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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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Inequalities
sqing   22
N 5 minutes ago by sqing
Let $ a,b>0 $ . Prove that
$$ \frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1} \leq \frac{2}{5} $$$$ \frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1} \leq \frac{2}{5} $$
22 replies
sqing
May 13, 2025
sqing
5 minutes ago
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Minimize
lgx57   3
N 14 minutes ago by MathRook7817
Minimize $\sqrt{\cos^2 x+(2-\sin x)^2}+\dfrac{1}{2}\sqrt{(\sqrt 3-\cos x)^2+(\sin x+1)^2}$
3 replies
lgx57
Friday at 1:29 PM
MathRook7817
14 minutes ago
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2019 PMO Qualifying Stage III.3
wonderboy807   2
N an hour ago by aops-g5-gethsemanea2
A sequence \{a_n\}_{n \geq 1} of positive integers is defined by a_1 = 2 and for integers n > 1,

\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_{n-1}} + \frac{n}{a_n} = 1.

Determine the value of \sum_{k=1}^{\infty} \frac{3^k(k+3)}{4^k a_k}.
2 replies
wonderboy807
an hour ago
aops-g5-gethsemanea2
an hour ago
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Serbian selection contest for the IMO 2025 - P4
OgnjenTesic   1
N an hour ago by nataliaonline75
Source: Serbian selection contest for the IMO 2025
For a permutation $\pi$ of the set $A = \{1, 2, \ldots, 2025\}$, define its colorfulness as the greatest natural number $k$ such that:
- For all $1 \le i, j \le 2025$, $i \ne j$, if $|i - j| < k$, then $|\pi(i) - \pi(j)| \ge k$.
What is the maximum possible colorfulness of a permutation of the set $A$? Determine how many such permutations have maximal colorfulness.

Proposed by Pavle Martinović
1 reply
OgnjenTesic
May 22, 2025
nataliaonline75
an hour ago
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Another OI geometry problem
chengbilly   7
N an hour ago by MathLuis
Source: own
Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$. $H$ the orthocenter of triangle $BIC$ . The incircle of $ABC$ touches side $AC,AB$ at $E,F$ respectively. Suppose that $\odot (AEF)$ and $\odot (AIO)$ intersects $\odot (ABC)$ at $S$ and $T$ respectively (differ from $A$). Prove that $T,H,I,S$ lies on a circle.

($\odot (ABC)$ denote the circumcircle of $ABC$)
7 replies
chengbilly
Apr 14, 2021
MathLuis
an hour ago
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a tst 2013 test
Math2030   6
N an hour ago by Math2030
Given the sequence $(a_n): a_1=1, a_2=11$ and $a_{n+2}=a_{n+1}+5a_{n}, n \geq 1$
. Prove that $a_n $not is a perfect square for all $n > 3$.
6 replies
+1 w
Math2030
Yesterday at 5:26 AM
Math2030
an hour ago
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Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   2
N an hour ago by nataliaonline75
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
2 replies
OgnjenTesic
May 22, 2025
nataliaonline75
an hour ago
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Infinite Pairs of Integers
steven_zhang123   0
an hour ago
Source: 2025 Spring NSMO P6
Given a positive integer \( k \), prove that there exist infinitely many pairs of positive integers \((m, n)\) (\(m < n\)) such that
\[ \tau(m^k) \tau(m^k + 1) \cdots \tau(n^k - 1) \tau(n^k) \]is a perfect square, where \(\tau(n)\) denotes the number of positive divisors of \(n\).
Proposed by Dong Zhenyu
0 replies
steven_zhang123
an hour ago
0 replies
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power sum system of equations in 3 variables
Stear14   0
an hour ago
Given that
$x^2+y^2+z^2=8\ ,$
$x^3+y^3+z^3=15\ ,$
$x^5+y^5+z^5=100\ .$

Find the value of $\ x+y+z\ .$
0 replies
Stear14
an hour ago
0 replies
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Game on 6 by 6 grid
billzhao   26
N 2 hours ago by Sleepy_Head
Source: USAMO 2004, problem 4
Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number in that row is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't. (If two squares share a vertex, Alice can draw a line from one to the other that stays in those two squares.) Find, with proof, a winning strategy for one of the players.
26 replies
billzhao
Apr 29, 2004
Sleepy_Head
2 hours ago
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USA GEO 2003
dreammath   21
N 2 hours ago by lpieleanu
Source: TST USA 2003
Let $ABC$ be a triangle and let $P$ be a point in its interior. Lines $PA$, $PB$, $PC$ intersect sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Prove that
\[ [PAF]+[PBD]+[PCE]=\frac{1}{2}[ABC] \]
if and only if $P$ lies on at least one of the medians of triangle $ABC$. (Here $[XYZ]$ denotes the area of triangle $XYZ$.)
21 replies
dreammath
Feb 16, 2004
lpieleanu
2 hours ago
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Balkan Mathematical Olympiad
ABCD1728   0
2 hours ago
Can anyone provide the PDF version of the book "Balkan Mathematical Olympiads" by Mircea Becheanu and Bogdan Enescu (published by XYZ press in 2014), thanks!
0 replies
ABCD1728
2 hours ago
0 replies
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An amazing functional equation over positive reals
ariopro1387   0
3 hours ago
Source: Iran Team selection test 2025 - P6
Find all functions $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$, such that:
$f(f(f(xy))+x^2)=f(y)(f(x)-f(x+y))$
for all $x, y>0$.
0 replies
ariopro1387
3 hours ago
0 replies
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Nice "if and only if" function problem
ICE_CNME_4   10
N 3 hours ago by maromex
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[ f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0. \](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )

Please do it at 9th grade level. Thank you!
10 replies
1 viewing
ICE_CNME_4
Friday at 7:23 PM
maromex
3 hours ago
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BABBAGE'S THEOREM EXTENSION
Mathgloggers   4
N May 6, 2025 by Mathgloggers
A few days ago I came across. this interesting result is someone interested in proving this.

$\boxed{\sum_{k=1}^{p-1} \frac{1}{k} \equiv \sum_{k=p+1}^{2p-1} \frac{1}{k} \equiv \sum_{k=2p+1}^{3p-1}\frac{1}{k} \equiv.....\sum_{k=p(p-1)+1}^{p^2-1}\frac{1}{k} \equiv 0(mod p^2)}$
4 replies
Mathgloggers
Apr 29, 2025
Mathgloggers
May 6, 2025
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BABBAGE'S THEOREM EXTENSION
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Mathgloggers
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Mathgloggers
#1 h  Apr 29, 2025, 12:18 PM • 1 Y
Y by orange0707
A few days ago I came across. this interesting result is someone interested in proving this.

$\boxed{\sum_{k=1}^{p-1} \frac{1}{k} \equiv \sum_{k=p+1}^{2p-1} \frac{1}{k} \equiv \sum_{k=2p+1}^{3p-1}\frac{1}{k} \equiv.....\sum_{k=p(p-1)+1}^{p^2-1}\frac{1}{k} \equiv 0(mod p^2)}$
This post has been edited 1 time. Last edited by Mathgloggers, Apr 29, 2025, 12:22 PM
Reason: D
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orange0707
2 posts
orange0707
#2 h  May 6, 2025, 3:46 AM • 1 Y
Y by Mathgloggers
Is this an extension of Wolstenholme's theorem?
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Mathgloggers
90 posts
Mathgloggers
#3 h  May 6, 2025, 4:18 AM
Y by
Yes a sort of because that connects with $\frac{1}{k} (mod p^2) $ congruences but this also follows from Babbages theorem which is:
$ \binom{aP}{bP} \equiv 0 (mod p) $ which act a sort of proof for Wolstenholme's theorem.
So when I was working I saw the congruences of of the other part because at $\equiv 0 (mod p) $ modulo inverse does not exist and I came up with this.
This post has been edited 1 time. Last edited by Mathgloggers, May 6, 2025, 4:19 AM
Reason: m
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orange0707
2 posts
orange0707
#4 h  May 6, 2025, 6:50 AM • 1 Y
Y by Mathgloggers
This is really an interesting idea. When I saw your conclusion, I happened to be studying the method of constructing polynomials and then proving Wolstenholme's theorem by using Lagrange's theorem and Fermat's little theorem, and found that it could also be used here. Additionally, may I ask when new users can upload images?Can complex mathematical formulas be directly entered using a LaTeX editor?
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Mathgloggers
90 posts
Mathgloggers
#5 h  May 6, 2025, 7:11 AM
Y by
Yes you can upload images through the pin image icon the extreme right of head bar where you send solution and yes you can upload complex mathematical formula .
I would recommend you the "LATEX FORUMS " on aops you can check that to form your formulas
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