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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.

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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
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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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Sum of Good Indices
raxu   25
N 12 minutes ago by cj13609517288
Source: TSTST 2015 Problem 1
Let $a_1, a_2, \dots, a_n$ be a sequence of real numbers, and let $m$ be a fixed positive integer less than $n$. We say an index $k$ with $1\le k\le n$ is good if there exists some $\ell$ with $1\le \ell \le m$ such that $a_k+a_{k+1}+...+a_{k+\ell-1}\ge0$, where the indices are taken modulo $n$. Let $T$ be the set of all good indices. Prove that $\sum\limits_{k \in T}a_k \ge 0$.

Proposed by Mark Sellke
25 replies
raxu
Jun 26, 2015
cj13609517288
12 minutes ago
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f(x^2+y^2+z^2)=f(xy)+f(yz)+f(zx)
dangerousliri   7
N 14 minutes ago by jasperE3
Source: FEOO, Shortlist A4
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}$ such that for any positive real numbers $x, y$ and $z$,
$$f(x^2+y^2+z^2)=f(xy)+f(yz)+f(zx)$$
Proposed by ThE-dArK-lOrD, and Papon Tan Lapate, Thailand
7 replies
dangerousliri
May 31, 2020
jasperE3
14 minutes ago
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IMO Genre Predictions
ohiorizzler1434   24
N 32 minutes ago by Miquel-point
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
24 replies
ohiorizzler1434
Yesterday at 6:51 AM
Miquel-point
32 minutes ago
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foldina a rectangle paper 3 times
parmenides51   1
N an hour ago by TheBaiano
Source: 2023 May Olympiad L2 p4
Matías has a rectangular sheet of paper $ABCD$, with $AB<AD$.Initially, he folds the sheet along a straight line $AE$, where $E$ is a point on the side $DC$ , so that vertex $D$ is located on side $BC$, as shown in the figure. Then folds the sheet again along a straight line $AF$, where $F$ is a point on side $BC$, so that vertex $B$ lies on the line $AE$; and finally folds the sheet along the line $EF$. Matías observed that the vertices $B$ and $C$ were located on the same point of segment $AE$ after making the folds. Calculate the measure of the angle $\angle DAE$.
IMAGE
1 reply
parmenides51
Mar 24, 2024
TheBaiano
an hour ago
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Divisibility NT
reni_wee   0
an hour ago
Source: Japan 1996, ONTCP
Let $m,n$ be relatively prime positive integers. Calculate $gcd(5^m+7^m, 5^n+7^n).$
0 replies
reni_wee
an hour ago
0 replies
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Modular arithmetic at mod n
electrovector   3
N 2 hours ago by Primeniyazidayi
Source: 2021 Turkey JBMO TST P6
Integers $a_1, a_2, \dots a_n$ are different at $\text{mod n}$. If $a_1, a_2-a_1, a_3-a_2, \dots a_n-a_{n-1}$ are also different at $\text{mod n}$, we call the ordered $n$-tuple $(a_1, a_2, \dots a_n)$ lucky. For which positive integers $n$, one can find a lucky $n$-tuple?
3 replies
electrovector
May 24, 2021
Primeniyazidayi
2 hours ago
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Sequences problem
BBNoDollar   3
N 3 hours ago by BBNoDollar
Source: Mathematical Gazette Contest
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
3 replies
BBNoDollar
Yesterday at 5:53 PM
BBNoDollar
3 hours ago
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Arbitrary point on BC and its relation with orthocenter
falantrng   33
N 3 hours ago by Thapakazi
Source: Balkan MO 2025 P2
In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus
33 replies
falantrng
Apr 27, 2025
Thapakazi
3 hours ago
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Inequality
lgx57   4
N 3 hours ago by GeoMorocco
Source: Own
$a,b,c>0,ab+bc+ca=1$. Prove that

$$\sum \sqrt{8ab+1} \ge 5$$
(I don't know whether the equality holds)
4 replies
lgx57
Yesterday at 3:14 PM
GeoMorocco
3 hours ago
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Rubber bands
v_Enhance   5
N 3 hours ago by lpieleanu
Source: OTIS Mock AIME 2024 #12
Let $\mathcal G_n$ denote a triangular grid of side length $n$ consisting of $\frac{(n+1)(n+2)}{2}$ pegs. Charles the Otter wishes to place some rubber bands along the pegs of $\mathcal G_n$ such that every edge of the grid is covered by exactly one rubber band (and no rubber band traverses an edge twice). He considers two placements to be different if the sets of edges covered by the rubber bands are different or if any rubber band traverses its edges in a different order. The ordering of which bands are over and under does not matter.
For example, Charles finds there are exactly $10$ different ways to cover $\mathcal G_2$ using exactly two rubber bands; the full list is shown below, with one rubber band in orange and the other in blue.
IMAGE
Let $N$ denote the total number of ways to cover $\mathcal G_4$ with any number of rubber bands. Compute the remainder when $N$ is divided by $1000$.

Ethan Lee
5 replies
v_Enhance
Jan 16, 2024
lpieleanu
3 hours ago
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Geometry with orthocenter config
thdnder   6
N 4 hours ago by ohhh
Source: Own
Let $ABC$ be a triangle, and let $AD, BE, CF$ be its altitudes. Let $H$ be its orthocenter, and let $O_B$ and $O_C$ be the circumcenters of triangles $AHC$ and $AHB$. Let $G$ be the second intersection of the circumcircles of triangles $FDO_B$ and $EDO_C$. Prove that the lines $DG$, $EF$, and $A$-median of $\triangle ABC$ are concurrent.
6 replies
thdnder
Apr 29, 2025
ohhh
4 hours ago
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Strange Inequality
anantmudgal09   40
N 4 hours ago by starchan
Source: INMO 2020 P4
Let $n \geqslant 2$ be an integer and let $1<a_1 \le a_2 \le \dots \le a_n$ be $n$ real numbers such that $a_1+a_2+\dots+a_n=2n$. Prove that$$a_1a_2\dots a_{n-1}+a_1a_2\dots a_{n-2}+\dots+a_1a_2+a_1+2 \leqslant a_1a_2\dots a_n.$$
Proposed by Kapil Pause
40 replies
anantmudgal09
Jan 19, 2020
starchan
4 hours ago
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Finding Solutions
MathStudent2002   22
N 4 hours ago by ihategeo_1969
Source: Shortlist 2016, Number Theory 5
Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\]Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.
22 replies
MathStudent2002
Jul 19, 2017
ihategeo_1969
4 hours ago
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USAMO 2000 Problem 3
MithsApprentice   10
N 4 hours ago by HamstPan38825
A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
10 replies
MithsApprentice
Oct 1, 2005
HamstPan38825
4 hours ago
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Number Theory
TUAN2k8   4
N Apr 19, 2025 by vsamc
Prove that there are infinitely many of positive integers m such that $m+1 | 3^m+1$
4 replies
TUAN2k8
Apr 18, 2025
vsamc
Apr 19, 2025
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Number Theory
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fun nt
number theory
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TUAN2k8
11 posts
TUAN2k8
#1 h  Apr 18, 2025, 3:25 PM
Y by
Prove that there are infinitely many of positive integers m such that $m+1 | 3^m+1$
This post has been edited 2 times. Last edited by TUAN2k8, Apr 18, 2025, 11:24 PM
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Roger.Moore
5 posts
Roger.Moore
#2 h  Apr 18, 2025, 6:06 PM
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TUAN2k8
11 posts
TUAN2k8
#3 h  Apr 18, 2025, 11:08 PM
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Any solution?
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epl1
156 posts
epl1
#4 h  Apr 19, 2025, 2:44 AM
Y by
Woah, self made?
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vsamc
3789 posts
vsamc
#5 h  Apr 19, 2025, 2:51 AM • 1 Y
Y by kiyoras_2001
$m = 3^{3^k}$ works for all $k\geq 0$, since $$3^{3^k} + 1 \mid (3^{3^k})^{3^{3^k-k}} + 1^{3^{3^k-k}} = 3^{3^{3^k}} + 1.$$
This post has been edited 2 times. Last edited by vsamc, Apr 19, 2025, 2:51 AM
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