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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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IOQM P26 2024
SomeonecoolLovesMaths   5
N 24 minutes ago by SomeonecoolLovesMaths
The sum of $\lfloor x \rfloor$ for all real numbers $x$ satisfying the equation $16 + 15x + 15x^2 = \lfloor x \rfloor ^3$ is:
5 replies
SomeonecoolLovesMaths
Sep 8, 2024
SomeonecoolLovesMaths
24 minutes ago
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IOQM 2023-24 P-4
lifeismathematics   8
N 33 minutes ago by SomeonecoolLovesMaths
Let $x, y$ be positive integers such that
$$ x^4=(x-1)\left(y^3-23\right)-1 . $$Find the maximum possible value of $x+y$.
8 replies
lifeismathematics
Sep 3, 2023
SomeonecoolLovesMaths
33 minutes ago
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Inspired by qrxz17
sqing   7
N 42 minutes ago by sqing
Source: Own
Let $a, b,c>0 ,(a^2+b^2+c^2)^2 - 2(a^4+b^4+c^4) = 27 $. Prove that $$a+b+c\geq 3\sqrt {3}$$
7 replies
sqing
5 hours ago
sqing
42 minutes ago
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Geometry problem
Whatisthepurposeoflife   2
N an hour ago by Whatisthepurposeoflife
Source: Derived from MEMO 2024 I3
Triangle ∆ABC is scalene the circle w that goes through the points A and B intersects AC at E BC at D let the Lines BE and AD intersect at point F. And let the tangents A and B of circle w Intersect at point G.
Prove that C F and G are collinear
2 replies
Whatisthepurposeoflife
Yesterday at 1:45 PM
Whatisthepurposeoflife
an hour ago
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A Sequence of +1's and -1's
ike.chen   36
N an hour ago by maromex
Source: ISL 2022/C1
A $\pm 1$-sequence is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
36 replies
ike.chen
Jul 9, 2023
maromex
an hour ago
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Basic ideas in junior diophantine equations
Maths_VC   1
N an hour ago by grupyorum
Source: Serbia JBMO TST 2025, Problem 3
Determine all positive integers $a, b$ and $c$ such that
$2$ $\cdot$ $10^a + 5^b = 2025^c$
1 reply
Maths_VC
Tuesday at 7:54 PM
grupyorum
an hour ago
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Prove atleast one from a,b,c is 2
Darealzolt   1
N an hour ago by pooh123
Let \(a,b,c\) be real numbers, such that
\[ a^2+b^2+c^2+abc=5 \]\[ a+b+c=3 \]Prove that atleast one of the numbers \(a,b,c\) is equal to \( 2\).
1 reply
Darealzolt
2 hours ago
pooh123
an hour ago
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IOQM 2023-24 P-2
lifeismathematics   7
N an hour ago by SomeonecoolLovesMaths
Find the number of elements in the set
$$ \left\{(a, b) \in \mathbb{N}: 2 \leq a, b \leq 2023, \log _a(b)+6 \log _b(a)=5\right\} $$
7 replies
lifeismathematics
Sep 3, 2023
SomeonecoolLovesMaths
an hour ago
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Inspired by qrxz17
sqing   3
N an hour ago by sqing
Source: Own
Let $ a,b,c $ be reals such that $ (a^2+b^2)^2 + (b^2+c^2)^2 +(c^2+a^2)^2 = 28 $ and $ (a^2+b^2+c^2)^2 =16. $ Find the value of $ a^2(a^2-1) + b^2(b^2-1)+c^2(c^2-1).$
3 replies
sqing
5 hours ago
sqing
an hour ago
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Hardest in ARO 2008
discredit   30
N an hour ago by Phat_23000245
Source: ARO 2008, Problem 11.8
In a chess tournament $ 2n+3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.
30 replies
discredit
Jun 11, 2008
Phat_23000245
an hour ago
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Find the value
sqing   12
N 2 hours ago by Phat_23000245
Source: 2024 China Fujian High School Mathematics Competition
Let $f(x)=a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0,$ $a_i\in\{-1,1\} ,i=0,1,2,\cdots,6 $ and $f(2)=-53 .$ Find the value of $f(1).$
12 replies
sqing
Jun 22, 2024
Phat_23000245
2 hours ago
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Midpoints of arcs form a similar triangle
v_Enhance   19
N 2 hours ago by Adywastaken
Source: USA TSTST 2013, Problem 1
Let $ABC$ be a triangle and $D$, $E$, $F$ be the midpoints of arcs $BC$, $CA$, $AB$ on the circumcircle. Line $\ell_a$ passes through the feet of the perpendiculars from $A$ to $DB$ and $DC$. Line $m_a$ passes through the feet of the perpendiculars from $D$ to $AB$ and $AC$. Let $A_1$ denote the intersection of lines $\ell_a$ and $m_a$. Define points $B_1$ and $C_1$ similarly. Prove that triangle $DEF$ and $A_1B_1C_1$ are similar to each other.
19 replies
1 viewing
v_Enhance
Aug 13, 2013
Adywastaken
2 hours ago
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find question
mathematical-forest   4
N 2 hours ago by GreekIdiot
Are there any contest questions that seem simple but are actually difficult? :-D
4 replies
mathematical-forest
3 hours ago
GreekIdiot
2 hours ago
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4 var inequality
SunnyEvan   1
N 2 hours ago by SunnyEvan
Source: Own
Let $ x,y,z,t \in R^+ ,$ such that : $ (x+y+z+t)^2 = x+y+z+t + (x+z)(y+t) $ and $ x \geq y \geq z \geq t .$
Try to prove or disprove : $$ \frac{2 \sqrt{x+y+z+t +(x+t)(y+z)}}{x^2+y^2+z^2+t^2 +3xz+3yt+xt+yz} \geq \frac{11(x+y)(z+t)-(x+y+z+t)}{x+y+z+t +(x+z)(y+t)} $$
1 reply
SunnyEvan
Today at 7:07 AM
SunnyEvan
2 hours ago
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ez problem....
Cobedangiu   4
N Apr 19, 2025 by iniffur
Let $x,y \in Z$ and $xy \cancel \vdots7$
Find $n \in Z^+$.
$x^2+y^2+xy=7^n$
4 replies
Cobedangiu
Apr 18, 2025
iniffur
Apr 19, 2025
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ez problem....
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Cobedangiu
70 posts
Cobedangiu
#1 h  Apr 18, 2025, 11:07 AM
Y by
Let $x,y \in Z$ and $xy \cancel \vdots7$
Find $n \in Z^+$.
$x^2+y^2+xy=7^n$
This post has been edited 1 time. Last edited by Cobedangiu, Apr 18, 2025, 11:08 AM
Reason: .
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Cobedangiu
70 posts
Cobedangiu
#2 h  Apr 19, 2025, 7:43 AM
Y by
Cobedangiu wrote:
Let $x,y \in Z$ and $xy \cancel \vdots7$
Find $n \in Z^+$.
$x^2+y^2+xy=7^n$

no :<?
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Lankou
1406 posts
Lankou
#3 h  Apr 19, 2025, 4:06 PM
Y by
What does the symbol $\cancel\vdots$ mean?
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giangtruong13
148 posts
giangtruong13
#4 h  Apr 19, 2025, 4:09 PM • 1 Y
Y by Lankou
Lankou wrote:
What does the symbol $\cancel\vdots$ mean?
It means $ 7 \cancel | xy $
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iniffur
538 posts
iniffur
#5 h  Apr 19, 2025, 11:22 PM
Y by
$$x^2+xy+y^2=7^n~~~~~~~~~~~~~~~~~~(1)$$
In its present wording the problem does not require to find out the triplets $~(x,~y,n)~$,

but rather the powers of seven which can be represented as $~~x^2+xy+y^2,~~$ with the proviso that

$~7\nmid xy~~$

It can be derived from the literature regarding $~~x^2+xy+y^2=N~~$ that if $~(x,y)=(a,b),~(c,d)~~$ are

two solutions to $~x^2+xy+y^2=7^n,~~~(x,y)=(ac+bc+bd,~ad-bc)~~$ is a further solution by virtue

of the multiplicative property of numbers under the form $~~x^2+xy+y^2=N~~$ (see (*) below)

Therefore, it is sufficient to depart from two solutions, one selected from the group consisting of:

$x^2+xy+y^2=7~~\Longrightarrow (a,b)=(1,2), (2,1), (-1,3), (-2,3), (-3,2), (-3,1), (-2, -1), (-1,-2), (1,-3), (2, -3), (3,-2),$

$(3, -1)$

and the second one selected from the group consisting of:

$ x^2+xy+y^2=49~~\Longrightarrow (c,d)= (3,5), (5,3), (-3,8), (-5,8), (-8,5), (-8,3), (-5,-3),(-3,-5), (-8,3), (8,-5)$

to derive further solutions corresponding to any power of seven by applying the formula:

$(x,y)=(ac+bc+bd,~ad-bc)$

Of course, care should be taken that $~~7\nmid ac+bc+bd,~ad-bc~~$

Example 1

$a=1, b=2, c=3, d=5\Longrightarrow (1*3+2*3+2*5=19,~~1*5-2*3=-1)\Longrightarrow 19^2-19+1=343=7^3$

Example 2

$a=-3, b=1, c=19, d=-1\Longrightarrow (-39,-16)\Longrightarrow 39^2+39*16+16^2=2401=7^4$

Example 3

$a=8, b=-5, c=19, d=-1\Longrightarrow (62,~87)\Longrightarrow 62^2+62*87+87^2=16807=7^5$

And so on.

It might be possible to generalize this approach to the power of other primes (at first sight 3 and 13 could

work).



(*)Click to reveal hidden text
This post has been edited 1 time. Last edited by iniffur, Apr 20, 2025, 1:12 PM
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