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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
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Problems for v_p(n)
xytunghoanh   1
N 3 minutes ago by whwlqkd
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 :>
1 reply
xytunghoanh
38 minutes ago
whwlqkd
3 minutes ago
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Parallel Lines and Q Point
taptya17   13
N 8 minutes ago by everythingpi3141592
Source: India EGMO TST 2025 Day 1 P3
Let $\Delta ABC$ be an acute angled scalene triangle with circumcircle $\omega$. Let $O$ and $H$ be the circumcenter and orthocenter of $\Delta ABC,$ respectively. Let $E,F$ and $Q$ be points on segments $AB,AC$ and $\omega$, respectively, such that
$$\angle BHE=\angle CHF=\angle AQH=90^\circ.$$Prove that $OQ$ and $AH$ intersect on the circumcircle of $\Delta AEF$.

Proposed by Antareep Nath
13 replies
taptya17
Dec 13, 2024
everythingpi3141592
8 minutes ago
J
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Cool combinatorial problem (grid)
Anto0110   0
8 minutes ago
Suppose you have an $m \cdot n$ grid with $m$ rows and $n$ columns, and each square of the grid is filled with a non-negative integer. Let $a$ be the average of all the numbers in the grid. Prove that if $m >(10a+10)^n$ the there exist two identical rows (meaning same numbers in the same order).
0 replies
Anto0110
8 minutes ago
0 replies
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Cards and combi
AlephG_64   0
15 minutes ago
Source: 2025 Finals Portuguese Mathematical Olympiad P6
Maria wants to solve a challenge with a deck of cards, each with a different figure. Initially, the cards are distributed randomly into two piles, not necessarily in equal parts. Maria's goal is to get all the cards into the same pile.

On each turn, Maria takes the top card from each pile and compares them. In the rule book, there's a table that indicates, for each card match, which of the two wins. Both cards are then placed on the bottom of the winning card in the order Maria chooses. The challenge ends when all the cards are in one pile.

Show that it is always possible for Maria to solve the challenge. Regardless of the initial distribution of the cards and the table in the rule book.
0 replies
+1 w
AlephG_64
15 minutes ago
0 replies
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Cool number condition
AlephG_64   0
23 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
23 minutes ago
0 replies
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four variables inequality
JK1603JK   1
N 28 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
28 minutes ago
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Geo to make por people happy
AlephG_64   0
29 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
1 viewing
AlephG_64
29 minutes ago
0 replies
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Cool sequence problem
AlephG_64   0
32 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
32 minutes ago
0 replies
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Table with cells
RagvaloD   8
N 34 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
RagvaloD
May 3, 2017
Radin_
34 minutes ago
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H,G,O geometry
m4thbl3nd3r   2
N 39 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
an hour ago
m4thbl3nd3r
39 minutes ago
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Prime number and composite number
mingzhehu   2
N 40 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
40 minutes ago
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P2 Geo that most of contestants died
AlephG_64   0
43 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
43 minutes ago
0 replies
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Squence problem
AlephG_64   0
an hour 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
an hour ago
0 replies
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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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<PXI + <PYI = 90^o wanted, 3 incenters, starting with int. tangent circles
parmenides51   3
N Jan 27, 2023 by PNT
Source: 2014 Armenia NMO XII p6
Circles $\Omega$ and $\omega$ are tangent at point $P$ and $\omega $ is inside $\Omega$. The chord $AB$ of the circle $\Omega$ is tangent to the circle $\omega$ at the point $C $, and the line $PC$ intersects $\Omega$ for the second time at point $Q$. The chords $QR$ and $QS$ of the circle $\Omega$ are tangent to $\omega$. Let $I,X$ and $Y$ be the centers of the circles inscribed in the triangles $APB$, $ARB$ and $ASB$, respectively. Prove that $\angle PXI + \angle PYI = 90 ^\circ $.
3 replies
parmenides51
Aug 16, 2021
PNT
Jan 27, 2023
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<PXI + <PYI = 90^o wanted, 3 incenters, starting with int. tangent circles
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Reveal topic tags
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angles
circles
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Source: 2014 Armenia NMO XII p6
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parmenides51
30629 posts
parmenides51
#1 h  Aug 16, 2021, 10:08 PM • 1 Y
Y by PNT
Circles $\Omega$ and $\omega$ are tangent at point $P$ and $\omega $ is inside $\Omega$. The chord $AB$ of the circle $\Omega$ is tangent to the circle $\omega$ at the point $C $, and the line $PC$ intersects $\Omega$ for the second time at point $Q$. The chords $QR$ and $QS$ of the circle $\Omega$ are tangent to $\omega$. Let $I,X$ and $Y$ be the centers of the circles inscribed in the triangles $APB$, $ARB$ and $ASB$, respectively. Prove that $\angle PXI + \angle PYI = 90 ^\circ $.
This post has been edited 1 time. Last edited by parmenides51, Aug 16, 2021, 10:10 PM
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PNT
320 posts
PNT
#2 h  Jan 25, 2023, 1:55 PM
Y by
It's not hard to see $Q$ is the midpoint or arc $AB$ and the incenters all lie on the circle with center $Q$ and radius $QA$. Furthermore the the two circles $\omega $ and $\Omega$ are orthogonal. Therefore $X$ and $Y$ lie on $\omega$. They're precisely the points of tangency so we can completely ignore $R$ and $S$. Let’s angle chase,

Let $a=\angle MPY=\angle MYQ$ and $b=\angle MPX=\angle QXM$. We have $$\angle XIY=180^{\circ}-\frac{1}{2}\angle XQY=180^{\circ}-\frac{1}{2}(180^{\circ}-2(a+b))=90+a+b$$
Therefore $\angle XIM+\angle MIY=90^{\circ}+a+b$ i.e. $\angle XIM =90^{\circ}+a+b-\angle MIY.$

Now $\angle XIP+\angle PIY=360^{\circ}-\angle XIY=360^{\circ}-90^{\circ}-a-b=270^{\circ}-a-b$ meaning $\angle XIP = 270^{\circ}-a-b -\angle PIY$

Finally $$\angle PXI=180^{\circ}-\angle XIP-\angle XPI=180^{\circ}-(270^{\circ}-a-b -\angle PIY )-b$$$$=-90^{\circ}+a+b+\angle PIY-b=a+\angle PIY-90^{\circ}=180^{\circ}-\angle IYP-90^{\circ}=90^{\circ}-\angle IYP.$$
Attachments:
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teomihai
2949 posts
teomihai
#3 h  Jan 27, 2023, 7:00 PM
Y by
i seems it is corect!
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PNT
320 posts
PNT
#4 h  Jan 27, 2023, 7:05 PM
Y by
teomihai wrote:
i seems it is corect!

Thank you.
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