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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
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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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Alice in Wonderland
ilovemath04   26
N 2 minutes ago by atdaotlohbh
Source: ISL 2019 C8
Alice has a map of Wonderland, a country consisting of $n \geq 2$ towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns.

Alice wants to know whether there is at least one town in Wonderland with at most one outgoing road. Prove that she can always find out by asking at most $4n$ questions.
26 replies
ilovemath04
Sep 22, 2020
atdaotlohbh
2 minutes ago
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Geometry
Captainscrubz   0
19 minutes ago
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
0 replies
+1 w
Captainscrubz
19 minutes ago
0 replies
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Pebble Game
oVlad   6
N an hour ago by The5within
Source: KöMaL A. 790
Andrew and Barry play the following game: there are two heaps with $a$ and $b$ pebbles, respectively. In the first round Barry chooses a positive integer $k,$ and Andrew takes away $k$ pebbles from one of the two heaps (if $k$ is bigger than the number of pebbles in the heap, he takes away the complete heap). In the second round, the roles are reversed: Andrew chooses a positive integer and Barry takes away the pebbles from one of the two heaps. This goes on, in each round the two players are reversing the roles. The player that takes the last pebble loses the game.

Which player has a winning strategy?

Submitted by András Imolay, Budapest
6 replies
oVlad
Mar 24, 2022
The5within
an hour ago
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Equality with Fermat Point
nsato   13
N an hour ago by Nari_Tom
Source: 2012 Baltic Way, Problem 11
Let $ABC$ be a triangle with $\angle A = 60^\circ$. The point $T$ lies inside the triangle in such a way that $\angle ATB = \angle BTC = \angle CTA = 120^\circ$. Let $M$ be the midpoint of $BC$. Prove that $TA + TB + TC = 2AM$.
13 replies
nsato
Nov 22, 2012
Nari_Tom
an hour ago
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China 2017 TSTST1 Day 2 Geometry Problem
HuangZhen   46
N an hour ago by ihategeo_1969
Source: China 2017 TSTST1 Day 2 Problem 5
In the non-isosceles triangle $ABC$,$D$ is the midpoint of side $BC$,$E$ is the midpoint of side $CA$,$F$ is the midpoint of side $AB$.The line(different from line $BC$) that is tangent to the inscribed circle of triangle $ABC$ and passing through point $D$ intersect line $EF$ at $X$.Define $Y,Z$ similarly.Prove that $X,Y,Z$ are collinear.
46 replies
HuangZhen
Mar 7, 2017
ihategeo_1969
an hour ago
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Cool combinatorial problem (grid)
Anto0110   1
N 2 hours ago by Anto0110
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).
1 reply
Anto0110
Yesterday at 1:57 PM
Anto0110
2 hours ago
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one nice!
teomihai   2
N 2 hours ago by teomihai
3 girls and 4 boys must be seated at a round table. In how many distinct ways can they be seated so that the 3 girls do not sit next to each other and there can be a maximum of 2 girls next to each other. (The table is round so the seats are not numbered.)
2 replies
teomihai
Yesterday at 7:32 PM
teomihai
2 hours ago
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Find the constant
JK1603JK   0
2 hours ago
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 $a,b,c\ge 0.$
0 replies
1 viewing
JK1603JK
2 hours ago
0 replies
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IMO ShortList 1999, number theory problem 1
orl   61
N 2 hours ago by cursed_tangent1434
Source: IMO ShortList 1999, number theory problem 1
Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.
61 replies
orl
Nov 13, 2004
cursed_tangent1434
2 hours ago
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(2^n + 1)/n^2 is an integer (IMO 1990 Problem 3)
orl   105
N 3 hours ago by cursed_tangent1434
Source: IMO 1990, Day 1, Problem 3, IMO ShortList 1990, Problem 23 (ROM 5)
Determine all integers $ n > 1$ such that
\[ \frac {2^n + 1}{n^2} \]is an integer.
105 replies
orl
Nov 11, 2005
cursed_tangent1434
3 hours ago
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Maximum angle ratio
miiirz30   2
N 3 hours ago by zuzuzu222
Source: 2025 Euler Olympiad, Round 1
Given any arc $AB$ on a circle and points $C$ and $D$ on segment $AB$, such that $$CD = DB = 2AC.$$Find the ratio $\frac{CM}{MD}$, where $M$ is a point on arc $AB$, such that $\angle CMD$ is maximized.

IMAGE

Proposed by Andria Gvaramia, Georgia
2 replies
miiirz30
Mar 31, 2025
zuzuzu222
3 hours ago
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S must be the incentre of triangle ABC
WakeUp   3
N 3 hours ago by Nari_Tom
Source: Baltic Way 2007
In triangle $ABC$ let $AD,BE$ and $CF$ be the altitudes. Let the points $P,Q,R$ and $S$ fulfil the following requirements:
i) $P$ is the circumcentre of triangle $ABC$.
ii) All the segments $PQ,QR$ and $RS$ are equal to the circumradius of triangle $ABC$.
iii) The oriented segment $PQ$ has the same direction as the oriented segment $AD$. Similarly, $QR$ has the same direction as $BE$, and $Rs$ has the same direction as $CF$.
Prove that $S$ is the incentre of triangle $ABC$.
3 replies
WakeUp
Nov 30, 2010
Nari_Tom
3 hours ago
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Two midpoints and the circumcenter are collinear.
ricarlos   1
N 4 hours ago by Luis González
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point on the perpendicular bisector of $AB$ (see figure) and $Q$, $R$ be the intersections of the perpendicular bisectors of $AC$ and $BC$, respectively, with $PA$ and $PB$. Prove that the midpoints of $PC$ and $QR$ and the point $O$ are collinear.

1 reply
ricarlos
Yesterday at 5:52 PM
Luis González
4 hours ago
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¿10^n-1 is a divisor of 11^n-1?
EmersonSoriano   2
N 4 hours ago by Giant_PT
Source: 2017 Peru Southern Cone TST P2
Determine if there exists a positive integer $n$ such that $10^n - 1$ is a divisor of $11^n - 1$.
2 replies
EmersonSoriano
Yesterday at 6:32 PM
Giant_PT
4 hours ago
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computational in space, rectangle related
parmenides51   3
N Sep 22, 2018 by NikoIsLife
Source: Vyacheslav Yasinsky Geometry Olympiad 2017 X-XI p5 [Ukranie]
$ABCD$ is a rectangle. The segment $MA$ is perpendicular to plane $ABC$ . $MB= 15$ , $MC=24$ , $MD=20$. Find the length of $MA$ .
3 replies
parmenides51
Sep 22, 2018
NikoIsLife
Sep 22, 2018
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computational in space, rectangle related
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Reveal topic tags
geometry
rectangle
3-Dimensional Geometry
3D geometry
solid geometry
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Source: Vyacheslav Yasinsky Geometry Olympiad 2017 X-XI p5 [Ukranie]
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parmenides51
30629 posts
parmenides51
#1 h  Sep 22, 2018, 4:41 AM • 2 Y
Y by Adventure10, Mango247
$ABCD$ is a rectangle. The segment $MA$ is perpendicular to plane $ABC$ . $MB= 15$ , $MC=24$ , $MD=20$. Find the length of $MA$ .
This post has been edited 2 times. Last edited by parmenides51, Sep 22, 2018, 9:28 AM
Reason: latex
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sunken rock
4379 posts
sunken rock
#2 h  Sep 22, 2018, 8:37 AM • 2 Y
Y by Adventure10, Mango247
$M$ may be any point in space!

Best regards,
sunken rock
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little-fermat
147 posts
little-fermat
#3 h  Sep 22, 2018, 9:08 AM • 2 Y
Y by Adventure10, Mango247
let $AB=CD=a,AD=CB=b,AM=x$ then by pythagorian theorem we have :
$x^2+a^2=(15)^2,x^2+b^2=(20)^2,x^2+a^2+b^2=(24)^2\implies x^2=(15)^2+(20)^2-(24)^2=49\implies AM=7$
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NikoIsLife
9657 posts
NikoIsLife
#4 h  Sep 22, 2018, 9:13 AM • 2 Y
Y by Adventure10, Mango247
Use the British Flag Theorem.

What sunken rock said was true, $M$ may be any point in space. $AM$ does not have to be perpendicular to plane $ABC$.
$$MA^2+MC^2=MB^2+MD^2\implies MA=\boxed7$$
This post has been edited 1 time. Last edited by NikoIsLife, Sep 22, 2018, 9:43 AM
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