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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 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
J
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Cool number condition
AlephG_64   0
7 minutes ago
Source: 2025 Finals Portuguese Mathematical Olympiad P5
An integer number $n \geq 2$ is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of $n$ less than $n$ is the difference between two numbers on the sheet, and no other positive number is.
Find all the feirense numbers.
0 replies
AlephG_64
7 minutes ago
0 replies
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four variables inequality
JK1603JK   1
N 12 minutes ago by arqady
Source: unknown?
Prove that $$27(a^4+b^4+c^4+d^4)+148abcd\ge (a+b+c+d)^4,\ \ \forall a,b,c,d\ge 0.$$
1 reply
JK1603JK
Yesterday at 4:26 PM
arqady
12 minutes ago
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Geo to make por people happy
AlephG_64   0
13 minutes ago
Source: 2025 Finals Portuguese Mathematical Olympiad P4
Let $ABCD$ be a square with $2cm$ side length and with center $T$. A rhombus $ARTE$ is drawn where point $E$ lies on line $DC$. What is the area of $ARTE$?
0 replies
AlephG_64
13 minutes ago
0 replies
J
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Cool sequence problem
AlephG_64   0
17 minutes ago
Source: 2025 Finals Portuguese Mathematical Olympiad P3
A computer science teacher has asked his students to write a program that, given a list of $n$ numbers $a_1, a_2, ..., a_n$, calculates the list $b_1, b_2, ..., b_n$ where $b_k$ is the number of times the number $a_k$ appears in the list. So, for example, for the list $1,2,3,1$, the program returns the list $2,1,1,2$.

Next, the teacher asked Alexandre to run the program for a list of $2025$ numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to $k$ appears in the list. Find the largest value of $k$ for which, whatever the initial list of $2025$ positive integers $a_1, a_2, ..., a_{2025}$, it is possible for Alexander to do what the teacher asked him to do.
0 replies
AlephG_64
17 minutes ago
0 replies
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Table with cells
RagvaloD   8
N 18 minutes ago by Radin_
Source: All Russian Olympiad 2017,Day2,grade 9,P8
Every cell of $100\times 100$ table is colored black or white. Every cell on table border is black. It is known, that in every $2\times 2$ square there are cells of two colors. Prove, that exist $2\times 2$ square that is colored in chess order.
8 replies
+1 w
RagvaloD
May 3, 2017
Radin_
18 minutes ago
J
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Problems for v_p(n)
xytunghoanh   0
22 minutes ago
Hello everyone. I need some easy problems for $v_p(n)$ (use in a problem for junior) to practice. Can anyone share to me?
Thanks :>
0 replies
xytunghoanh
22 minutes ago
0 replies
J
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H,G,O geometry
m4thbl3nd3r   2
N 23 minutes ago by m4thbl3nd3r
Source: own
Let $ABC$ be a non-isosceles triangle with orthocenter $H$, circumcenter $O$. Let $M,N,P$ be midpoints of $BC,CA,AB$ and $A',B',C'$ be reflections of $M,N,P$ across $HO$. Prove that triangle $ABC,A'B'C'$ have the same centroid.
2 replies
m4thbl3nd3r
33 minutes ago
m4thbl3nd3r
23 minutes ago
J
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Prime number and composite number
mingzhehu   2
N 24 minutes ago by mingzhehu
I have one topic on how to identify Prime Number and Composite Number quickly? Maybe the number is more than 100 or 1000.......!
If there are some formula that can be used to verify the number easily, it will be highly appreciated.
Does anybody has any good idea for that?

2 replies
mingzhehu
6 hours ago
mingzhehu
24 minutes ago
J
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P2 Geo that most of contestants died
AlephG_64   0
27 minutes ago
Source: 2025 Finals Portuguese Mathematical Olympiad P2
Let $ABCD$ be a quadrilateral such that $\angle A$ and $\angle D$ are acute and $\overline{AB} = \overline{BC} = \overline{CD}$. Suppose that $\angle BDA = 30^\circ$, prove that $\angle DAC= 30^\circ$.
0 replies
AlephG_64
27 minutes ago
0 replies
J
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Squence problem
AlephG_64   0
31 minutes ago
Source: 2025 Finals Portuguese Math Olympiad P1
Francisco wrote a sequence of numbers starting with $25$. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was $8$, what is the fifth term of the sequence?
0 replies
AlephG_64
31 minutes ago
0 replies
J
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Beautiful problem
luutrongphuc   1
N an hour ago by aidenkim119
Let triangle $ABC$ be circumscribed about circle $(I)$, and let $H$ be the orthocenter of $\triangle ABC$. The circle $(I)$ touches line $BC$ at $D$. The tangent to the circle $(BHC)$ at $H$ meets $BC$ at $S$. Let $J$ be the midpoint of $HI$, and let the line $DJ$ meet $(I)$ again at $X$. The tangent to $(I)$ parallel to $BC$ meets the line $AX$ at $T$. Prove that $ST$ is tangent to $(I)$.
1 reply
luutrongphuc
Yesterday at 5:35 AM
aidenkim119
an hour ago
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inequality
pennypc123456789   4
N an hour ago by sqing
Let \( x, y \) be positive real numbers satisfying \( x + y = 2 \). Prove that

\[ 3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}. \]
4 replies
pennypc123456789
Mar 24, 2025
sqing
an hour ago
J
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Convex lattice polygon
Oksutok   2
N an hour ago by Oksutok
Let $f(n)$ be the maximal number of the vertices of a convex lattice polygon with exactly $n$ lattice points in the interior. Show that:
a) $f(n) \le 2n$ for $n \ge 3$
b)$f(n)<Cn^{1/3}$ for some constant $C \in \mathbb{R}_{>0}$.
2 replies
Oksutok
Sep 29, 2024
Oksutok
an hour ago
J
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inquequality
ngocthi0101   11
N an hour ago by sqing
let $a,b,c > 0$ prove that
$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$
11 replies
ngocthi0101
Sep 26, 2014
sqing
an hour ago
J
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J
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2 points with distance <3, when 6 points are inside a rectangle 4x5
parmenides51   1
N Sep 29, 2020 by RagvaloD
Source: Lithuania TST 2012 p5
Inside a rectangle with sides $4$ and $5$, there are six points. Prove that there are two points with distance less than $3$.
1 reply
parmenides51
Sep 29, 2020
RagvaloD
Sep 29, 2020
J
2 points with distance <3, when 6 points are inside a rectangle 4x5
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Reveal topic tags
geometry
rectangle
distance
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G H BBookmark kLocked kLocked NReply
Source: Lithuania TST 2012 p5
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parmenides51
30629 posts
parmenides51
#1 h  Sep 29, 2020, 12:00 PM
Y by
Inside a rectangle with sides $4$ and $5$, there are six points. Prove that there are two points with distance less than $3$.
Z K Y
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RagvaloD
4894 posts
RagvaloD
#2 h  Sep 29, 2020, 12:38 PM
Y by
Divide rectangle into 2 rectangles with sides $2.5 \times 1.6$ and 3 rectangles with sides $\frac{5}{3} \times 2.4$ Every such rectangle has diagonal less than $3$, and by pigeon principle, at least one of this rectangles contains $2$ points.
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