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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

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0 replies
jwelsh
Jul 1, 2025
0 replies
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
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
Inequality
SunnyEvan   3
N 21 minutes ago by SunnyEvan
Find the smallest positive real number \( k \) such that the following inequality holds:
\[
x^k y^k z^k (x^2 + y^2 + z^2) \leq 3
\]for all positive real numbers \( x, y, z \) satisfying the condition \( x + y + z = 3 \)
Click to reveal hidden text

Find the smallest positive real number \( k \) such that the following inequality holds:
\[
x^k y^k z^k (x^2 + y^2 + z^2) \leq xy+yz+zx
\]for all positive real numbers \( x, y, z \) satisfying the condition \( x + y + z = 3 \)
Click to reveal hidden text
3 replies
SunnyEvan
Yesterday at 11:19 AM
SunnyEvan
21 minutes ago
Perpendicularity in Two Tangent Circles
steven_zhang123   1
N an hour ago by aaravdodhia
Source: 2025 Hope League Test 2 P3
Circle \(O_1\) and circle \(O_2\) are externally tangent at point \(T\). From a point \(X\) on circle \(O_2\), a tangent is drawn intersecting circle \(O_1\) at points \(A\) and \(B\). The line \(XT\) is extended to intersect circle \(O_1\) at point \(S\). A point \(C\) is taken on the arc \(TS\) of circle \(O_1\). The line \(SC\) is extended to intersect the angle bisector of \(\angle BAC\) at point \(I\). The circle passing through points \(A, T, X\) and the circle passing through points \(C, T, I\) intersect at another point \(E\). Prove that \(EO_2 \perp XI\).
Proposed by Luo Haoyu
1 reply
steven_zhang123
Yesterday at 8:24 AM
aaravdodhia
an hour ago
Peru IMO TST 2022
diegoca1   0
an hour ago
Source: Peru IMO TST 2022 D1 P4
Let $\Omega$ be the circumcircle of triangle $ABC$, with $\angle BAC > 90^\circ $ and $ AB > AC $. The tangents to $\Omega$ at points $B$ and $C$ intersect at $D$. The tangent to $\Omega$ at point $A$ intersects line $BC$ at $E$. The line through $D$ parallel to $AE$ intersects line $BC$ at $F$. The circumference $\Gamma$ with diameter $EF$ intersects line $AB$ at points $P$ and $Q$, and line $AC$ at points $X$ and $Y$.
Prove that one of the angles $\angle AEB$, $\angle PEQ$, $\angle XEY$ is equal to the sum of the other two.
0 replies
diegoca1
an hour ago
0 replies
number theory
Hoapham235   5
N an hour ago by Jjesus
Let $x >  y$ be positive integer such that \[ \text{LCM}(x+2, y+2)+\text{LCM}(x, y)=2\text{LCM}(x+1, y+1).\]Prove that $x$ is divisible by $y$.
5 replies
Hoapham235
Wednesday at 4:51 AM
Jjesus
an hour ago
sumcif..
teomihai   1
N Yesterday at 6:08 PM by Royal_mhyasd
Let $a=123456789^{123456789}$ ,$a_{1}=sumcif\{a\}$ ,$a_{2}=sumcifa\{1\}$...
Find number $a_{k}$ with one digit.
1 reply
teomihai
Yesterday at 5:11 PM
Royal_mhyasd
Yesterday at 6:08 PM
Jbmo 2011 Problem 2
Eukleidis   10
N Jul 23, 2025 by LeYohan
Source: Jbmo 2011
Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$
10 replies
Eukleidis
Jun 21, 2011
LeYohan
Jul 23, 2025
Sasha guessing X <= 100 in 7 questions
v_Enhance   8
N Jul 22, 2025 by NicoN9
Source: All-Russian MO 2000
Tanya chose a natural number $X\le100$, and Sasha is trying to guess this number. He can select two natural numbers $M$ and $N$ less than $100$ and ask about $\gcd(X+M,N)$. Show that Sasha can determine Tanya's number with at most seven questions.
8 replies
v_Enhance
Dec 30, 2012
NicoN9
Jul 22, 2025
a^2-3a-19 not divisible by 289
keyree10   19
N Jul 19, 2025 by brainfertiIzer
Source: (Indian) RMO 2009 Problem 2
Show that there is no integer $ a$ such that $ a^2 - 3a - 19$ is divisible by $ 289$.
19 replies
keyree10
Nov 29, 2009
brainfertiIzer
Jul 19, 2025
18th ibmo - argentina 2003/q6
carlosbr   8
N Jul 18, 2025 by OronSH
Source: Spanish Communities
The sequences $(a_n),(b_n)$ are defined by $a_0=1,b_0=4$ and for $n\ge 0$
\[a_{n+1}=a_n^{2001}+b_n,\ \ b_{n+1}=b_n^{2001}+a_n\]
Show that $2003$ is not divisor of any of the terms in these two sequences.
8 replies
carlosbr
Apr 9, 2006
OronSH
Jul 18, 2025
dividing sets into triples with distinct sums
Stear14   1
N Jul 16, 2025 by Stear14

(a) $\ $ Find nine integers $ \ a_1<a_2<...<a_9\ $ such that

$\bullet \  $ no matter how we partition them into 3 triples, the three sums of numbers in each triple will differ from each other.
$\bullet \  $ the difference $\ a_9-a_1\ $ is the smallest possible

(b) $\ $ Same question but allowing equalities: $ \ a_1\le a_2\le ...\le a_9\ $.

(c), (d) $\ $ Same questions for twelve numbers which we partition into four triples.

The problem is proposed by renowned puzzle master Wei-Hwa Huang. Solutions to (a) and (b) are known, whereas (c) and (d) remain open.
1 reply
Stear14
Jul 15, 2025
Stear14
Jul 16, 2025
Ratio of magical to non-magical numbers
Timta27   0
Jul 15, 2025
Source: own
Let us call a natural number $n$ magical if there exists a divisor $k$ of $n$ ($k \neq 1$ and $k \neq n$), such that $k^{k} \equiv k \hspace{0.5em} (mod \hspace{0.5em} n)$.

Is it true that the ratio of magical to non-magical numbers in the interval $[1; t]$ trends to some limit as $t \to \infty$?
0 replies
Timta27
Jul 15, 2025
0 replies
Diophantine Equation
ThAzN1   7
N Jul 13, 2025 by Jupiterballs
Source: USA TST, 2001 #8
Find all pairs of nonnegative integers $(m,n)$ such that \[(m+n-5)^2=9mn.\]
7 replies
ThAzN1
Jan 27, 2005
Jupiterballs
Jul 13, 2025
Median dividing an angle
krithikrokcs   2
N Jul 7, 2025 by krithikrokcs
Source: KNMC Sample Problem P6
Let $\triangle$$ABC$ be a triangle with integer side lengths, and let $M$ be the midpoint of $BC.$ If $k=\frac{\angle BAM}{\angle MAC},$ and $k\in\mathbb Q,$ find, with proof, all possible values of $k.$

this problem is way harder than it looks :(
2 replies
krithikrokcs
Jul 7, 2025
krithikrokcs
Jul 7, 2025
Sum of the numbers is divisible by k
colosimo   3
N Jul 4, 2025 by blug
Source: LXII Polish Olympiad 2011, Problem 1
Find all integers $n\geq 1$ such that there exists a permutation $(a_1,a_2,...,a_n)$ of $(1,2,...,n)$ such that $a_1+a_2+...+a_k$ is divisible by $k$ for $k=1,2,...,n$
3 replies
colosimo
May 24, 2011
blug
Jul 4, 2025
Euler line problem
m4thbl3nd3r   2
N Jun 6, 2025 by m4thbl3nd3r
Let $O,H$ be the circumcenter and orthocenter of triangle $ABC$ and $E,F$ be intersections of $OH$ with $AB,AC$. Let $H',O'$ be orthocenter and circumcenter of triangle $AEF$. Prove that $O'H'\parallel BC.$
2 replies
m4thbl3nd3r
Jun 6, 2025
m4thbl3nd3r
Jun 6, 2025
Euler line problem
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m4thbl3nd3r
310 posts
#1
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Let $O,H$ be the circumcenter and orthocenter of triangle $ABC$ and $E,F$ be intersections of $OH$ with $AB,AC$. Let $H',O'$ be orthocenter and circumcenter of triangle $AEF$. Prove that $O'H'\parallel BC.$
This post has been edited 1 time. Last edited by m4thbl3nd3r, Jun 6, 2025, 9:09 AM
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Ducksohappi
27 posts
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(Zeeman theorem). Suppose that $E \in AB, F \in AC$
Let EF meet BC at S, P be the intersection of (O) and (O'). Then P is miquel point of BCFE.
Which just lead to $\widehat{FSC}=\widehat{FPC}$. Easy to check that $\triangle PO'O\sim \triangle PFC$ so $\widehat{O'PO}=\widehat{FSC}$
Easy to check that $\widehat{O''AO}=\widehat{O''PO}$ so $\widehat{O'AO}=\widehat{FSC}$
Let K be the symmetric point of H' through AH.
Easy to check that $\frac{AH}{AO}=2cos\widehat{BAC}=\frac{AH'}{AO'}$, and $\widehat{H'AH}=\widehat{O'AO}$ so that $\triangle H'AH \sim \triangle O'AO$
which just lead to $\triangle AHK\sim\triangle AOO'\to$ A is the miquel point of HOO'K. Let O'K meet OH at L. Then LOO'A inscribed so $\widehat{O'LO}=\widehat{O'AO} \to \widehat{O'LO}= \widehat{FSC} \to O'K \parallel BC$. But $H'K\parallel BC so H'O' \parallel BC$ QED
P/s. đoạn điểm miquel là dùng vị tự quay nhé
This post has been edited 2 times. Last edited by Ducksohappi, Jun 6, 2025, 9:58 AM
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m4thbl3nd3r
310 posts
#3
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Ducksohappi wrote:
(Zeeman theorem). Suppose that $E \in AB, F \in AC$
Let EF meet BC at S, P be the intersection of (O) and (O'). Then P is miquel point of BCFE.
Which just lead to $\widehat{FSC}=\widehat{FPC}$. Easy to check that $\triangle PO'O\sim \triangle PFC$ so $\widehat{O'PO}=\widehat{FSC}$
Easy to check that $\widehat{O''AO}=\widehat{O''PO}$ so $\widehat{O'AO}=\widehat{FSC}$
Let K be the symmetric point of H' through AH.
Easy to check that $\frac{AH}{AO}=2cos\widehat{BAC}=\frac{AH'}{AO'}$, and $\widehat{H'AH}=\widehat{O'AO}$ so that $\triangle H'AH \sim \triangle O'AO$
which just lead to $\triangle AHK\sim\triangle AOO'\to$ A is the miquel point of HOO'K. Let O'K meet OH at L. Then LOO'A inscribed so $\widehat{O'LO}=\widehat{O'AO} \to \widehat{O'LO}= \widehat{FSC} \to O'K \parallel BC$. But $H'K\parallel BC so H'O' \parallel BC$ QED
P/s:Xin chào hsgser, xin in4 nhé. đoạn điểm miquel là dùng vị tự quay nhé

Thanks for your solution

p/s. Mình đã PM bạn rồi. Mình chưa phải hsgser đâu, không biết có đủ điểm đỗ không nữa.
This post has been edited 1 time. Last edited by m4thbl3nd3r, Jun 6, 2025, 9:57 AM
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