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k a August Highlights and 2025 AoPS Online Class Information
jwelsh   0
Aug 1, 2025
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!:
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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.

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0 replies
jwelsh
Aug 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
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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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Quadrilateral OI geometry problem
chengbilly   0
18 minutes ago
Source: 2025 IND-IRN-SGP-TWN Friendly Math Competition P2, 2025 IMOC Day2-G
Let $ABCD$ be a quadrilateral with both an incircle and a circumcircle. $I$ and $O$ be the incenter and circumcenter of $ABCD$, respectively. Let $E$ be the intersection of lines $AB$ and $CD$, and let $F$ be the intersection of lines $BC$ and $DA$. Let $X$ and $Y$ be the intersections of the line $FI$ with lines $AB$ and $CD$, respectively. Prove that the circumcircle of $\triangle FEI$, the circumcircle of $\triangle EXY$, and the line $FO$ are concurrent.

proposed by chengbilly
0 replies
chengbilly
18 minutes ago
0 replies
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Four circles are concurrent
Ktoan07   0
23 minutes ago
Source: Own
Let \(ABC\) be an acute triangle, and let points \(A_1 \in BC\), \(B_1 \in CA\), \(C_1 \in AB\) vary along the sides of triangle \(ABC\) such that
\[ \frac{A_1B}{A_1C} = \frac{B_1C}{B_1A} = \frac{C_1A}{C_1B}. \]Proof that three circles \((AB_1C_1)\), \((BC_1A_1)\), and \((CA_1B_1)\) are concurrent at a point lies on the Brocard circle of triangle \(ABC\).
0 replies
Ktoan07
23 minutes ago
0 replies
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Nesbitt refined for 4 var
mihaig   0
27 minutes ago
Source: VL
Let $u\ge v\ge w\ge z\ge0,$ with $v>0$ and $u+v+w+z=S.$ We set:
$$\frac u{S-u}:=a;\frac v{S-v}:=b;\frac w{S-w}:=c;\frac z{S-z}:=d.$$Prove
$$a+b+c+d+\frac1{ab+ac+ad+bc+bd+cd}\geq\frac{17}6.$$When do we have equality?
0 replies
mihaig
27 minutes ago
0 replies
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Symmedian and centroid
m4thbl3nd3r   0
29 minutes ago
Let $L$ be the symmedian point of triangle $ABC$ and $D$ be the intersection of $AL$ and $BC$. Let $E$ be the the second intersection of $(CLD)$ and $AC$, $F$ be the second intersection of $(BLD)$ and $AB$. Prove that $L$ is the centroid of $\triangle DEF$.
0 replies
m4thbl3nd3r
29 minutes ago
0 replies
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inequality
aktyw19   2
N 29 minutes ago by ehuseyinyigit
Let $x,y>0$ and $xy<1$. Prove $\left(\frac{2x}{1+x^{2}}\right)^{2}+\left(\frac{2y}{1+y^{2}}\right)^{2}\le\frac{1}{1-xy}$.
2 replies
aktyw19
Dec 19, 2012
ehuseyinyigit
29 minutes ago
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a tst pre VietNam
Math2030   3
N 34 minutes ago by ektorasmiliotis
Given the sequence $(a_n): a_1=1, a_2=11$ and $a_{n+2}=a_{n+1}+5a_{n}, n \geq 1$
. Prove that $a_n $not is a perfect square for all $n > 3$.
3 replies
Math2030
Today at 1:39 AM
ektorasmiliotis
34 minutes ago
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IMO Shortlist 2008, Geometry problem 4
April   101
N 37 minutes ago by happypi31415
Source: IMO Shortlist 2008, Geometry problem 4, German TST 5, P3, 2009
In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$.

Proposed by Davood Vakili, Iran
101 replies
April
Jul 9, 2009
happypi31415
37 minutes ago
J
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Twice continuously differntiable function
enter16180   6
N 44 minutes ago by Dragon2311
Source: IMC 2025, Problem 2
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice continuously differentiable function, and suppose that $\int_{-1}^1 f(x) \mathrm{d} x=0$ and $f(1)=f(-1)=1$. Prove that
$$ \int_{-1}^1\left(f^{\prime \prime}(x)\right)^2 \mathrm{~d} x \geq 15 $$and find all such functions for which equality holds.
6 replies
enter16180
Jul 30, 2025
Dragon2311
44 minutes ago
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Interesting inequality
sqing   0
44 minutes ago
Source: Own
Let $ a,b,c \in[-1,1] . $ Prove that
$$ (a^2+ b-bc) (2b^2+3c-4ca ) \leq \frac{64}{7}$$$$(a^2+ b - bc) ( b^2+2c-3ca-4ab ) \leq \frac{45}{4}$$$$ (a^2+ b - bc) (2b^2+3c-4ca-5ab ) \leq \frac{63}{4}$$$$ (a^2+ b - bc) ( b^2+2c-3ca-4ab-5bc ) \leq \frac{4(846+55\sqrt{330})}{441}$$$$ (a^2+ b - bc) (2b^2+3c-4ca-5ab-6bc ) \leq \frac{2(9863+535\sqrt{535})}{2187}$$
0 replies
sqing
44 minutes ago
0 replies
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An integral inequality
Anabcde   1
N an hour ago by Hello_Kitty
A function $f$ is continuous on [0, 1] and diffrentiable on (0, 1). Given that $f(0)=0$ and $0 \le f'(x) \le 1, \forall 0 \le x \le 1$. Prove:
$$(\int_{0}^{1} f(x) \,dx )^2 \ge \int_{0}^{1} (f(x))^3 \,dx $$
1 reply
Anabcde
2 hours ago
Hello_Kitty
an hour ago
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Find sum over permutations
JustPostNorthKoreaTST   1
N an hour ago by Lil_flip38
Source: 2016 North Korea TST P1
Given an odd positive integer $n$. Find the value of
$$ \sum_\pi \prod_{i=1}^n (\pi(i)-i), $$where $\{\pi(i)\}_{i=1}^n$ is a permutation of $\{1,2,\ldots,n\}$, and the summation runs over all such permutations.
1 reply
JustPostNorthKoreaTST
3 hours ago
Lil_flip38
an hour ago
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hard 3 vars symetric
perfect_square   7
N an hour ago by ehuseyinyigit
Let $a,b,c \ge 0$ which satisfy:
$ \begin{cases} a+b+c=4 \\ a^4+b^4+c^4 =18 \end{cases} $
Prove that: $ab+bc+ca \le 5$
7 replies
perfect_square
2 hours ago
ehuseyinyigit
an hour ago
J
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prove that there exists \xi
Peter   22
N 6 hours ago by Mathloops
Source: IMC 1998 day 1 problem 4
The function $f: \mathbb{R}\rightarrow\mathbb{R}$ is twice differentiable and satisfies $f(0)=2,f'(0)=-2,f(1)=1$.
Prove that there is a $\xi \in ]0,1[$ for which we have $f(\xi)\cdot f'(\xi)+f''(\xi)=0$.
22 replies
Peter
Nov 1, 2005
Mathloops
6 hours ago
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Identity
Saucepan_man02   4
N Today at 8:25 AM by ray66
Could anyone provide an elegant proof for this identity?

$\sum_{k=0}^{n} (-1)^k \binom{2n+1}{k} = (-1)^n \binom{2n}{n}$
4 replies
Saucepan_man02
Friday at 4:19 PM
ray66
Today at 8:25 AM
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Investigating functions
mikejoe   1
N May 14, 2025 by Mathzeus1024
Source: Edwards and Penney
Investigate the function $f(x) = (x-2) \sqrt{x+1}$
Also determine its domain and range.
1 reply
mikejoe
Nov 2, 2012
Mathzeus1024
May 14, 2025
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Investigating functions
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Reveal topic tags
function
algebra
domain
calculus
calculus computations
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Source: Edwards and Penney
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mikejoe
8 posts
mikejoe
#1 h  Nov 2, 2012, 8:02 PM • 2 Y
Y by Adventure10, Mango247
Investigate the function $f(x) = (x-2) \sqrt{x+1}$
Also determine its domain and range.
Z K Y
The post below has been deleted. Click to close.
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Mathzeus1024
1089 posts
Mathzeus1024
#2 h  May 14, 2025, 1:08 PM
Y by
The domain is clearly $x \in [-1, \infty)$ due to the function's radicand. As for the range, taking the derivative of $f$ equal to zero yields the critical value:

$f'(x) = \sqrt{x+1}+\frac{x-2}{2\sqrt{x+1}}=0 \Rightarrow 2(x+1) = 2-x \Rightarrow x = 0$;

and second derivative check of $f$ against this critical value yields:

$f''(x) = \frac{1}{2\sqrt{x+1}} + \frac{x+4}{4(x+1)^{3/2}} \Rightarrow f''(0) = \frac{3}{2} > 0$ (a global minimum).

Thus, the range of $f$ is $[-2,\infty)$.
This post has been edited 1 time. Last edited by Mathzeus1024, May 14, 2025, 1:11 PM
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