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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
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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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Chessboard pattern
BR1F1SZ   2
N 11 minutes ago by Ubfo
Source: 2018 Argentina TST P3
In a $100 \times 100$ board, each square is colored either white or black, with all the squares on the border of the board being black. Additionally, no $2 \times 2$ square within the board has all four squares of the same color. Prove that the board contains a $2 \times 2$ square colored like a chessboard.
2 replies
BR1F1SZ
Dec 27, 2024
Ubfo
11 minutes ago
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IMO ShortList 2001, geometry problem 2
orl   48
N 16 minutes ago by legogubbe
Source: IMO ShortList 2001, geometry problem 2
Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.
48 replies
orl
Sep 30, 2004
legogubbe
16 minutes ago
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Projective Electrostatistics
drago.7437   0
24 minutes ago
Source: Own
Given two charges of any magnitude , a third charge collinear with them , exists such that it is in equillibirum , Prove that if a fourth charge in the same line exists such that it is in equillibrium , then the 3rd charge and the fourth charge are harmonic conjugates with respect to the two fixed charges . , For example if two +q charges are fixed then if in their midpoint placed a charge -q , it is in equillibrium , also if the same charge -q is placed at infinity the system is again in equillibrium , and the midpoint and the point at infinity are harmonic conjugates .
0 replies
drago.7437
24 minutes ago
0 replies
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A very nice inequality
KhuongTrang   3
N 24 minutes ago by Mathdreams
Source: own
Problem. Let $a,b,c\in \mathbb{R}:\ a+b+c=3.$ Prove that $$\color{black}{\sqrt{5a^{2}-ab+5b^{2}}+\sqrt{5b^{2}-bc+5c^{2}}+\sqrt{5c^{2}-ca+5a^{2}}\le 2(a^2+b^2+c^2)+ab+bc+ca.}$$When does equality hold?
3 replies
KhuongTrang
2 hours ago
Mathdreams
24 minutes ago
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Find an angle
socrates   3
N 30 minutes ago by Nari_Tom
Source: Baltic Way 2016, Problem 18
Let $ABCD$ be a parallelogram such that $\angle BAD = 60^{\circ}.$ Let $K$ and $L$ be the midpoints of $BC$ and $CD,$ respectively. Assuming that $ABKL$ is a cyclic quadrilateral, find $\angle ABD.$
3 replies
socrates
Nov 5, 2016
Nari_Tom
30 minutes ago
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ISI 2019 : Problem #7
integrated_JRC   12
N 41 minutes ago by Levieee
Source: I.S.I. 2019
Let $f$ be a polynomial with integer coefficients. Define $$a_1 = f(0)~,~a_2 = f(a_1) = f(f(0))~,$$and $~a_n = f(a_{n-1})$ for $n \geqslant 3$.

If there exists a natural number $k \geqslant 3$ such that $a_k = 0$, then prove that either $a_1=0$ or $a_2=0$.
12 replies
integrated_JRC
May 5, 2019
Levieee
41 minutes ago
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Problem 1 of the HMO 2025
GreekIdiot   4
N an hour ago by eric201291
Let $P(x)=x^4+5x^3+mx^2+5nx+4$ have $2$ distinct real roots, which sum up to $-5$. If $m,n \in \mathbb {Z_+}$, find the values of $m,n$ and their corresponding roots.
4 replies
GreekIdiot
Feb 22, 2025
eric201291
an hour ago
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Nordic 2025 P1
anirbanbz   5
N an hour ago by eric201291
Source: Nordic 2025
Let $n$ be a positive integer greater than $2$. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying:
$(f(x+y))^{n} = f(x^{n})+f(y^{n}),$ for all integers $x,y$
5 replies
anirbanbz
Mar 25, 2025
eric201291
an hour ago
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Two Orthocenters and an Invariant Point
Mathdreams   1
N an hour ago by RANDOM__USER
Source: 2025 Nepal Mock TST Day 1 Problem 3
Let $\triangle{ABC}$ be a triangle, and let $P$ be an arbitrary point on line $AO$, where $O$ is the circumcenter of $\triangle{ABC}$. Define $H_1$ and $H_2$ as the orthocenters of triangles $\triangle{APB}$ and $\triangle{APC}$. Prove that $H_1H_2$ passes through a fixed point which is independent of the choice of $P$.

(Kritesh Dhakal, Nepal)
1 reply
Mathdreams
2 hours ago
RANDOM__USER
an hour ago
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Geometry
youochange   2
N an hour ago by youochange
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 $\triangle ANK$.
2 replies
+1 w
youochange
4 hours ago
youochange
an hour ago
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Beautiful problem
luutrongphuc   20
N an hour ago by r7di048hd3wwd3o3w58q
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 $(I)$.
20 replies
luutrongphuc
Apr 4, 2025
r7di048hd3wwd3o3w58q
an hour ago
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Stereotypical Diophantine Equation
Mathdreams   2
N an hour ago by grupyorum
Source: 2025 Nepal Mock TST Day 2 Problem 1
Find all solutions in the nonnegative integers to $2^a3^b5^c7^d - 1 = 11^e$.

(Shining Sun, USA)
2 replies
Mathdreams
2 hours ago
grupyorum
an hour ago
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Common tangent to diameter circles
Stuttgarden   1
N an hour ago by jrpartty
Source: Spain MO 2025 P2
The cyclic quadrilateral $ABCD$, inscribed in the circle $\Gamma$, satisfies $AB=BC$ and $CD=DA$, and $E$ is the intersection point of the diagonals $AC$ and $BD$. The circle with center $A$ and radius $AE$ intersects $\Gamma$ in two points $F$ and $G$. Prove that the line $FG$ is tangent to the circles with diameters $BE$ and $DE$.
1 reply
Stuttgarden
Mar 31, 2025
jrpartty
an hour ago
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Two Functional Inequalities
Mathdreams   2
N an hour ago by kokcio
Source: 2025 Nepal Mock TST Day 2 Problem 2
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x) \le x^3$$and $$f(x + y) \le f(x) + f(y) + 3xy(x + y)$$for any real numbers $x$ and $y$.

(Miroslav Marinov, Bulgaria)
2 replies
Mathdreams
2 hours ago
kokcio
an hour ago
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Tangent to two parabolas
ancamagelqueme   4
N Feb 4, 2023 by ancamagelqueme
Two parabolas have the same tangent at their vertices and are in different half planes with respect to said tangent. Construct the other common tangent.
4 replies
ancamagelqueme
Feb 4, 2023
ancamagelqueme
Feb 4, 2023
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Tangent to two parabolas
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Reveal topic tags
geometry
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parabola
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ancamagelqueme
104 posts
ancamagelqueme
#1 h  Feb 4, 2023, 6:16 PM
Y by
Two parabolas have the same tangent at their vertices and are in different half planes with respect to said tangent. Construct the other common tangent.
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PROF65
2016 posts
PROF65
#2 h  Feb 4, 2023, 7:06 PM
Y by
Since they are tangent to each other they have only one common tangent at finite point ( plus projectively the line at infinity )
best regards
RH HAS
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ancamagelqueme
104 posts
ancamagelqueme
#3 h  Feb 4, 2023, 7:18 PM
Y by
The vertices of the given parabolas do not have to coincide.
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PROF65
2016 posts
PROF65
#4 h  Feb 4, 2023, 8:16 PM • 1 Y
Y by archimedes26
If so then $S$, the point of intersection of $ FF'$ with the common tangent is their center of homothety where $F,F'$ are their foci ; to draw the second common tangent draw the other tangent to one of them through $S$.
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ancamagelqueme
104 posts
ancamagelqueme
#5 h  Feb 4, 2023, 8:29 PM
Y by
You're right. Thank you RH HAS.
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