Plan ahead for the next school year. Schedule your class today!

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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
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
I need your help
Mr_adjective   22
N 5 minutes ago by Dragon_Yang
how do you use aops?
22 replies
Mr_adjective
Thursday at 6:37 PM
Dragon_Yang
5 minutes ago
an elegant geometric configuration
Mr_adjective   1
N 16 minutes ago by OronSH
Let \(ABC\) be a triangle with incenter \(I\), and let \(AD\) be an altitude (where \(D\) lies on \(BC\)). Denote by \(A'\) the point diametrically opposite to \(A\) on the circumcircle of \(\triangle ABC\). Prove that \(\angle ADI + \angle AIA' = 180^\circ\).
1 reply
+1 w
Mr_adjective
44 minutes ago
OronSH
16 minutes ago
Peru Ibero TST 2022
diegoca1   0
38 minutes ago
Source: Peru Ibero TST 2022 D1 P4
Let $\Gamma$ be the circumcircle of triangle $ABC$, and let $M$ be the midpoint of side $AC$. The line $BM$ intersects $\Gamma$ at point $D$ (with $D \ne B$). The circumcircles of triangles $AMD$ and $BMC$ intersect at point $E$ (with $E \ne M$). The circumcircles of triangles $AMB$ and $CMD$ intersect at point $F$ (with $F \ne M$). Prove that the circumcircle of triangle $BEF$ is tangent to $\Gamma$.
0 replies
diegoca1
38 minutes ago
0 replies
Peru Ibero TST 2022
diegoca1   0
43 minutes ago
Source: Peru Ibero TST 2022 D1 P1
Let $P$ be a degree 6 polynomial with integer coefficients. Prove that there exists a real number $a \in [0,3]$ such that
\[
|P(a)| \geq \frac{45}{64}.
\]
0 replies
diegoca1
43 minutes ago
0 replies
Excircles produce homothetic lines
sanyalarnab   3
N 43 minutes ago by endless_abyss
Source: Sharygin Finals 2022 9.1
Let $BH$ be an altitude of right angled triangle $ABC$($\angle B = 90^o$). An excircle of triangle $ABH$ opposite to $B$ touches $AB$ at point $A_1$; a point $C_1$ is defined similarly. Prove that $AC // A_1C_1$.
3 replies
sanyalarnab
Jul 31, 2022
endless_abyss
43 minutes ago
NT problem
RedFireTruck   1
N an hour ago by ItzsleepyXD
Find all positive integers $n$ such that there exist pairwise distinct positive integers $a_1, \dots, a_n$ with $a_i|(a_1+...+a_n)$ for all integers $1\le i\le n$.
1 reply
RedFireTruck
2 hours ago
ItzsleepyXD
an hour ago
Trig Sum
MithsApprentice   18
N an hour ago by ray66
Source: USAMO 1996
Prove that the average of the numbers $n \sin n^{\circ} \; (n = 2,4,6,\ldots,180)$ is $\cot 1^{\circ}$.
18 replies
MithsApprentice
Oct 22, 2005
ray66
an hour ago
harder than ever
perfect_square   4
N an hour ago by perfect_square
Let $a,b,c$ which satisfy:
$  \begin{cases}
a+b+c=4 \\
ab+bc+ca=5 \end{cases} $
and $w=abc$
a. Prove that: $ \frac{50}{27} \le w \le 2$
b. Given $ n \in N, n \ge 3$. Prove that: $ a^n+b^n+c^n=f(w)$, which is increasing function.
4 replies
perfect_square
Jul 16, 2025
perfect_square
an hour ago
Bogus Proof Marathon
pifinity   7760
N an hour ago by IamCurlyLizard39
Hi!
I'd like to introduce the Bogus Proof Marathon.

In this marathon, simply post a bogus proof that is middle-school level and the next person will find the error. You don't have to post the real solution :P

Use classic Marathon format:
[hide=P#]a1b2c3[/hide]
[hide=S#]a1b2c3[/hide]


Example posts:

P(x)
-----
S(x)
P(x+1)
-----
Let's go!! Just don't make it too hard!
7760 replies
pifinity
Mar 12, 2018
IamCurlyLizard39
an hour ago
Determine the minimum value
April   20
N 2 hours ago by lpieleanu
Source: CGMO 2007 P3
Let $ n$ be an integer greater than $ 3$, and let $ a_1, a_2, \cdots, a_n$ be non-negative real numbers with $ a_1 + a_2 + \cdots + a_n = 2$. Determine the minimum value of \[ \frac{a_1}{a_2^2 + 1}+ \frac{a_2}{a^2_3 + 1}+ \cdots + \frac{a_n}{a^2_1 + 1}.\]
20 replies
April
Dec 28, 2008
lpieleanu
2 hours ago
3 Var (?)
SunnyEvan   4
N 2 hours ago by SunnyEvan
Source: Own
Let $ a,b,c \in R^+ $, such that :$ab+bc+ca+abc=2$.
Prove that: $$ \frac{1}{3\sqrt{a^2(b+1)(c+1)+abc+2}}+\frac{1}{3\sqrt{b^2(c+1)(a+1)+abc+2}}+\frac{1}{5\sqrt{c^2(a+1)(b+1)+abc+2}} < \frac{43}{90} $$$$ \frac{1}{3\sqrt{a^2(b+1)(c+1)+abc+2}}+\frac{1}{4\sqrt{b^2(c+1)(a+1)+abc+2}}+\frac{1}{5\sqrt{c^2(a+1)(b+1)+abc+2}} < \frac{5}{12}  $$
4 replies
SunnyEvan
Jul 18, 2025
SunnyEvan
2 hours ago
2-var inequality
sqing   0
2 hours ago
Source: Own
Let $ a,b  \geq 0 , \frac {a+1} {a^2+ab+1}+ \frac {b+1} {b^2+ab+1} \leq 1  . $ Prove that
$$ a+b\geq 1+\sqrt 3$$Let $ a,b \geq 0, \frac {a+1} {a^2+ab+1}+ \frac {b+1} {b^2+ab+1} \geq \frac {4} {3}   . $ Prove that
$$ a+b\leq \frac {3+\sqrt{17}} {2}$$Let $ a,b \geq 0, \frac {a} {a^2+ab+1}+ \frac {b} {b^2+ab+1} \geq \frac {2} {5}  . $ Prove that
$$ a+b\leq \frac {5-\sqrt{17}} {2}$$
0 replies
sqing
2 hours ago
0 replies
Chinese chess!
EthanWYX2009   0
2 hours ago
Source: 2024 February 谜之竞赛-5
The Cannon is a chess piece. A move consists of the following: if the Cannon and two other pieces \( A \) and \( B \) lie consecutively on a line parallel to the edge of the board, then the Cannon can jump over \( A \) to capture \( B \), landing on \( B \)'s original position.

Determine the smallest positive integer \( N \) such that for any integers \( m, n \geq 2024 \), no matter where the Cannon is initially placed on an \( m \times n \) chessboard (with all other positions occupied by pieces), it is guaranteed that after a finite number of moves, at most \( N \) pieces will remain on the board.

Created by Yuhang Li, Chengdu Shude High School and Kaiyu Li, Renmin University Affiliated High School
0 replies
EthanWYX2009
2 hours ago
0 replies
Should I reset Alcumus?
SunnieBunnie   17
N 2 hours ago by SunnieBunnie
My Alcumus rating is pretty low, 59.5, and I only did about 90% of my problems correct out of 328. Should I reset?
17 replies
SunnieBunnie
Yesterday at 6:52 PM
SunnieBunnie
2 hours ago
Estimation Problems
slightly_irrational   3
N May 25, 2025 by CJB19
Is there a general strategy for problems like this where you find the first nonzero digit of some number?

Mathcounts 2025 Chapter Sprint 28
What is the first non-zero digit of $\sqrt{75^2 + 1}$ that appears after the decimal point?
3 replies
slightly_irrational
May 24, 2025
CJB19
May 25, 2025
Estimation Problems
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G H BBookmark kLocked kLocked NReply
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slightly_irrational
9 posts
#1
Y by
Is there a general strategy for problems like this where you find the first nonzero digit of some number?

Mathcounts 2025 Chapter Sprint 28
What is the first non-zero digit of $\sqrt{75^2 + 1}$ that appears after the decimal point?
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aidan0626
2080 posts
#2 • 1 Y
Y by efx
well for that specific type of square root problem there is a strategy
note that $\left(75+\frac1{150}\right)^2=75^2+1+\left(\frac1{150}\right)^2$, which is just barely greater than what is in the square root
so the answer is very slightly less than $75+\frac{1}{150}=75.00\overline{6}$, giving an answer of $6$
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pingpongmerrily
4055 posts
#3
Y by
aidan0626 wrote:
well for that specific type of square root problem there is a strategy
note that $\left(75+\frac1{150}\right)^2=75^2+1+\left(\frac1{150}\right)^2$, which is just barely greater than what is in the square root
so the answer is very slightly less than $75+\frac{1}{150}=75.00\overline{6}$, giving an answer of $6$

orz sol
yeah I would add that for radicals like that you can either try factoring or completing the square depending on the problem
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CJB19
178 posts
#4
Y by
aidan0626 wrote:
well for that specific type of square root problem there is a strategy
note that $\left(75+\frac1{150}\right)^2=75^2+1+\left(\frac1{150}\right)^2$, which is just barely greater than what is in the square root
so the answer is very slightly less than $75+\frac{1}{150}=75.00\overline{6}$, giving an answer of $6$

Ohhhh someone tried explaining this to me in words but seeing it in writing makes more sense.
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