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k a August Highlights and 2025 AoPS Online Class Information
jwelsh   0
Yesterday at 2:14 PM
CONGRATULATIONS to all the competitors at this year’s International Mathematical Olympiad (IMO)! The US Team took second place with 5 gold medals and 1 silver - we are proud to say that each member of the 2025 IMO team has participated in an AoPS WOOT (Worldwide Online Olympiad Training) class!

"As a parent, I'm deeply grateful to AoPS. Tiger has taken very few math courses outside of AoPS, except for a local Math Circle that doesn't focus on Olympiad math. AoPS has been one of the most important resources in his journey. Without AoPS, Tiger wouldn't be where he is today — especially considering he's grown up in a family with no STEM background at all."
— Doreen Dai, parent of IMO US Team Member Tiger Zhang

Interested to learn more about our WOOT programs? Check out the course page here or join a Free Scheduled Info Session. Early bird pricing ends August 19th!:
CodeWOOT Code Jam - Monday, August 11th
ChemWOOT Chemistry Jam - Wednesday, August 13th
PhysicsWOOT Physics Jam - Thursday, August 14th
MathWOOT Math Jam - Friday, August 15th

There is still time to enroll in our last wave of summer camps that start in August at the Virtual Campus, our video-based platform, for math and language arts! From Math Beasts Camp 6 (Prealgebra Prep) to AMC 10/12 Prep, you can find an informative 2-week camp before school starts. Plus, our math camps don’t have homework and cover cool enrichment topics like graph theory. Our language arts courses will build the foundation for next year’s challenges, such as Language Arts Triathlon for levels 5-6 and Academic Essay Writing for high school students.

Lastly, Fall is right around the corner! 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. We’ve opened new Academy locations in San Mateo, CA, Pasadena, CA, Saratoga, CA, Johns Creek, GA, Northbrook, IL, and Upper West Side (NYC), New York.

Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.
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0 replies
jwelsh
Yesterday at 2:14 PM
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
Find the largest value of p
Darealzolt   7
N 9 minutes ago by P0tat0b0y
It is known that
\[
\sqrt{x-3}+\sqrt{6-x} \leq p
\]In which \(x \in \mathbb{R}\), hence find the largest value of \(p\).
7 replies
Darealzolt
Jun 6, 2025
P0tat0b0y
9 minutes ago
Circle geometry proof
littleduckysteve   1
N 11 minutes ago by littleduckysteve
Suppose 3 circles are drawn in the 2-dimensional grid such that no two circles are of the same radius. Now we draw the 2 lines which are both tangent to the smallest circle and the median circle, and call their intersection, A. Now we do the same thing for the biggest circle and the smallest, and finally the biggest and the median circles. Now assume that we call these two points, B and C. Prove that A, B, and C are all colinear regardless of where the circles are.
1 reply
littleduckysteve
Today at 7:29 AM
littleduckysteve
11 minutes ago
15 dropped AIME problems from 1983-88 #1 37 | 37abc, 37bca, 37cab
parmenides51   6
N 36 minutes ago by littleduckysteve
Determine the number of five digit integers $37abc$ (in base $10$), such that $37abc, 37bca, 37cab$ is divisible by $37$.
6 replies
+1 w
parmenides51
Jan 22, 2024
littleduckysteve
36 minutes ago
|x_1-x_2|+|x_2-x_3|+....+|x_1-x_{10}|/x_1+...+x_10
nhathhuyyp5c   1
N 38 minutes ago by littleduckysteve
Given positive real numbers \( x_1, x_2, \dots, x_{10} \) satisfying \( 1 < x_i < 10 \). Find the maximum value of the expression:

\[
P = \frac{|x_1 - x_{2}| + |x_2 - x_3| + |x_3 - x_4| + \dots + |x_9 - x_{10}| + |x_{10} - x_1|}{x_1 + x_2 + \dots + x_{10}}.
\]
1 reply
nhathhuyyp5c
4 hours ago
littleduckysteve
38 minutes ago
About X(252)
LuxusN   2
N Today at 7:47 AM by LuxusN
In triangle $ABC$, let $H$ be the orthocenter and $N$ be the nine-point center (Euler center). Prove that the lines symmetric to $AH$, $BH$, and $CH$ with respect to $AN$, $BN$, and $CN$, respectively, are concurrent.
2 replies
LuxusN
Jul 27, 2025
LuxusN
Today at 7:47 AM
Lemoine Point Reflection
ND_   0
Today at 4:05 AM
Source: Sharygin Final Round 10.4
Let $M, H, L$ be the centroid, the orthocenter, and the Lemoine point respectively of a triangle $ABC$. A point $S$ is such that the circles $SLH, SML$ touch $MH$, and $L'$ is the reflection of $L$ about the circumcircle of the triangle. Prove that $SL' \parallel MH$.
0 replies
ND_
Today at 4:05 AM
0 replies
Cyclic Pentagon
Siddharthmaybe   2
N Yesterday at 11:33 AM by ND_
Source: Sharygin Final 2025 Grade 8 Day 1, problem 1
A cyclic pentagon $ABCDE$ is given. The diagonals $AC$ and $CE$ are equal and meet at $BD$ at points $M$ and $N$ respectively. It is known that $BM = ND, BC \neq CD$. Prove that the reflection of $C$ about the midpoint of $BD$ lies on $AE$.
2 replies
Siddharthmaybe
Jul 31, 2025
ND_
Yesterday at 11:33 AM
Show that (DEN) passes through the midpoint of BC
v_Enhance   26
N Yesterday at 5:26 AM by Ilikeminecraft
Source: Sharygin First Round 2013, Problem 21
Chords $BC$ and $DE$ of circle $\omega$ meet at point $A$. The line through $D$ parallel to $BC$ meets $\omega$ again at $F$, and $FA$ meets $\omega$ again at $T$. Let $M = ET \cap BC$ and let $N$ be the reflection of $A$ over $M$. Show that $(DEN)$ passes through the midpoint of $BC$.
26 replies
v_Enhance
Apr 7, 2013
Ilikeminecraft
Yesterday at 5:26 AM
IMO 2009, Problem 2
orl   151
N Jul 30, 2025 by lksb
Source: IMO 2009, Problem 2
Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP = OQ.$

Proposed by Sergei Berlov, Russia
151 replies
orl
Jul 15, 2009
lksb
Jul 30, 2025
Relate to IMO 2025 P2
hn111009   1
N Jul 30, 2025 by hn111009
Source: Le Viet An
Let $\Omega$ and $\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, so that $C, M, N, D$ lie on $MN$ in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ meets $\Omega$ again at $E\neq A$ and meets $\Gamma$ again at $F\neq A$. Let $t$ be the line tangent to $\Omega$ and $\Gamma$ near $B$ than $A.$
Prove that the reflection of $AP$ through $t$ tangent to $\odot(BEF).$
1 reply
hn111009
Jul 29, 2025
hn111009
Jul 30, 2025
Miquel circles and a beautiful similarity
pohoatza   52
N Jul 30, 2025 by Kempu33334
Source: IMO Shortlist 2006, Geometry 9, AIMO 2007, TST 2, P3
Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.
52 replies
pohoatza
Jun 28, 2007
Kempu33334
Jul 30, 2025
Asian Pacific Mathematical Olympiad 2010 Problem 4
Goutham   70
N Jul 29, 2025 by Kempu33334
Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.
70 replies
Goutham
May 7, 2010
Kempu33334
Jul 29, 2025
Steiner line and isogonal lines
flower417477   2
N Jul 29, 2025 by Melid
$\odot O$ is the circumcircle of $\triangle ABC$,$H$ is the orthocenter of $\triangle ABC$
$D$ is an arbitrary point on $\odot O$
$E$ is the reflection point of $D$ wrt $BC$,$EH$ meet $OD$ at $F$.
$K$ is the reflection point of $A$ wrt $OH$.
$P$ is a point on $\odot O$ such that $PK$ is parallel to $BC$,$Q$ is a point on $OH$ such that $PQ$ is parallel to $EH$.
$N$ is the circumcenter of $\triangle PQK$
Prove that $AF,AN$ is a pair of isogonal lines wrt $\angle BAC$
2 replies
flower417477
Jul 18, 2025
Melid
Jul 29, 2025
IMO Shortlist 2010 - Problem G1
Amir Hossein   144
N Jul 28, 2025 by Yiyj1
Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$

Proposed by Christopher Bradley, United Kingdom
144 replies
Amir Hossein
Jul 17, 2011
Yiyj1
Jul 28, 2025
Original Problem: Geometry and Functions
wonderboy807   2
N May 25, 2025 by LilKirb
For any positive integer n, let F(n) be the number of interior diagonals in a convex polygon with n+3 sides. Find 1/(F(1)) + 1/(F(2)) + ... + 1/F(10))

Answer: Click to reveal hidden text
2 replies
wonderboy807
May 24, 2025
LilKirb
May 25, 2025
Original Problem: Geometry and Functions
G H J
G H BBookmark kLocked kLocked NReply
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wonderboy807
30 posts
#1
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For any positive integer n, let F(n) be the number of interior diagonals in a convex polygon with n+3 sides. Find 1/(F(1)) + 1/(F(2)) + ... + 1/F(10))

Answer: Click to reveal hidden text
Z K Y
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wonderboy807
30 posts
#2
Y by
Solution: Click to reveal hidden text
This post has been edited 1 time. Last edited by wonderboy807, May 25, 2025, 12:08 AM
Reason: I forgot a period.
Z K Y
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LilKirb
67 posts
#3
Y by
Cool problem
Solution
Z K Y
N Quick Reply
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