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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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IMO Shortlist 2013, Combinatorics #3
lyukhson   31
N 4 minutes ago by Maximilian113
Source: IMO Shortlist 2013, Combinatorics #3
A crazy physicist discovered a new kind of particle wich he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time.
(i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it.
(ii) At any moment, he may double the whole family of imons in the lab by creating a copy $I'$ of each imon $I$. During this procedure, the two copies $I'$ and $J'$ become entangled if and only if the original imons $I$ and $J$ are entangled, and each copy $I'$ becomes entangled with its original imon $I$; no other entanglements occur or disappear at this moment.

Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled.
31 replies
lyukhson
Jul 9, 2014
Maximilian113
4 minutes ago
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A wizard kidnaps 101 people
Leicich   14
N 8 minutes ago by MathCosine
Source: Argentina TST 2011, Problem 2
A wizard kidnaps $31$ members from party $A$, $28$ members from party $B$, $23$ members from party $C$, and $19$ members from party $D$, keeping them isolated in individual rooms in his castle, where he forces them to work.
Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with $4$ or more people involved).

a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong.
b) Find all quartets of positive integers that add up to $101$ that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.
14 replies
Leicich
Aug 29, 2014
MathCosine
8 minutes ago
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PAMO Problem 4: Perpendicular lines
DylanN   11
N 27 minutes ago by ATM_
Source: 2019 Pan-African Mathematics Olympiad, Problem 4
The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.
11 replies
DylanN
Apr 9, 2019
ATM_
27 minutes ago
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Number Theory
fasttrust_12-mn   5
N an hour ago by GreekIdiot
Source: Pan African Mathematics Olympiad p6
Find all integers $n$ for which $n^7-41$ is the square of an integer
5 replies
1 viewing
fasttrust_12-mn
Aug 16, 2024
GreekIdiot
an hour ago
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Complex Numbers Question
franklin2013   2
N Today at 1:59 PM by Xx_BABAI_xX
Hello everyone! This is one of my favorite complex numbers questions. Have fun!

$f(z)=z^{720}-z^{120}$. How many complex numbers $z$ are there such that $|z|=1$ and $f(z)$ is an integer.

Hint
2 replies
franklin2013
Apr 20, 2025
Xx_BABAI_xX
Today at 1:59 PM
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Inequalities
sqing   17
N Today at 1:26 PM by sqing
Let $ a,b,c> 0 $ and $ ab+bc+ca\leq 3abc . $ Prove that
$$ a+ b^2+c\leq a^2+ b^3+c^2 $$$$ a+ b^{11}+c\leq a^2+ b^{12}+c^2 $$
17 replies
sqing
Yesterday at 1:54 PM
sqing
Today at 1:26 PM
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Geometric inequality
ReticulatedPython   1
N Today at 12:43 PM by vanstraelen
Let $A$ and $B$ be points on a plane such that $AB=n$, where $n$ is a positive integer. Let $S$ be the set of all points $P$ such that $\frac{AP^2+BP^2}{(AP)(BP)}=c$, where $c$ is a real number. The path that $S$ traces is continuous, and the value of $c$ is minimized. Prove that $c$ is rational for all positive integers $n.$
1 reply
ReticulatedPython
Yesterday at 5:12 PM
vanstraelen
Today at 12:43 PM
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Binomial Sum
P162008   0
Today at 12:34 PM
Compute $\sum_{r=0}^{n} \sum_{k=0}^{r} (-1)^k (k + 1)(k + 2) \binom {n + 5}{r - k}$
0 replies
P162008
Today at 12:34 PM
0 replies
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Triple Sum
P162008   0
Today at 12:24 PM
Find the value of

$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^m}{k.2^n + 2m + 1}$
0 replies
P162008
Today at 12:24 PM
0 replies
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Binomial Sum
P162008   0
Today at 12:03 PM
The numbers $p$ and $q$ are defined in the following manner:

$p = 99^{98} - \frac{99}{1} 98^{98} + \frac{99.98}{1.2} 97^{98} - \frac{99.98.97}{1.2.3} 96^{98} + .... + 99$

$q = 99^{100} - \frac{99}{1} 98^{100} + \frac{99.98}{1.2} 97^{100} - \frac{99.98.97}{1.2.3} 96^{100} + .... + 99$

If $p + q = k(99!)$ then find the value of $\frac{k}{10}.$
0 replies
P162008
Today at 12:03 PM
0 replies
J
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Polynomial Limit
P162008   0
Today at 11:55 AM
If $P_{n}(x) = \prod_{k=0}^{n} \left(x + \frac{1}{2^k}\right) = \sum_{k=0}^{n} a_{k} x^k$ then find the value of $\lim_{n \to \infty} \frac{a_{n - 2}}{a_{n - 4}}.$
0 replies
P162008
Today at 11:55 AM
0 replies
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Telescopic Sum
P162008   0
Today at 11:40 AM
Compute the value of $\Omega = \sum_{r=1}^{\infty} \frac{14 - 9r - 90r^2 - 36r^3}{7^r r(r + 1)(r + 2)(4r^2 - 1)}$
0 replies
P162008
Today at 11:40 AM
0 replies
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CHINA TST 2017 P6 DAY1
lingaguliguli   0
Today at 9:03 AM
When i search the china TST 2017 problem 6 day I i crossed out this lemme, but don't know to prove it, anyone have suggestion? tks
Given a fixed number n, and a prime p. Let f(x)=(x+a_1)(x+a_2)...(x+a_n) in which a_1,a_2,...a_n are positive intergers. Show that there exist an interger M so that 0<v_p((f(M))< n + v_p(n!)
0 replies
lingaguliguli
Today at 9:03 AM
0 replies
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Combinatoric
spiderman0   1
N Today at 6:44 AM by MathBot101101
Let $ S = \{1, 2, 3, \ldots, 2024\}.$ Find the maximum positive integer $n \geq 2$ such that for every subset $T \subset S$ with n elements, there always exist two elements a, b in T such that:

$|\sqrt{a} - \sqrt{b}| < \frac{1}{2} \sqrt{a - b}$
1 reply
spiderman0
Yesterday at 7:46 AM
MathBot101101
Today at 6:44 AM
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Regarding Maaths olympiad prepration
omega2007   13
N Apr 5, 2025 by omega2007
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compiled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your perspective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
13 replies
omega2007
Apr 4, 2025
omega2007
Apr 5, 2025
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Regarding Maaths olympiad prepration
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Reveal topic tags
L
G H BBookmark kLocked kLocked NReply
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omega2007
10 posts
omega2007
#1 h  Apr 4, 2025, 3:13 PM
Y by
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compiled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your perspective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
This post has been edited 1 time. Last edited by omega2007, Apr 5, 2025, 2:48 AM
Reason: Spelling error
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GreekIdiot
175 posts
GreekIdiot
#2 h  Apr 4, 2025, 3:46 PM • 1 Y
Y by omega2007
Sure... Try Chen's EGMO and Khurmi's MONT. Great books and very thorough, albeit extremely challenging at some points. Keep in mind to study in an organized manner, not learning random interesting theorems. Im at your age too and this is something I have personally struggled with, thats why I am mentioning it.
You can find them online easily in pdf format, but I suggest printing the books or buying them. Helps me study a lot more.
This post has been edited 1 time. Last edited by GreekIdiot, Apr 4, 2025, 3:47 PM
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omega2007
10 posts
omega2007
#3 h  Apr 5, 2025, 2:40 AM
Y by
Thanks for your support.
EGMO AND MONT I HAVE SOLVE ALREADY
In Aops things are not arranged //
So I need this things
But do you have compiled resources

Like >>>> List of books
List of handouts
Pdf
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MathleteMystic
7 posts
MathleteMystic
#4 h  Apr 5, 2025, 5:44 AM
Y by
omega2007 wrote:
Thanks for your support.
EGMO AND MONT I HAVE SOLVE ALREADY
In Aops things are not arranged //
So I need this things
But do you have compiled resources

Like >>>> List of books
List of handouts
Pdf

Didn't u say u r starting ur preparation from scratch..??..I don't think it can be called starting from scratch if u hv already solved those 2 books :|
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omega2007
10 posts
omega2007
#5 h  Apr 5, 2025, 5:53 AM
Y by
Yeah

Don't you have compiled resources
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Captainscrubz
57 posts
Captainscrubz
#6 h  Apr 5, 2025, 8:16 AM • 1 Y
Y by MrdiuryPeter
Visit evan chens website theres all material and resources you need
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mikkymini2
13 posts
mikkymini2
#7 h  Apr 5, 2025, 8:38 AM
Y by
Hi, this is a separate doubt.. But how did you guys start with EGMO? Like do I just pick up the book and start solving if my Geo basics are good or do I cover other supplement materials before I start with this? And lastly how do I proceed if I am stuck with a question (and preferably could you explain how you guys overcame that)...To my knowledge some hints and some solutions are provided, they often require so much effort to try on your own :wacko:
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omega2007
10 posts
omega2007
#8 h  Apr 5, 2025, 8:43 AM
Y by
It totally depends on your understanding
Like in case of geometry
Are your basic clear for example are you familiar
and can you proof theorems of class 8,9,10,11,12 maths ( geometry section) >>> if yes >> than you can proceed with EGMO> GRASP THEORY and solve problems.

Here are some resources for geometry
Geometry Resources
Euclidean Geometry in Mathematical Olympiads - Evan Chen - Both book - good book. By far the greatest geometry book to prepare for olympiads. if you had to choose one book, its definitely this one
Muricaa- It is book/handout discussing popular configs in geo mostly pointed out to be well-known by aops-ers. It has 100 problems at the end with roughly the difficulty increasing as you move forward. It can be found here
Handouts on Projective Geometry and Moving Points A really nice handout written by Rohan Goyal, starting from basics and going upto nukes like DIT and DDIT can be found here. Another nice handout on learning the method of moving points and it's application in problems can be found here
AOPS Geo Mocks: https://artofproblemsolving.com/community/c1668102_aops_geo_mocks
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omega2007
10 posts
omega2007
#9 h  Apr 5, 2025, 8:45 AM
Y by
mikkymini2 wrote:
Hi, this is a separate doubt.. But how did you guys start with EGMO? Like do I just pick up the book and start solving if my Geo basics are good or do I cover other supplement materials before I start with this? And lastly how do I proceed if I am stuck with a question (and preferably could you explain how you guys overcame that)...To my knowledge some hints and some solutions are provided, they often require so much effort to try on your own :wacko:

For questions solving I guess you can use AI like chat gpt and my favourite deepseek
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mikkymini2
13 posts
mikkymini2
#10 h  Apr 5, 2025, 8:56 AM • 1 Y
Y by omega2007
@omega, thx for the help, I will look into those books. Will try using AI in the future, I did not try it before thinking they would have been inaccurate for challenging math problems...
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omega2007
10 posts
omega2007
#11 h  Apr 5, 2025, 9:00 AM
Y by
Do you all compiled resources which are shared on Aops for maths olympiad
Like handouts , books like that
Because there are many .....
I want in one place compiled so that I can use

I hope someone will give me
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kjhgyuio
50 posts
kjhgyuio
#12 h  Apr 5, 2025, 9:09 AM
Y by
omega2007 wrote:
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compiled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your perspective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.

i also want to ask the same but for 8th grade
This post has been edited 1 time. Last edited by kjhgyuio, Apr 5, 2025, 9:11 AM
Reason: nil
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mikkymini2
13 posts
mikkymini2
#13 h  Apr 5, 2025, 9:37 AM
Y by
I think you could try this...
https://artofproblemsolving.com/community/c13_contest_collections
It has complied questions for various Olympiads PYQs
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omega2007
10 posts
omega2007
#14 h  Apr 5, 2025, 9:48 AM
Y by
I don't need pyq
I need olympiad articles
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