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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.

Summer camps are starting next 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 an enriching 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)!

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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
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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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9 Pythagorean Triples
ZMB038   52
N an hour ago by pieMax2713
Please put some of the ones you know, and try not to troll/start flame wars! Thank you :D
52 replies
ZMB038
May 19, 2025
pieMax2713
an hour ago
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n-term Sequence
MithsApprentice   15
N an hour ago by Ilikeminecraft
Source: USAMO 1996, Problem 4
An $n$-term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a binary sequence of length $n$. Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.
15 replies
MithsApprentice
Oct 22, 2005
Ilikeminecraft
an hour ago
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Drawing Triangles Against Your Clone
pieater314159   19
N an hour ago by Ilikeminecraft
Source: 2019 ELMO Shortlist C1
Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.)

Proposed by Milan Haiman
19 replies
pieater314159
Jun 27, 2019
Ilikeminecraft
an hour ago
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Odd digit multiplication
JuanDelPan   12
N 2 hours ago by Ilikeminecraft
Source: Pan-American Girls' Mathematical Olympiad 2021, P4
Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$.

$\textit{Proposed by Pablo Serrano, Ecuador}$
12 replies
JuanDelPan
Oct 6, 2021
Ilikeminecraft
2 hours ago
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Cup of Combinatorics
M11100111001Y1R   7
N 2 hours ago by MathematicalArceus
Source: Iran TST 2025 Test 4 Problem 2
There are \( n \) cups labeled \( 1, 2, \dots, n \), where the \( i \)-th cup has capacity \( i \) liters. In total, there are \( n \) liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.

$a)$ Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most \( \frac{4n}{3} \) steps.

$b)$ Prove that in at most \( \frac{5n}{3} \) steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
7 replies
M11100111001Y1R
May 27, 2025
MathematicalArceus
2 hours ago
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Inequality
knm2608   17
N 2 hours ago by Adywastaken
Source: JBMO 2016 shortlist
If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that
$$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$When does the equality occur?

Proposed by Dorlir Ahmeti, Albania
17 replies
knm2608
Jun 25, 2017
Adywastaken
2 hours ago
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Easy number theory
britishprobe17   35
N 2 hours ago by Andyluo
The number of factors from 2024 that are greater than $\sqrt{2024}$ are
35 replies
britishprobe17
Oct 16, 2024
Andyluo
2 hours ago
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A Variety of Math Problems to solve
FJH07   45
N 2 hours ago by FJH07
Hi, so people can post different math problems that they think are hard, and I will post some (I think middle school math level) problems so that the community can help solve them. :)
45 replies
FJH07
May 22, 2025
FJH07
2 hours ago
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How many combinations of ABCDEFGHI can be placed in a 3x3 grid
Darealzolt   2
N 2 hours ago by LXC007
Find the number of ways that the letters in \(ABCDEFGHI\) can be placed in a 3x3 grid, such that each letter can only be used once, and that \(A\) is not next to \(B\) or \(C\) and \(F\) is not next to \(G\) ,\(H\),or \(I\).
2 replies
Darealzolt
Today at 2:11 PM
LXC007
2 hours ago
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Circumcircle of XYZ is tangent to circumcircle of ABC
mathuz   39
N 2 hours ago by zuat.e
Source: ARMO 2013 Grade 11 Day 2 P4
Let $ \omega $ be the incircle of the triangle $ABC$ and with centre $I$. Let $\Gamma $ be the circumcircle of the triangle $AIB$. Circles $ \omega $ and $ \Gamma $ intersect at the point $X$ and $Y$. Let $Z$ be the intersection of the common tangents of the circles $\omega$ and $\Gamma$. Show that the circumcircle of the triangle $XYZ$ is tangent to the circumcircle of the triangle $ABC$.
39 replies
mathuz
May 22, 2013
zuat.e
2 hours ago
J
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Arc Midpoints Form Cyclic Quadrilateral
ike.chen   57
N 3 hours ago by cj13609517288
Source: ISL 2022/G2
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
57 replies
ike.chen
Jul 9, 2023
cj13609517288
3 hours ago
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Complex number
ronitdeb   0
3 hours ago
Let $z_1, ... ,z_5$ be vertices of regular pentagon inscribed in a circle whose radius is $2$ and center is at $6+i8$. Find all possible values of $z_1^2+z_2^2+...+z_5^2$
0 replies
ronitdeb
3 hours ago
0 replies
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Elementary Problems Compilation
Saucepan_man02   29
N 3 hours ago by Electrodynamix777
Could anyone send some elementary problems, which have tricky and short elegant methods to solve?

For example like this one:
Solve over reals: $$a^2 + b^2 + c^2 + d^2 -ab-bc-cd-d +2/5=0$$
29 replies
Saucepan_man02
May 26, 2025
Electrodynamix777
3 hours ago
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Find all possible values of BT/BM
va2010   54
N 4 hours ago by lpieleanu
Source: 2015 ISL G4
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
54 replies
va2010
Jul 7, 2016
lpieleanu
4 hours ago
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State target p8 sol
EaZ_Shadow   116
N Apr 18, 2025 by derekwang2048
This solution was so educational! It helped me understand how to solve this problem in many ways I couldn't, and I feel so enlightened!
116 replies
EaZ_Shadow
Apr 6, 2025
derekwang2048
Apr 18, 2025
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State target p8 sol
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EaZ_Shadow
1286 posts
EaZ_Shadow
#1 h  Apr 6, 2025, 3:53 PM • 3 Y
Y by PikaPika999, Soupboy0, jkim0656
This solution was so educational! It helped me understand how to solve this problem in many ways I couldn't, and I feel so enlightened!
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greenplanet2050
1329 posts
greenplanet2050
#2 h  Apr 6, 2025, 3:55 PM • 1 Y
Y by PikaPika999
ikr i literally kinda guessed this one correct
This post has been edited 1 time. Last edited by greenplanet2050, Apr 6, 2025, 3:55 PM
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EaZ_Shadow
1286 posts
EaZ_Shadow
#3 h  Apr 6, 2025, 3:56 PM • 1 Y
Y by PikaPika999
greenplanet2050 wrote:
ikr i literally kinda guessed this one correct

Lowk mathcounts was def cooking with that sol!
This post has been edited 1 time. Last edited by EaZ_Shadow, Apr 6, 2025, 3:56 PM
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evt917
2425 posts
evt917
#4 h  Apr 6, 2025, 3:56 PM • 1 Y
Y by PikaPika999
i didn't take mathcounts cuz i'm only in 5th grade

but i also checked the problems

the solution was way too good
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DreamineYT
298 posts
DreamineYT
#5 h  Apr 6, 2025, 4:01 PM • 1 Y
Y by PikaPika999
EaZ_Shadow wrote:
This solution was so educational! It helped me understand how to solve this problem in many ways I couldn't, and I feel so enlightened!

our team got p1 team wrong :noo:
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sadas123
1327 posts
sadas123
#7 h  Apr 6, 2025, 7:39 PM • 1 Y
Y by PikaPika999
It is actually a good solution and I am not being sarcastic skull
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maxamc
585 posts
maxamc
#8 h  Apr 6, 2025, 7:47 PM • 1 Y
Y by PikaPika999
i did all 10 team round in 19 min but failed target p8 and i am not being sarcastic skull
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sadas123
1327 posts
sadas123
#9 h  Apr 6, 2025, 8:19 PM • 1 Y
Y by PikaPika999
maxamc wrote:
i did all 10 team round in 19 min but failed target p8 and i am not being sarcastic skull

skull i did 11 team rounds and got a perfect on target and sprint to get 46 overall ¿como estas?
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DottedCaculator
7357 posts
DottedCaculator
#10 h  Apr 6, 2025, 8:41 PM • 3 Y
Y by PikaPika999, jkim0656, scannose
https://www.eecs.qmul.ac.uk/~norman/papers/probability_puzzles/equal_sequences.shtml
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maxamc
585 posts
maxamc
#11 h  Apr 6, 2025, 9:03 PM • 1 Y
Y by PikaPika999
DottedCaculator wrote:
https://www.eecs.qmul.ac.uk/~norman/papers/probability_puzzles/equal_sequences.shtml

https://www.youtube.com/watch?v=t8xqMxlZz9Y
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huajun78
76 posts
huajun78
#12 h  Apr 6, 2025, 9:44 PM • 1 Y
Y by PikaPika999
Quote:
The probability of getting consecutive coin tosses with the pattern HTT is (2/3)(1/3)(1/3)=2/27 . The expectation value of the number of tosses needed to first achieve that pattern is the reciprocal of that probability, thus 27/2.

I don't get the solution (why can you just reciprocate it?)
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Craftybutterfly
583 posts
Craftybutterfly
#13 h  Apr 6, 2025, 9:51 PM • 1 Y
Y by PikaPika999
Is MATHCOUNTS only for middle school students or also high school students or elementary students as well?
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DottedCaculator
7357 posts
DottedCaculator
#14 h  Apr 6, 2025, 9:56 PM • 4 Y
Y by PikaPika999, huajun78, iwastedmyusername, vincentwant
huajun78 wrote:
Quote:
The probability of getting consecutive coin tosses with the pattern HTT is (2/3)(1/3)(1/3)=2/27 . The expectation value of the number of tosses needed to first achieve that pattern is the reciprocal of that probability, thus 27/2.

I don't get the solution (why can you just reciprocate it?)

Consider doing many coin tosses. Each appearance of HTT is disjoint, and the total number of HTTs is 2/27 times the number of tosses, so the expected distance to the next one is 27/2.
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mdk2013
645 posts
mdk2013
#15 h  Apr 7, 2025, 12:19 AM • 1 Y
Y by ChristianYoo
i did it like dottedcaculator did, everyone was like did you p8 and they were like god that problem was actually hard, and i was like bruh its just that the probability is 2/3*1/3*1/3= 2/27 so the expected number is 27/2 lol you guys shouldve seen my friends expression when i told him that
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scrabbler94
7555 posts
scrabbler94
#16 h  Apr 7, 2025, 12:27 AM
Y by
DottedCaculator wrote:
Consider doing many coin tosses. Each appearance of HTT is disjoint, and the total number of HTTs is 2/27 times the number of tosses, so the expected distance to the next one is 27/2.

I like how this solution fails if HTT is replaced with, say, TT. The expected number of flips is not 4 in this case.
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