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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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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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Hard functional equation
Hopeooooo   33
N 43 minutes ago by jasperE3
Source: IMO shortlist A8 2020
Let $R^+$ be the set of positive real numbers. Determine all functions $f:R^+$ $\rightarrow$ $R^+$ such that for all positive real numbers $x$ and $y:$
\[f(x+f(xy))+y=f(x)f(y)+1\]
Ukraine
33 replies
Hopeooooo
Jul 20, 2021
jasperE3
43 minutes ago
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series and factorials?
jenishmalla   9
N an hour ago by mpcnotnpc
Source: 2025 Nepal ptst p4 of 4
Find all pairs of positive integers \( n \) and \( x \) such that
\[ 1^n + 2^n + 3^n + \cdots + n^n = x! \]
(Petko Lazarov, Bulgaria)
9 replies
jenishmalla
Mar 15, 2025
mpcnotnpc
an hour ago
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Geo with unnecessary condition
egxa   8
N 2 hours ago by ErTeeEs06
Source: Turkey Olympic Revenge 2024 P4
Let the circumcircle of a triangle $ABC$ be $\Gamma$. The tangents to $\Gamma$ at $B,C$ meet at point $E$. For a point $F$ on line $BC$ which is not on the segment $BC$, let the midpoint of $EF$ be $G$. Lines $GB,GC$ meet $\Gamma$ again at points $I,H$ respectively. Let $M$ be the midpoint of $BC$. Prove that the points $F,I,H,M$ lie on a circle.

Proposed by Mehmet Can Baştemir
8 replies
egxa
Aug 6, 2024
ErTeeEs06
2 hours ago
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USAMO 2000 Problem 3
MithsApprentice   9
N 3 hours ago by Anto0110
A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
9 replies
MithsApprentice
Oct 1, 2005
Anto0110
3 hours ago
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Problem 4
blug   1
N 3 hours ago by Filipjack
Source: Polish Math Olympiad 2025 Finals P4
A positive integer $n\geq 2$ and a set $S$ consisting of $2n$ disting positive integers smaller than $n^2$ are given. Prove that there exists a positive integer $r\in \{1, 2, ..., n\}$ that can be written in the form $r=a-b$, for $a, b\in \mathbb{S}$ in at least $3$ different ways.
1 reply
blug
Yesterday at 11:59 AM
Filipjack
3 hours ago
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Strange angle condition and concyclic points
lminsl   126
N 4 hours ago by cj13609517288
Source: IMO 2019 Problem 2
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.

Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.

Proposed by Anton Trygub, Ukraine
126 replies
lminsl
Jul 16, 2019
cj13609517288
4 hours ago
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Functional equations
hanzo.ei   19
N 4 hours ago by GreekIdiot
Source: Greekldiot
Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$
19 replies
hanzo.ei
Mar 29, 2025
GreekIdiot
4 hours ago
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Functional Equation
AnhQuang_67   3
N 4 hours ago by GreekIdiot
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+y(x), \forall x, y \in \mathbb{R} $$











3 replies
AnhQuang_67
Yesterday at 4:50 PM
GreekIdiot
4 hours ago
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Geometry :3c
popop614   4
N 4 hours ago by goaoat
Source: MINE :<
Quadrilateral $ABCD$ has an incenter $I$ Suppose $AB > BC$. Let $M$ be the midpoint of $AC$. Suppose that $MI \perp BI$. $DI$ meets $(BDM)$ again at point $T$. Let points $P$ and $Q$ be such that $T$ is the midpoint of $MP$ and $I$ is the midpoint of $MQ$. Point $S$ lies on the plane such that $AMSQ$ is a parallelogram, and suppose the angle bisectors of $MCQ$ and $MSQ$ concur on $IM$.

The angle bisectors of $\angle PAQ$ and $\angle PCQ$ meet $PQ$ at $X$ and $Y$. Prove that $PX = QY$.
4 replies
1 viewing
popop614
Thursday at 12:19 AM
goaoat
4 hours ago
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$f(xy)=xf(y)+yf(x)$
yumeidesu   2
N 5 hours ago by jasperE3
Find $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y)=f(x)+f(y), \forall x, y \in \mathbb{R}$ and $f(xy)=xf(y)+yf(x), \forall x, y \in \mathbb{R}.$
2 replies
yumeidesu
Apr 14, 2020
jasperE3
5 hours ago
J
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Pythagorean journey on the blackboard
sarjinius   1
N 6 hours ago by alfonsoramires
Source: Philippine Mathematical Olympiad 2025 P2
A positive integer is written on a blackboard. Carmela can perform the following operation as many times as she wants: replace the current integer $x$ with another positive integer $y$, as long as $|x^2 - y^2|$ is a perfect square. For example, if the number on the blackboard is $17$, Carmela can replace it with $15$, because $|17^2 - 15^2| = 8^2$, then replace it with $9$, because $|15^2 - 9^2| = 12^2$. If the number on the blackboard is initially $3$, determine all integers that Carmela can write on the blackboard after finitely many operations.
1 reply
sarjinius
Mar 9, 2025
alfonsoramires
6 hours ago
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Assisted perpendicular chasing
sarjinius   4
N Yesterday at 6:01 PM by X.Allaberdiyev
Source: Philippine Mathematical Olympiad 2025 P7
In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$.
(a) Show that $AP$ and $BR$ are perpendicular.
(b) Show that $FM$ and $BM$ are perpendicular.
4 replies
sarjinius
Mar 9, 2025
X.Allaberdiyev
Yesterday at 6:01 PM
J
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Problem 2
SlovEcience   1
N Yesterday at 5:42 PM by Primeniyazidayi
Let \( a, n \) be positive integers and \( p \) be an odd prime such that:
\[ a^p \equiv 1 \pmod{p^n}. \]Prove that:
\[ a \equiv 1 \pmod{p^{n-1}}. \]
1 reply
SlovEcience
Yesterday at 3:52 PM
Primeniyazidayi
Yesterday at 5:42 PM
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H not needed
dchenmathcounts   45
N Yesterday at 5:21 PM by EpicBird08
Source: USEMO 2019/1
Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar.

Robin Son
45 replies
dchenmathcounts
May 23, 2020
EpicBird08
Yesterday at 5:21 PM
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regular diagonal
myceliumful   1
N Aug 17, 2011 by KingSmasher3
In the convex pentagon ABCDE, the sides BC, CD and DE are equal and each diagonal is parallel to a side.
Prove that ABCDE is a regular pentagon.
1 reply
myceliumful
Aug 17, 2011
KingSmasher3
Aug 17, 2011
J
regular diagonal
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Reveal topic tags
geometry
trapezoid
trigonometry
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myceliumful
264 posts
myceliumful
#1 h  Aug 17, 2011, 1:22 AM • 2 Y
Y by Adventure10, Mango247
In the convex pentagon ABCDE, the sides BC, CD and DE are equal and each diagonal is parallel to a side.
Prove that ABCDE is a regular pentagon.
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KingSmasher3
1399 posts
KingSmasher3
#2 h  Aug 17, 2011, 4:06 AM • 3 Y
Y by math07, Adventure10, Mango247
Consider side $AB$ of pentagon $ABCDE.$ Diagonal $CE$ is the only diagonal that does not contain either $A$ or $B.$ Thus, for any side, only one diagonal could possibly be parallel to it.

Now, consider the three given sides, $BC=CD=DE.$ Without loss of generality, let them all be $1.$ Since $BE \| CD,$ let $\angle BCD = \angle EDC = \alpha.$ Let $k$ be the line through $D$ such that $k \| BC.$ Let $\ell$ be the line through $C$ such that $\ell \| DE.$ Let $m$ be the line through $E$ such that $m \| BD.$ Let $n$ be the line through $B$ such that $n \| CE.$ We must prove that $k \cap \ell = m \cap n.$

Since $BCDE$ is an isosceles trapezoid, we have that $BE= 2\sin \left(\alpha-\frac{\pi}{2}\right) + 1$ and that the distance between $BE$ and $CD$ (let this value be $x$) is $\cos \left(\alpha - \frac{\pi}{2} \right).$ Look at the altitude from $A$ to $CD,$ we have the equation

\[\frac{1}{2}BE \cdot \tan \left( \frac{\pi - \alpha}{2} \right) + x = \frac{1}{2} \cdot \tan (\pi - \alpha)\]

Plugging in the values of $BE$ and $x,$ we find that $x=108^\circ ,$ which implies that the given pentagon is regular.
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