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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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[*]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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Fun challange problem :)
TigerSenju   32
N 4 minutes ago by maxamc
Scenario:

Master Alchemist Aurelius is renowned for his mastery of elemental fusion. He works with seven fundamental, yet mysterious, elements: Ignis (Fire), Aqua (Water), Terra (Earth), Aer (Air), Lux (Light), Umbra (Shadow), and Aether (Spirit). Each element possesses a unique 'potency' value, a positive integer crucial for his most complex fusions

Aurelius has lost his master log of these potencies. All he has left are seven cryptic scrolls, each containing a precise relationship between the potencies of various elements. He needs these values to complete his Grand Device. Can you help him deduce the exact potency of each element?

The Elements and Their Potencies:

Let I represent the potency of Ignis (Fire).
Let A represent the potency of Aqua (Water).
Let T represent the potency of Terra (Earth).
Let R represent the potency of Aer (Air).
Let L represent the potency of Lux (Light).
Let U represent the potency of Umbra (Shadow).
Let E represent the potency of Aether (Spirit).
The Cryptic Scrolls (System of Equations):

Aurelius's scrolls reveal the following relationships:

The combined potency of Ignis, Aqua, and Terra is equal to the potency of Aer plus Lux, plus a constant of two.

If you sum the potencies of Aqua and Umbra, it precisely equals the sum of Lux and Aether, minus one.

The sum of Terra and Aer potencies is the same as the sum of Ignis, Aqua, and Aether potencies, minus one.

Three times the potency of Ignis, plus the potency of Aer, is equal to the sum of Aqua, Terra, and Aether potencies, plus five.

The difference between Lux and Ignis potencies is identical to the difference between Umbra and Aqua potencies.

The sum of Umbra and Aether potencies, when decreased by the potency of Terra, results in twice the potency of Aqua.

The potency of Ignis added to Lux, minus the potency of Aer, is equivalent to the potency of Aether minus Umbra, plus one.

The Grand Challenge:

Using only the information from the cryptic scrolls, set up and solve the system of seven linear equations to determine the unique positive integer potency value for each of the seven elements: I,A,T,R,L,U,E.

good luck, and whoever finds the potencies first, gets a title of The SYSTEMS OF EQUATIONS MASTER

p.s. Yes, I did just come up with a whole story of words to make a ridiculously long problem, but hey, you're reading this, so you probably have nothing better to be doing. ;)
32 replies
+1 w
TigerSenju
May 18, 2025
maxamc
4 minutes ago
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Warning!
VivaanKam   40
N an hour ago by ayeshaaq
This problem will try to trick you! :!:

40 replies
VivaanKam
May 5, 2025
ayeshaaq
an hour ago
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Worst Sillies of All Time
pingpongmerrily   50
N 3 hours ago by bbojy
Share the worst sillies you have ever made!

Mine was probably on the 2024 MathCounts State Target Round Problem 8, where I wrote my answer as a fraction instead of a percent, which cost me a trip to Nationals that year.
50 replies
pingpongmerrily
Yesterday at 12:34 PM
bbojy
3 hours ago
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MathDash help
Spacepandamath13   8
N 3 hours ago by Yiyj
AkshajK ORZ by the way invited me to do MathDash a few months ago and I did try it one day but haven't done it much after (Sorry). Now, I'm getting back into it and finding the format kind of weird. When selecting certain problem type sometimes it lets me pick immediately, other times not. Any fixes?
8 replies
Spacepandamath13
May 29, 2025
Yiyj
3 hours ago
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Easy P4 combi game with nt flavour
Maths_VC   1
N 3 hours ago by p.lazarov06
Source: Serbia JBMO TST 2025, Problem 4
Two players, Alice and Bob, play the following game, taking turns. In the beginning, the number $1$ is written on the board. A move consists of adding either $1$, $2$ or $3$ to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is $10$, then in a move a player can't choose the number $2$, but he can choose either $1$ or $3$). The player who first writes a perfect square on the board loses. Prove that one of the players has a winning strategy and determine who wins in the game.
1 reply
Maths_VC
May 27, 2025
p.lazarov06
3 hours ago
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Central sequences
EeEeRUT   14
N 4 hours ago by HamstPan38825
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 $a_n = b_n$.
14 replies
EeEeRUT
Apr 16, 2025
HamstPan38825
4 hours ago
J
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Elementary Problems Compilation
Saucepan_man02   32
N 5 hours ago by atdaotlohbh
Could anyone send some elementary problems, which have tricky and short elegant methods to solve?

For example like this one:
Solve over reals: $$a^2 + b^2 + c^2 + d^2 -ab-bc-cd-d +2/5=0$$
32 replies
Saucepan_man02
May 26, 2025
atdaotlohbh
5 hours ago
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Random Points = Problem
kingu   5
N 5 hours ago by happypi31415
Source: Chinese Geometry Handout
Let $ABC$ be a triangle. Let $\omega$ be a circle passing through $B$ intersecting $AB$ at $D$ and $BC$ at $F$. Let $G$ be the intersection of $AF$ and $\omega$. Further, let $M$ and $N$ be the intersections of $FD$ and $DG$ with the tangent to $(ABC)$ at $A$. Now, let $L$ be the second intersection of $MC$ and $(ABC)$. Then, prove that $M$ , $L$ , $D$ , $E$ and $N$ are concyclic.
5 replies
kingu
Apr 27, 2024
happypi31415
5 hours ago
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Combo resources
Fly_into_the_sky   1
N 5 hours ago by Fly_into_the_sky
Ok so i never did combinatorics in my life :oops: and i am willing to be able to do P1/P4 combos (or even more)
So yeah how can i start from scratch?
Remark:i don't want compuational combo resources :noo:
1 reply
Fly_into_the_sky
5 hours ago
Fly_into_the_sky
5 hours ago
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Very odd geo
Royal_mhyasd   2
N 5 hours ago by Royal_mhyasd
Source: own (i think)
nevermind
2 replies
Royal_mhyasd
Yesterday at 6:10 PM
Royal_mhyasd
5 hours ago
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Polynomial Application Sequences and GCDs
pieater314159   46
N 6 hours ago by cursed_tangent1434
Source: ELMO 2019 Problem 1, 2019 ELMO Shortlist N1
Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$.

Proposed by Milan Haiman and Carl Schildkraut
46 replies
pieater314159
Jun 19, 2019
cursed_tangent1434
6 hours ago
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c^a + a = 2^b
Havu   10
N 6 hours ago by Havu
Find $a, b, c\in\mathbb{Z}^+$ such that $a,b,c$ coprime, $a + b = 2c$ and $c^a + a = 2^b$.
10 replies
Havu
May 10, 2025
Havu
6 hours ago
J
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Own made functional equation
JARP091   0
Today at 4:10 PM
Source: Own (Maybe?)
\[ \text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\ f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right) \]
0 replies
JARP091
Today at 4:10 PM
0 replies
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Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   16
N Today at 3:56 PM by JARP091
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
16 replies
OgnjenTesic
May 22, 2025
JARP091
Today at 3:56 PM
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mathcounts state discussion
Soupboy0   65
N Apr 20, 2025 by ERMSCoach
les goo its finally april
65 replies
Soupboy0
Apr 1, 2025
ERMSCoach
Apr 20, 2025
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mathcounts state discussion
G H J
Reveal topic tags
MATHCOUNTS
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iwastedmyusername
185 posts
iwastedmyusername
#52 h  Apr 3, 2025, 7:22 PM
Y by
was there a non bashy way to do target #4
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jb2015007
1973 posts
jb2015007
#53 h  Apr 3, 2025, 7:23 PM
Y by
yes js let the side lengths be a and b, and the hypotneuse is 40-a-b, and then solve
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iwastedmyusername
185 posts
iwastedmyusername
#54 h  Apr 3, 2025, 7:26 PM
Y by
wouldnt you have to square 40-a-b though
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jb2015007
1973 posts
jb2015007
#55 h  Apr 3, 2025, 7:28 PM
Y by
i js timed myself and did it in 15 secs
its not that big of a deal
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Bummer12345
159 posts
Bummer12345
#56 h  Apr 3, 2025, 8:28 PM
Y by
cheese sol
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gamma_pi
153 posts
gamma_pi
#57 h  Apr 3, 2025, 9:14 PM • 1 Y
Y by Eddie_tiger
jb2015007 wrote:
nah it was 27/2 if i remember correctly

yeah i think it is
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EightYearLetterman
16 posts
EightYearLetterman
#58 h  Apr 3, 2025, 9:17 PM
Y by
iwastedmyusername wrote:
wouldnt you have to square 40-a-b though

a^2 + b^2 = 1600 - 80(a+b) + (a+b)^2
0 = 1600 - 80(40-c) + 2ab
-80c = 1600 - 80*40 + 80
c = -20 + 40 - 1
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Pi_isCool31415
25 posts
Pi_isCool31415
#59 h  Apr 4, 2025, 12:30 AM
Y by
target 6 u could just induct and not prove it lol
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Pi_isCool31415
25 posts
Pi_isCool31415
#60 h  Apr 4, 2025, 12:31 AM
Y by
Bummer12345 wrote:
cheese sol

bruh even worse sol, calc bash and guess based on answer form
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mathprodigy2011
350 posts
mathprodigy2011
#61 h  Apr 4, 2025, 11:24 PM
Y by
iwastedmyusername wrote:
was there a non bashy way to do target #4

Click to reveal hidden text
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chesswillow
256 posts
chesswillow
#62 h  Apr 5, 2025, 3:00 AM
Y by
How were you supposed to do target 6 though

I randomly added up all the squares from 1 to 2025 and divided that by the sum of all integers 1-2025 (I don't even know why) and somehow got the right answer after doing a bunch of random other things.
This post has been edited 1 time. Last edited by chesswillow, Apr 25, 2025, 3:27 PM
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Soupboy0
499 posts
Soupboy0
#63 h  Apr 5, 2025, 3:01 AM
Y by
i just used arithmetico geometric sequence
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sadas123
1330 posts
sadas123
#64 h  Apr 5, 2025, 11:09 AM
Y by
chesswillow wrote:
How were you supposed to do target 6 though

I randomly added up all the squares from 1 to 2025 and divided that by the sum of all integers 1-2025 (I don't even know why) and somehow got the right answer.

I used a method kind of guessing but we can see the number of subsets that have 2025 as the greateset is (Not Blank)

2^2025-1

Then the number of subsets that have 2024 as the greatest is:

2^2024 -1

Then the numbers of subsets that have 2023 as the greatest is:

2^2023-1

We establish a pattern similar to an geometric sequence but not exact.... 1/2, 1/4, 1/8, 1/16, 1/32,1/64,1/128,1/256 So with this pattern we already know that the numbers that count the most are the first 3 numbers that I listed but if you look at it carefully you can see that when you subtract some integer n from n_1+n_2 we see what it is always 1/2^n so we can use this finding to see that is has to be somewhere between 2024>=x>=2023
but we again see that 1/4-1/8=1/8 so we are done and our final answer is $2024$
This post has been edited 1 time. Last edited by sadas123, Apr 5, 2025, 11:09 AM
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Andyluo
1008 posts
Andyluo
#65 h  Apr 5, 2025, 11:41 AM
Y by
you can also try cases with n=3-6 which is what I did to convince myself that $n-1$ was the answer
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ERMSCoach
76 posts
ERMSCoach
#66 h  Apr 20, 2025, 7:00 AM
Y by
Add one set with just element 0.
First two sets have average 1/2
Average with two sets of average 2 makes four sets with average 1+1/4
Average with four sets of average 3 makes eight sets with average 2+1/8
...
average is 2024 + 1/ 2^2025,
not needed but you can adjust by removing 0 set to get exact average
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