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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
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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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Inspired by old results
sqing   8
N 35 minutes ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a^2+ab+b^2=2$ . Prove that
$$ (a+b-ab)\left( \frac{1}{a+1} + \frac{1}{b+1}\right)\leq 2 $$$$ (a+b-ab)\left( \frac{a}{b+1} + \frac{2b}{a+2}\right)\leq 2 $$$$ (a+b-ab)\left( \frac{a}{b+1} + \frac{2b}{a+1}\right)\leq 4$$Let $ a,b $ be reals such that $ a^2+b^2=2$ . Prove that
$$ (a+b)\left( \frac{1}{a+1} + \frac{1}{b+1}\right)= 2 $$$$ (a+b)\left( \frac{a}{b+1} + \frac{b}{a+1}\right)=2 $$
8 replies
1 viewing
sqing
Today at 2:42 AM
sqing
35 minutes ago
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Transform the sequence
steven_zhang123   1
N 38 minutes ago by vgtcross
Given a sequence of \( n \) real numbers \( a_1, a_2, \ldots, a_n \), we can select a real number \( \alpha \) and transform the sequence into \( |a_1 - \alpha|, |a_2 - \alpha|, \ldots, |a_n - \alpha| \). This transformation can be performed multiple times, with each chosen real number \( \alpha \) potentially being different
(i) Prove that it is possible to transform the sequence into all zeros after a finite number of such transformations.
(ii) To ensure that the above result can be achieved for any given initial sequence, what is the minimum number of transformations required?
1 reply
steven_zhang123
Today at 3:57 AM
vgtcross
38 minutes ago
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NEPAL TST DAY 2 PROBLEM 2
Tony_stark0094   5
N 42 minutes ago by ThatApollo777
Kritesh manages traffic on a $45 \times 45$ grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the $45 \times 45$ grid.

Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove $k$ cars from the $45 \times 45$ grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.

What is the minimum value of $k$ that guarantees that Kritesh's job is possible?

$\textbf{Proposed by Shining Sun. USA}$
5 replies
Tony_stark0094
Yesterday at 8:37 AM
ThatApollo777
42 minutes ago
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Product of distinct integers in arithmetic progression -- ever a perfect power ?
adityaguharoy   1
N an hour ago by Mathzeus1024
Source: Well known (the gen. is more difficult, but may be not this one -- so this is here)
Let $a_1,a_2,a_3,a_4$ be four positive integers in arithmetic progression (that is, $a_1-a_2=a_2-a_3=a_3-a_4$) and with $a_1 \ne a_2$. Can the product $a_1 \cdot a_2 \cdot a_3 \cdot a_4$ ever be a number of the form $n^k$ for some $n \in \mathbb{N}$ and some $k \in \mathbb{N}$, with $k \ge 2$ ?
1 reply
adityaguharoy
Aug 31, 2019
Mathzeus1024
an hour ago
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For positive integers \( a, b, c \), find all possible positive integer values o
Jackson0423   2
N an hour ago by ATM_
For positive integers \( a, b, c \), find all possible positive integer values of
\[ \frac{a}{b} + \frac{b}{c} + \frac{c}{a}. \]
2 replies
Jackson0423
2 hours ago
ATM_
an hour ago
J
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Isosceles Triangle Geo
oVlad   2
N an hour ago by SomeonesPenguin
Source: Romania Junior TST 2025 Day 1 P2
Consider the isosceles triangle $ABC$ with $\angle A>90^\circ$ and the circle $\omega$ of radius $AC$ centered at $A.$ Let $M$ be the midpoint of $AC.$ The line $BM$ intersects $\omega$ a second time at $D.$ Let $E$ be a point on $\omega$ such that $BE\perp AC.$ Let $N$ be the intersection of $DE$ and $AC.$ Prove that $AN=2\cdot AB.$
2 replies
oVlad
Yesterday at 9:38 AM
SomeonesPenguin
an hour ago
J
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IMO ShortList 1998, number theory problem 5
orl   63
N an hour ago by ATM_
Source: IMO ShortList 1998, number theory problem 5
Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.
63 replies
orl
Oct 22, 2004
ATM_
an hour ago
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IMO Shortlist 2013, Number Theory #1
lyukhson   150
N an hour ago by MuradSafarli
Source: IMO Shortlist 2013, Number Theory #1
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
150 replies
lyukhson
Jul 10, 2014
MuradSafarli
an hour ago
J
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Interesting inequality
A_E_R   0
an hour ago
Let a,b,c,d are positive real numbers, if the following inequality holds ab^2+ac^2>=5bcd. Find the minimum value of explanation: (a^2+b^2+c^2+d^2)(1/a^2+1/b^2+1/c^2+1/d^2)
0 replies
A_E_R
an hour ago
0 replies
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abcd is not a perfect square if a,b,c,d are in arithmetic progression
adityaguharoy   1
N an hour ago by Mathzeus1024
If $a,b,c,d$ are in arithmetic progression and $a \ne b , b \ne c , c \ne d,$ then show that the product $a\cdot b \cdot c \cdot d$ is not a perfect square
1 reply
adityaguharoy
Jun 24, 2022
Mathzeus1024
an hour ago
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Doubt on a math problem
AVY2024   10
N 4 hours ago by Yiyj1
Solve for x and y given that xy=923, x+y=84
10 replies
AVY2024
Apr 8, 2025
Yiyj1
4 hours ago
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Website to learn math
hawa   22
N 4 hours ago by RedChameleon
Hi, I'm kinda curious what website do yall use to learn math, like i dont find any website thats fun to learn math
22 replies
hawa
Apr 9, 2025
RedChameleon
4 hours ago
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I think I regressed at math
PaperMath   36
N Today at 4:35 AM by pb0975
I found the slip of paper a few days ago that I think I wrote when I was in kindergarten. It is just a sequence of numbers and you have to find the next number, the pattern is $1,2,5,40,1280,?$. I couldn't solve this and was wondering if any of you can find the pattern
36 replies
PaperMath
Mar 8, 2025
pb0975
Today at 4:35 AM
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9 Was the 2025 AMC 8 harder or easier than last year?
Sunshine_Paradise   180
N Today at 4:27 AM by aichinara
Also what will be the DHR?
180 replies
Sunshine_Paradise
Jan 30, 2025
aichinara
Today at 4:27 AM
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STATE SOLUTIONS AND STUFF DROPPED!!!
Soupboy0   53
N Apr 7, 2025 by fruitmonster97
https://www.mathcounts.org/resources/past-competitions
53 replies
Soupboy0
Apr 4, 2025
fruitmonster97
Apr 7, 2025
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STATE SOLUTIONS AND STUFF DROPPED!!!
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Reveal topic tags
MATHCOUNTS
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Soupboy0
317 posts
Soupboy0
#40 h  Apr 5, 2025, 1:08 PM
Y by
i memorized sprint 20 before the competition:skull:
This post has been edited 1 time. Last edited by Soupboy0, Apr 5, 2025, 1:09 PM
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MathPerson12321
3684 posts
MathPerson12321
#41 h  Apr 5, 2025, 3:12 PM
Y by
@3above correct, this works for one time things and mostly is an intro to expected value.

You have to consider the other factors as well.
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superhuman233
473 posts
superhuman233
#42 h  Apr 5, 2025, 5:59 PM
Y by
i was so lucky that i guessed sprint 20 right otherwise i wouldn't have made nats

Click to reveal hidden text
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ilikemath247365
237 posts
ilikemath247365
#43 h  Apr 5, 2025, 6:00 PM
Y by
Bro, sprint 20 took the most time for me in the first 20. I can't believe that I actually did coordinate geometry on the real test on this problem!

I also almost trolled this problem. I miscalculated by somehow getting the square root of 49/25 but I clutched and caught my mistake to change it to the square root of 81/25 which is 9/5.
This post has been edited 1 time. Last edited by ilikemath247365, Apr 5, 2025, 6:01 PM
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Andyluo
916 posts
Andyluo
#44 h  Apr 5, 2025, 6:02 PM
Y by
superhuman233 wrote:
i was so lucky that i guessed sprint 20 right otherwise i wouldn't have made nats

Click to reveal hidden text

how is that guessing lmao, it's how you're supposed to solve it. honestly belongs in first 10 questions
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jb2015007
1845 posts
jb2015007
#45 h  Apr 5, 2025, 6:16 PM • 1 Y
Y by PikaPika999
20 was didnt take too long for me
js had to draw the altitude and do some coord bashing then pythag then ur done
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JamesYMath
12 posts
JamesYMath
#46 h  Apr 6, 2025, 12:34 AM • 1 Y
Y by PikaPika999
You could've also used the altitude theorem for #20 but I guess that's just similar triangles
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PikaPika999
1151 posts
PikaPika999
#47 h  Apr 6, 2025, 12:44 AM
Y by
MathPerson12321 wrote:
c_double_sharp wrote:
s11 was dumb i added wrong
s18 was dumb how did i not see that
s23 was dumb how did i not see that
s24 was dumb how am i not able to read properly
s27 was dumb like the official solution was assume rectangle
t6 was dumb like i did not calcbash enough terms
t8 was dumb how did i not see that

totally agree

fr tho

:rotfl:
This post has been edited 1 time. Last edited by PikaPika999, Apr 6, 2025, 12:44 AM
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giratina3
486 posts
giratina3
#48 h  Apr 6, 2025, 1:44 AM • 1 Y
Y by PikaPika999
I choked #7 on sprint because I thought for some stupid reason that the maximum area of a triangle with perimeter 24 was a right triangle :wallbash:

Probably could have gone to nationals with that question. Just saying...

Sigh... guess I just wasn't able to make nationals because I'm too nervous during the test... which is exactly what happened last year too.

So close to nationals... yet so far...
This post has been edited 1 time. Last edited by giratina3, Apr 6, 2025, 1:45 AM
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martianrunner
152 posts
martianrunner
#49 h  Apr 7, 2025, 4:17 AM
Y by
i had seen some extremely trivial solution for p4 target that didn't even require a calculator but i forgot it now

does anyone have a non-bash solution for that one
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derekwang2048
1209 posts
derekwang2048
#50 h  Apr 7, 2025, 4:23 AM
Y by
i just solved the system of equations and it was pretty straightforward, i realized how to do it with like 2 minutes left
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Bummer12345
132 posts
Bummer12345
#51 h  Apr 7, 2025, 1:01 PM
Y by
martianrunner wrote:
i had seen some extremely trivial solution for p4 target that didn't even require a calculator but i forgot it now

does anyone have a non-bash solution for that one

here are 5 stupid solutions to target #4

sol 1 (sort of bash, most logical)

sol 2 (cheese, no bash)

sol 3 (graphing calc)

sol 4 (even more cheese)

sol 5 (sol 2+3 equals 5)
This post has been edited 3 times. Last edited by Bummer12345, Apr 7, 2025, 1:09 PM
Reason: latex
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c_double_sharp
308 posts
c_double_sharp
#52 h  Apr 7, 2025, 1:31 PM • 1 Y
Y by MathPerson12321
p8 is 27/2 confirmed but why does the official solution work
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happypi31415
742 posts
happypi31415
#53 h  Apr 7, 2025, 4:48 PM • 2 Y
Y by Aaronjudgeisgoat, MathPerson12321
The official solution works because the coins follow a Binomial Distribution, which means that we have that $E[x]=np$ where $E[x]$ is the expected value, $p$ is the probability, and $n$ is the number of trials. This also follows from linearity of expectation on each of the groups of $3$. Then, setting $E[x]=1$ and $p=\frac{2}{27}$ gives the desired answer. :)

Alternatively, another easy solution is to use recursion: suppose that we want the expected number of rolls, $x$, until an event happens with probability $p$. Then, we have that $x=p+(x+1)(1-p)$ which rearranges to the desired $x=\frac{1}{p}$.

edit: thinking about it now im not completely sure this is valid because expected amount of rolls between HTH and TTT is different but whatever
This post has been edited 3 times. Last edited by happypi31415, Apr 7, 2025, 8:32 PM
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fruitmonster97
2459 posts
fruitmonster97
#54 h  Apr 7, 2025, 5:29 PM
Y by
target seems interesting
personal difficulty ratings: 1 is trivial, 10 is nats t8(from a year not named 2024 or 2023)
t1: 1, trivial, just average 9876 and 5432.
t2: 1.5 to 2, trivial but annoying.
First two seem hard to silly, esp. with 1 being so easy.
t3: 1 to 1.5, also trivial.
t4: 2 to 2.5, too easy for the placement.
Another very sweepable round, as intended. Nowhere near the difficulty of last year's t4(which was the only one i got wrong last year).
t5: 1 to 1.5, just floor(202/24)
t6: 6 to 7, imo perfect placement. nontrivial. Albeit you don't even have to find the intended solution, simply approximate 2^2025-1 to 2^2025 and plugging the first ~13 terms into a calc should give a good approximation.
t7: 3 to 3.5, fair placement. simple crt.
t8: 4 to 9, really just depends on your perception. there are definitely people who just assumed you can always reciprocate, but there are definitely people who tried much, much harder methods.
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