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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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Hard inequality
JK1603JK   0
11 minutes ago
Source: unknown?
Let $a,b,c>0$ and $a^2+b^2+c^2=2(a+b+c).$ Find the minimum $$P=(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$
0 replies
JK1603JK
11 minutes ago
0 replies
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Dou Fang Geometry in Taiwan TST
Li4   7
N 35 minutes ago by Korean_fish_Kaohsiung
Source: 2025 Taiwan TST Round 3 Mock P2
Let $\omega$ and $\Omega$ be the incircle and circumcircle of the acute triangle $ABC$, respectively. Draw a square $WXYZ$ so that all of its sides are tangent to $\omega$, and $X$, $Y$ are both on $BC$. Extend $AW$ and $AZ$, intersecting $\Omega$ at $P$ and $Q$, respectively. Prove that $PX$ and $QY$ intersects on $\Omega$.

Proposed by kyou46, Li4, Revolilol.
7 replies
+2 w
Li4
Apr 26, 2025
Korean_fish_Kaohsiung
35 minutes ago
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Proving bisection of an angle
Kyj9981   3
N 36 minutes ago by YaoAOPS
Source: A lemma used to prove 11th IGO advanced level P3
Given $\triangle{ABC}$ with Incenter $I$ and A-Excenter $I_{A}$, let $D$ be the foot of perpendicular from $A$ to $BC$. Prove that $BC$ bisects $\angle{IDI_{A}}$
3 replies
Kyj9981
an hour ago
YaoAOPS
36 minutes ago
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2019 Junior Balkan MO, Problem 1
Lukaluce   14
N 37 minutes ago by Adventure1000
Source: 2019 Junior Balkan MO
Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number

$x^p + y^p + z^p - x - y - z$

is a product of exactly three distinct prime numbers.
14 replies
Lukaluce
Jun 22, 2019
Adventure1000
37 minutes ago
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Indonesian Geometry Olympiad
somebodyyouusedtoknow   14
N 2 hours ago by Ihatecombin
Source: Indonesian National Mathematical Olympiad 2024, Problem 3
The triangle $ABC$ has $O$ as its circumcenter, and $H$ as its orthocenter. The line $AH$ and $BH$ intersect the circumcircle of $ABC$ for the second time at points $D$ and $E$, respectively. Let $A'$ and $B'$ be the circumcenters of triangle $AHE$ and $BHD$ respectively. If $A', B', O, H$ are not collinear, prove that $OH$ intersects the midpoint of segment $A'B'$.
14 replies
somebodyyouusedtoknow
Aug 28, 2024
Ihatecombin
2 hours ago
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Parallel lines with incircle
buratinogigle   2
N 2 hours ago by buratinogigle
Source: Own, test for the preliminary team of HSGS 2025
Let $ABC$ be a triangle with incircle $(I)$, which touches sides $CA$ and $AB$ at points $E$ and $F$, respectively. Choose points $M$ and $N$ on the line $EF$ such that $BM = BF$ and $CN = CE$. Let $P$ be the intersection of lines $CM$ and $BN$. Define $Q$ and $R$ as the intersections of $PN$ and $PM$ with lines $IC$ and $IB$, respectively. Assume that $J$ is the intersection of $QR$ and $BC$. Prove that $PJ \parallel MN$.
2 replies
buratinogigle
Sunday at 11:23 AM
buratinogigle
2 hours ago
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Greek gods flood the world
a1267ab   16
N 2 hours ago by awesomeming327.
Source: USA Winter TST for IMO 2020, Problem 3, by Nikolai Beluhov
Let $\alpha \geq 1$ be a real number. Hephaestus and Poseidon play a turn-based game on an infinite grid of unit squares. Before the game starts, Poseidon chooses a finite number of cells to be flooded. Hephaestus is building a levee, which is a subset of unit edges of the grid (called walls) forming a connected, non-self-intersecting path or loop*.

The game then begins with Hephaestus moving first. On each of Hephaestus’s turns, he adds one or more walls to the levee, as long as the total length of the levee is at most $\alpha n$ after his $n$th turn. On each of Poseidon’s turns, every cell which is adjacent to an already flooded cell and with no wall between them becomes flooded as well. Hephaestus wins if the levee forms a closed loop such that all flooded cells are contained in the interior of the loop — hence stopping the flood and saving the world. For which $\alpha$ can Hephaestus guarantee victory in a finite number of turns no matter how Poseidon chooses the initial cells to flood?
-----
*More formally, there must exist lattice points $\mbox{\footnotesize \(A_0, A_1, \dotsc, A_k\)}$, pairwise distinct except possibly $\mbox{\footnotesize \(A_0 = A_k\)}$, such that the set of walls is exactly $\mbox{\footnotesize \(\{A_0A_1, A_1A_2, \dotsc , A_{k-1}A_k\}\)}$. Once a wall is built it cannot be destroyed; in particular, if the levee is a closed loop (i.e. $\mbox{\footnotesize \(A_0 = A_k\)}$) then Hephaestus cannot add more walls. Since each wall has length $\mbox{\footnotesize \(1\)}$, the length of the levee is $\mbox{\footnotesize \(k\)}$.

Nikolai Beluhov
16 replies
a1267ab
Dec 16, 2019
awesomeming327.
2 hours ago
J
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f.e with finite number of f(t)=-t
jjkim0336   2
N 3 hours ago by jasperE3
Source: own
f:R->R
f(xf(y)+y)=yf(x)+f(f(y)) and there are finite number of t such that f(t)= - t
2 replies
jjkim0336
5 hours ago
jasperE3
3 hours ago
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inequalities
Cobedangiu   1
N 3 hours ago by pooh123
Source: UCT
Let $a,b,c>0$ and $a+b+c=3$ Prove that:
$\sum \dfrac{\sqrt{a^2+a+1}}{2a^2+10a+9}\ge \dfrac{\sqrt{3}}{7}$
1 reply
Cobedangiu
Yesterday at 4:14 PM
pooh123
3 hours ago
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Solve $\sin(17x)+\sin(13x)=\sin(7x)$
Speed2001   0
3 hours ago
How to solve the equation:
$$ \sin(17x)+\sin(13x)=\sin(7x),\;0<x<24^{\circ} $$Approach: I'm trying to factor $\sin(18x)$ to get $x=10^{\circ}$.

Any hint would be appreciated.
0 replies
Speed2001
3 hours ago
0 replies
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Symmedian line
April   91
N 4 hours ago by BS2012
Source: All Russian Olympiad - Problem 9.2, 10.2
Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.
91 replies
April
May 10, 2009
BS2012
4 hours ago
J
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Integral-Summation Duality
Mathandski   2
N 4 hours ago by Mathandski
Source: Friend at school gave it to me
Given a continuous function $f$ such that $f(2x) = 3 f(x)$ and $\int_0^1 f(x) \, dx = 1$, evaluate $\int_1^2 f(x) \, dx$.
2 replies
Mathandski
Yesterday at 7:58 PM
Mathandski
4 hours ago
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The old one is gone.
EeEeRUT   10
N 4 hours ago by Pompombojam
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.
10 replies
EeEeRUT
Apr 16, 2025
Pompombojam
4 hours ago
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poly roots equation
NicoN9   2
N 4 hours ago by alextheadventurer
Source: My acquaintance
Let $a, b, c, d$ be positive integers. Let $\alpha$, $\beta$, $\gamma$ be three roots of the polynomial $P(x)=ax^3+bx^2+cx+d$. Given that $\alpha^2+\beta^2+\gamma^2=-1$, $\alpha^3+\beta^3+\gamma^3=-3$, and $2a^2+c^2=27$, find the values of $a$, $b$, $c$, $d$.
2 replies
NicoN9
Sunday at 11:15 AM
alextheadventurer
4 hours ago
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<DEN=<CEM wanted, perpendicular passing through vertices of cyclic ABCD
parmenides51   0
May 15, 2021
Source: VII All-Ukrainian Tournament of Young Mathematicians, Final Battle p3
Let the quadrilateral $ABCD$ be inscribed in a circle. Through points $A$ and $D$ are drawn lines $\ell_A$ and $\ell_D$ perpendicular on the line $AD$ respectively; Through points $B$ and $C$ are drawn lines $\ell_B$ and $\ell_C$ perpendicular on the line $BC$, respectively. Let $\ell_A cap \ell_C=M$, $\ell_B cap \ell_D=N$, $E$ be the point of intersection of the lines AD and BC. Prove that $\angle DEN = \angle CEM$.
0 replies
parmenides51
May 15, 2021
0 replies
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<DEN=<CEM wanted, perpendicular passing through vertices of cyclic ABCD
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Reveal topic tags
geometry
cyclic quadrilateral
equal angles
Ukrainian TYM
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Source: VII All-Ukrainian Tournament of Young Mathematicians, Final Battle p3
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parmenides51
30650 posts
parmenides51
#1 h  May 15, 2021, 12:17 PM
Y by
Let the quadrilateral $ABCD$ be inscribed in a circle. Through points $A$ and $D$ are drawn lines $\ell_A$ and $\ell_D$ perpendicular on the line $AD$ respectively; Through points $B$ and $C$ are drawn lines $\ell_B$ and $\ell_C$ perpendicular on the line $BC$, respectively. Let $\ell_A cap \ell_C=M$, $\ell_B cap \ell_D=N$, $E$ be the point of intersection of the lines AD and BC. Prove that $\angle DEN = \angle CEM$.
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