would need to pick up your pencil from the paper. The actual limit turns out to be \[\lim_{x\to0}f(x)=\lim_{x\to0}x^2=0.\]Because $x^2$ is a polynomial, it is continuous; all polynomials are continuous. As well as all rational functions, over their domain. Anywhere a rational function is defined, it is continuous, except at points where its denominator is $0.$ sidenote
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Craftybutterfly
498 posts
Craftybutterfly
#11 h  Apr 10, 2025, 6:19 PM
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It is $\lim_{x\to8}$ not $\lim_{x\to-6}$
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yaxuan
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yaxuan
#12 h  Apr 10, 2025, 8:51 PM
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op wrote:
a kindergartener can solve this in 1 second
Bruh
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Craftybutterfly
498 posts
Craftybutterfly
#13 h  Apr 10, 2025, 9:27 PM
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Craftybutterfly wrote:
sol
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Oshawoot
134 posts
Oshawoot
#14 h  Apr 14, 2025, 2:27 AM
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whats lim again？。。。
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LawofCosine
837 posts
LawofCosine
#15 h  Apr 14, 2025, 2:40 AM
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a limit (concept in calculus)
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GentlePanda24
662 posts
GentlePanda24
#16 h  Apr 14, 2025, 1:11 PM
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Craftybutterfly wrote:
a kindergartener can solve this in 1 second

I think that is a bit exagerated
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GentlePanda24
662 posts
GentlePanda24
#17 h  Apr 14, 2025, 1:12 PM
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Craftybutterfly wrote:
a kindergartener can solve this in 1 second

I think that is a bit exagerated
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leahlyoung106
63 posts
leahlyoung106
#18 h  Apr 14, 2025, 2:12 PM
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Craftybutterfly
498 posts
Craftybutterfly
#19 h  Apr 15, 2025, 5:24 AM
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Question: what is an example of a with no limit or a limit of 0
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aidan0626
1912 posts
aidan0626
#20 h  Apr 15, 2025, 5:29 AM • 1 Y
Y by LawofCosine
Craftybutterfly wrote:
Question: what is an example of a with no limit or a limit of 0

x-> -8 has no limit
x-> -1/2 gives a limit of 0
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Craftybutterfly
498 posts
Craftybutterfly
#21 h  Apr 15, 2025, 5:35 AM
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Thx @bove
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