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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 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
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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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Beautiful problem
luutrongphuc   14
N 6 minutes 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 $(I)$.
14 replies
luutrongphuc
Apr 4, 2025
aidenkim119
6 minutes ago
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2025 - Turkmenistan National Math Olympiad
A_E_R   5
N 10 minutes ago by Filipjack
Source: Turkmenistan Math Olympiad - 2025
Let k,m,n>=2 positive integers and GCD(m,n)=1, Prove that the equation has infinitely many solutions in distict positive integers: x_1^m+x_2^m+⋯x_k^m=x_(k+1)^n
5 replies
A_E_R
3 hours ago
Filipjack
10 minutes ago
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Vector geometry with unusual points
Ciobi_   1
N 15 minutes ago by ericdimc
Source: Romania NMO 2025 9.2
Let $\triangle ABC$ be an acute-angled triangle, with circumcenter $O$, circumradius $R$ and orthocenter $H$. Let $A_1$ be a point on $BC$ such that $A_1H+A_1O=R$. Define $B_1$ and $C_1$ similarly.
If $\overrightarrow{AA_1} + \overrightarrow{BB_1} + \overrightarrow{CC_1} = \overrightarrow{0}$, prove that $\triangle ABC$ is equilateral.
1 reply
Ciobi_
Apr 2, 2025
ericdimc
15 minutes ago
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Collinearity with orthocenter
Retemoeg   9
N 28 minutes ago by X.Luser
Source: Own?
Given scalene triangle $ABC$ with circumcenter $(O)$. Let $H$ be a point on $(BOC)$ such that $\angle AOH = 90^{\circ}$. Denote $N$ the point on $(O)$ satisfying $AN \parallel BC$. If $L$ is the projection of $H$ onto $BC$, show that $LN$ passes through the orthocenter of $\triangle ABC$.
9 replies
1 viewing
Retemoeg
Mar 30, 2025
X.Luser
28 minutes ago
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Parallel Lines and Q Point
taptya17   14
N an hour ago by Haris1
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
14 replies
taptya17
Dec 13, 2024
Haris1
an hour ago
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The last nonzero digit of factorials
Tintarn   4
N an hour ago by Sadigly
Source: Bundeswettbewerb Mathematik 2025, Round 1 - Problem 2
For each integer $n \ge 2$ we consider the last digit different from zero in the decimal expansion of $n!$. The infinite sequence of these digits starts with $2,6,4,2,2$. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.
4 replies
Tintarn
Mar 17, 2025
Sadigly
an hour ago
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P2 Geo that most of contestants died
AlephG_64   2
N an hour ago by Tsikaloudakis
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 $\angle DAC= 30^\circ$.
2 replies
AlephG_64
Yesterday at 1:23 PM
Tsikaloudakis
an hour ago
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Geometry
youochange   0
an hour ago
m:}
Let $\triangle ABC$ be a triangle inscribed in a circle, where the tangents to the circle at points $B$ and $C$ intersect at the point $P$. Let $M$ be a point on the arc $AC$ (not containing $B$) such that $M \neq A$ and $M \neq C$. Let the lines $BC$ and $AM$ intersect at point $K$. Let $P'$ be the reflection of $P$ with respect to the line $AM$. The lines $AP'$ and $PM$ intersect at point $Q$, and $PM$ intersects the circumcircle of $\triangle ABC$ again at point $N$.

Prove that the point $Q$ lies on the circumcircle of $\triangle ANK$.
0 replies
youochange
an hour ago
0 replies
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comp. geo starting with a 90-75-15 triangle. <APB =<CPQ, <BQA =<CQP.
parmenides51   1
N an hour ago by Mathzeus1024
Source: 2013 Cuba 2.9
Let ABC be a triangle with $\angle A = 90^o$, $\angle B = 75^o$, and $AB = 2$. Points $P$ and $Q$ of the sides $AC$ and $BC$ respectively, are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$. Calculate the lenght of $QA$.
1 reply
parmenides51
Sep 20, 2024
Mathzeus1024
an hour ago
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Fridolin just can't get enough from jumping on the number line
Tintarn   2
N an hour ago by Sadigly
Source: Bundeswettbewerb Mathematik 2025, Round 1 - Problem 1
Fridolin the frog jumps on the number line: He starts at $0$, then jumps in some order on each of the numbers $1,2,\dots,9$ exactly once and finally returns with his last jump to $0$. Can the total distance he travelled with these $10$ jumps be a) $20$, b) $25$?
2 replies
Tintarn
Mar 17, 2025
Sadigly
an hour ago
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Geometry
Captainscrubz   2
N an hour ago by MrdiuryPeter
Source: Own
Let $D$ be any point on side $BC$ of $\triangle ABC$ .Let $E$ and $F$ be points on $AB$ and $AC$ such that $EB=ED$ and $FD=FC$ respectively. Prove that the locus of circumcenter of $(DEF)$ is a line.
Prove without using moving points :D
2 replies
Captainscrubz
3 hours ago
MrdiuryPeter
an hour ago
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inequality ( 4 var
SunnyEvan   4
N an hour ago by SunnyEvan
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$$$ a^4+b^4+c^4+d^4+ \frac{252}{25} \geq \frac{88}{25}(a^3+b^3+c^3+d^3) $$equality cases : ?
4 replies
SunnyEvan
Apr 4, 2025
SunnyEvan
an hour ago
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Find the constant
JK1603JK   1
N 2 hours ago by Quantum-Phantom
Source: unknown
Find all $k$ such that $$\left(a^{3}+b^{3}+c^{3}-3abc\right)^{2}-\left[a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a)\right]^{2}\ge 2k\cdot(a-b)^{2}(b-c)^{2}(c-a)^{2}$$forall $a,b,c\ge 0.$
1 reply
JK1603JK
5 hours ago
Quantum-Phantom
2 hours ago
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hard problem
Cobedangiu   15
N 2 hours ago by Nguyenhuyen_AG
problem
15 replies
Cobedangiu
Mar 27, 2025
Nguyenhuyen_AG
2 hours ago
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Graph vertices with degree
MetaphysicalWukong   1
N Mar 30, 2025 by truongphatt2668
Source: Qianrong Hao
For a graph $G=\left(V,E\right)$, what is the largest possible value of |V| if |E|=35 and $deg\left(v\right)\ge3$
1 reply
MetaphysicalWukong
Mar 30, 2025
truongphatt2668
Mar 30, 2025
J
Graph vertices with degree
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Reveal topic tags
graph theory
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Source: Qianrong Hao
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MetaphysicalWukong
260 posts
MetaphysicalWukong
#1 h  Mar 30, 2025, 6:47 AM
Y by
For a graph $G=\left(V,E\right)$, what is the largest possible value of |V| if |E|=35 and $deg\left(v\right)\ge3$
This post has been edited 1 time. Last edited by MetaphysicalWukong, Mar 30, 2025, 6:55 AM
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truongphatt2668
321 posts
truongphatt2668
#2 h  Mar 30, 2025, 7:07 AM
Y by
We have this familiar identity:
$$\sum \deg (v) = 2E = 70$$Each $\deg (v) \ge 3$ thus:
$$\sum \deg (v) \ge 3 |V|$$Then: $$3|V| \le 70 \Rightarrow |V| \le 23$$
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