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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
and train with the best! Please note that early bird pricing ends August 19th!
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.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

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0 replies
jwelsh
Jul 1, 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
J
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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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IMO ShortList 2001, geometry problem 1
orl   68
N 6 minutes ago by Kempu33334
Source: IMO ShortList 2001, geometry problem 1
Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.
68 replies
orl
Sep 30, 2004
Kempu33334
6 minutes ago
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IMO 2025 P2
sarjinius   46
N 7 minutes ago by shanelin-sigma
Source: 2025 IMO P2
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 $H$ be the orthocentre of triangle $PMN$.

Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.
46 replies
+7 w
sarjinius
Today at 3:38 AM
shanelin-sigma
7 minutes ago
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Sunny lines
sarjinius   33
N 10 minutes ago by juckter
Source: 2025 IMO P1
A line in the plane is called $sunny$ if it is not parallel to any of the $x$–axis, the $y$–axis, or the line $x+y=0$.

Let $n\ge3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following:
[list]
[*] for all positive integers $a$ and $b$ with $a+b\le n+1$, the point $(a,b)$ lies on at least one of the lines; and
[*] exactly $k$ of the $n$ lines are sunny.
[/list]
33 replies
+2 w
sarjinius
Today at 3:35 AM
juckter
10 minutes ago
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Bonza functions
KevinYang2.71   31
N 14 minutes ago by Adywastaken
Source: 2025 IMO P3
Let $\mathbb{N}$ denote the set of positive integers. A function $f\colon\mathbb{N}\to\mathbb{N}$ is said to be bonza if
\[ f(a)~~\text{divides}~~b^a-f(b)^{f(a)} \]for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\leqslant cn$ for all bonza functions $f$ and all positive integers $n$.
31 replies
+3 w
KevinYang2.71
Today at 3:38 AM
Adywastaken
14 minutes ago
J
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Partition set with equal sum and differnt cardinality
psi241   74
N 39 minutes ago by Namura
Source: IMO Shortlist 2018 C1
Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.
74 replies
psi241
Jul 17, 2019
Namura
39 minutes ago
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Areas of triangles AOH, BOH, COH
Arne   74
N an hour ago by Kempu33334
Source: APMO 2004, Problem 2
Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Prove that the area of one of the triangles $AOH$, $BOH$ and $COH$ is equal to the sum of the areas of the other two.
74 replies
Arne
Mar 23, 2004
Kempu33334
an hour ago
J
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Incenter and midpoint geom
sarjinius   96
N an hour ago by Berta_Arnlas
Source: 2024 IMO Problem 4
Let $ABC$ be a triangle with $AB < AC < BC$. Let the incenter and incircle of triangle $ABC$ be $I$ and $\omega$, respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$. Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$. Let $AI$ intersect the circumcircle of triangle $ABC$ at $P \ne A$. Let $K$ and $L$ be the midpoints of $AC$ and $AB$, respectively.
Prove that $\angle KIL + \angle YPX = 180^{\circ}$.

Proposed by Dominik Burek, Poland
96 replies
sarjinius
Jul 17, 2024
Berta_Arnlas
an hour ago
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Perpendicular line
Hung1092008   0
an hour ago
Let $ABCD$ be a quadrilateral with $AB=BC=AD$, $E=BC\cap AD$, $F=AB\cap CD$, $H=AC\cap BD$, $G$ the second intersection of $(EAB)$ and $(BCF)$, and $I$ the second intersection of $(HAD)$ and $(HBC)$; prove that $GH\perp IF$.
0 replies
Hung1092008
an hour ago
0 replies
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NICE Planimetry 3
prof.   0
5 hours ago
A circle is inscribed in triangle $ABC$, and tangents are drawn to this circle parallel to the sides of the triangle. In this way, three smaller triangles are cut off from triangle $ABC$, each with an incircle of radius $r_1,r_2,r_3$, respectively. If $r$ the radius of the incircle of triangle $ABC$, prove that $$r=r_1+r_2+r_3.$$
0 replies
prof.
5 hours ago
0 replies
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Prove two circles intersect on line BC
62861   78
N Today at 8:13 AM by ezpotd
Source: USA Winter TST for IMO 2019, Problem 1 and TST for EGMO 2019, Problem 2, by Merlijn Staps
Let $ABC$ be a triangle and let $M$ and $N$ denote the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Let $X$ be a point such that $\overline{AX}$ is tangent to the circumcircle of triangle $ABC$. Denote by $\omega_B$ the circle through $M$ and $B$ tangent to $\overline{MX}$, and by $\omega_C$ the circle through $N$ and $C$ tangent to $\overline{NX}$. Show that $\omega_B$ and $\omega_C$ intersect on line $BC$.

Merlijn Staps
78 replies
62861
Dec 10, 2018
ezpotd
Today at 8:13 AM
J
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Geometry finale: radical axis bisects D-altitude
v_Enhance   53
N Today at 6:57 AM by mira74
Source: USA TSTST 2016 Problem 6, by Danielle Wang
Let $ABC$ be a triangle with incenter $I$, and whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$, respectively. Let $K$ be the foot of the altitude from $D$ to $\overline{EF}$. Suppose that the circumcircle of $\triangle AIB$ meets the incircle at two distinct points $C_1$ and $C_2$, while the circumcircle of $\triangle AIC$ meets the incircle at two distinct points $B_1$ and $B_2$. Prove that the radical axis of the circumcircles of $\triangle BB_1B_2$ and $\triangle CC_1C_2$ passes through the midpoint $M$ of $\overline{DK}$.

Proposed by Danielle Wang
53 replies
v_Enhance
Jun 29, 2016
mira74
Today at 6:57 AM
J
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A geometry problem
MathMaxGreat   7
N Today at 1:56 AM by Joalro178
Source: 2025 Summer NSMO
In $\triangle ABC$, $D$ is on the angle bisector of $\angle BAC$, $ED\perp BC$ and $E$ is on $BC$.
Prove: the radius of the circumcircles of the pedal triangles from $E$ to $\triangle ABD$ and $\triangle ACD$ is equal.
7 replies
MathMaxGreat
Jul 9, 2025
Joalro178
Today at 1:56 AM
J
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Guessing Point is Hard
MarkBcc168   34
N Today at 12:26 AM by vincentwant
Source: IMO Shortlist 2023 G5
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$.

Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$.

Ivan Chan Kai Chin, Malaysia
34 replies
MarkBcc168
Jul 17, 2024
vincentwant
Today at 12:26 AM
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Circles cut from cardboard
BigSams   10
N Yesterday at 11:55 PM by mudkip42
Source: Canadian Mathematical Olympiad - 2009 - Problem 2.
Two circles of different radii are cut out of cardboard. Each circle is subdivided into $200$ equal sectors. On each circle $100$ sectors are painted white and the other $100$ are painted black. The smaller circle is then placed on top of the larger circle, so that their centers coincide. Show that one can rotate the small circle so that the sectors on the two circles line up and at least $100$ sectors on the small circle lie over sectors of the same color on the big circle.
10 replies
BigSams
May 4, 2011
mudkip42
Yesterday at 11:55 PM
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An easy combinatorics
fananhminh   2
N May 14, 2025 by fananhminh
Source: Vietnam
Let S={1, 2, 3,..., 65}. A is a subset of S such that the sum of A's elements is not divided by any element in A. Find the maximum of |A|.
2 replies
fananhminh
May 14, 2025
fananhminh
May 14, 2025
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An easy combinatorics
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Reveal topic tags
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Source: Vietnam
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fananhminh
2 posts
fananhminh
#1 h  May 14, 2025, 4:01 PM
Y by
Let S={1, 2, 3,..., 65}. A is a subset of S such that the sum of A's elements is not divided by any element in A. Find the maximum of |A|.
Z K Y
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Soupboy0
621 posts
Soupboy0
#2 h  May 14, 2025, 4:10 PM • 1 Y
Y by fananhminh
sol
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fananhminh
2 posts
fananhminh
#3 h  May 14, 2025, 4:23 PM
Y by
what if I do without calculator
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