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k a August Highlights and 2025 AoPS Online Class Information
jwelsh   0
Friday 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.

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0 replies
jwelsh
Friday at 2:14 PM
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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Swiss Mathematical Olympiad 2011 Final Round
radioactiverascal90210   1
N 10 minutes ago by nudinhtien
For a given rational number $r$ find all integers $z$ such that
$2^z + 2= r^2$
1 reply
radioactiverascal90210
23 minutes ago
nudinhtien
10 minutes ago
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Perimeter bisectors
JustPostNorthKoreaTST   0
16 minutes ago
Source: 2016 North Korea TST P4
Given a triangle $ABC$, if the line connecting a point $X$ on a side of $\triangle ABC$ and its corresponding vertex bisects the perimeter, we write $(X \to \triangle ABC)$.

In a convex quadrilateral $ABCD$, let $X,Y,Z,M$ be points on sides $AB,AD,DC,CB$, respectively, such that $BM=CM$, $AX=AY$, $YD=DZ$, $ZC=BX$, and $(X \to \triangle ABM)$, $(Y \to \triangle AMD)$, $(Z \to \triangle CDM)$.

Prove that $\triangle ABM \cong \triangle MDA \cong \triangle DMC$.
0 replies
1 viewing
JustPostNorthKoreaTST
16 minutes ago
0 replies
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Classic inequality in retrospect
JustPostNorthKoreaTST   0
22 minutes ago
Source: 2016 North Korea TST P3
Let $a_1,a_2,\ldots,a_n$ be positive real numbers, and denote $a_{n+1}=a_1$. Prove that
$$ \sum_{i=1}^n \frac{a_{i+1}}{a_i} \ge \sum_{i=1}^n \sqrt{\frac{a_{i+1}^2+1}{a_i^2+1}}. $$
0 replies
JustPostNorthKoreaTST
22 minutes ago
0 replies
J
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Infinitely many prime divisors of a recurrence sequence
JustPostNorthKoreaTST   0
26 minutes ago
Source: 2016 North Korea TST P2
Given a sequence $\{a_n\}_{n \ge 1}$ of positive integers, such that $a_1 \in \mathbb{N}_+$, and for $n \ge 1$,
$$ a_{n+1}=\sum_{i=2}^{n+1} \lfloor \sqrt[i]{a_n} \rfloor. $$Show that for any prime $p$, there are infinitely many terms in $\{a_n\}_{n \ge 1}$ that are divisible by $p$.
0 replies
JustPostNorthKoreaTST
26 minutes ago
0 replies
J
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Find sum over permutations
JustPostNorthKoreaTST   0
30 minutes ago
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.
0 replies
JustPostNorthKoreaTST
30 minutes ago
0 replies
J
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Set family with special conditions
JustPostNorthKoreaTST   0
33 minutes ago
Source: 2015 North Korea Mathematical Olympiad P6
Let $S$ be a set with $n$ ($n \ge 3$) elements. Find all possible integers $n$ such that there exists a family $\mathcal{F}$ of three-element subsets of $S$, satisfying the following conditions:
(1) For any $a,b \in S$, $a \neq b$, exactly one element of $\mathcal{F}$ contains $\{a,b\}$.
(2) For any $a,b,c,x,y,z \in S$, if $\{a,b,z\}, \{b,c,x\}, \{c,a,y\} \in \mathcal{F}$, then $\{x,y,z\} \in \mathcal{F}$.
0 replies
JustPostNorthKoreaTST
33 minutes ago
0 replies
J
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About Cauchy Inequality
lgx57   5
N 36 minutes ago by mudkip42
How does $\dfrac{x}{y+z}+\dfrac{4x}{2x+y+z} \ge \dfrac{9x}{2(x+y+z)}$ prove by Cauchy Inequality?
5 replies
lgx57
Jun 1, 2025
mudkip42
36 minutes ago
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Inequalities
sqing   18
N 39 minutes ago by nudinhtien
Let $ a,b \geq 0,a +b =1 . $ Prove that
$$ \sqrt{a^4 + \frac{3}{4}ab} + \sqrt{b^4 + \frac{3}{4}ab} \geq 1$$Let $ a,b \geq 0,a +b =4 . $ Prove that
$$ \sqrt{a^4 + 12ab} + \sqrt{b^4 + 12ab} \geq 16$$Let $ a,b,c \geq 0,a +b +c=1 . $ Prove that
$$ a^2 + b^2 + c^2 + \frac{1}{2}\sqrt{3abc} \geq \frac{1}{2}$$Let $ a,b,c \geq 0,a +b +c=3 . $ Prove that
$$ a^2 + b^2 + c^2 +\frac{3}{2} \sqrt{abc} \geq \frac{9}{2}$$
18 replies
sqing
Aug 1, 2025
nudinhtien
39 minutes ago
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inequality
aktyw19   1
N 42 minutes ago by Mathzeus1024
Let $x,y>0$ and $xy<1$. Prove $\left(\frac{2x}{1+x^{2}}\right)^{2}+\left(\frac{2y}{1+y^{2}}\right)^{2}\le\frac{1}{1-xy}$.
1 reply
aktyw19
Dec 19, 2012
Mathzeus1024
42 minutes ago
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Easy geometry with orthocenter
JustPostNorthKoreaTST   0
43 minutes ago
Source: 2015 North Korea Mathematical Olympiad P4
Let $ABC$ be a scalene triangle with circumcircle $\odot O$ and orthocenter $H$, and let $AH,BH,CH$ intersects $\odot O$ at $A_1,B_1,C_1$, respectively. The line passing through $A_1$ and parallel to $BC$ intersects $\odot O$ at $A_2$. Define $B_2,C_2$ similarly. Let $AC_2$ intersects $BC_1$ at $M$, $BA_2$ intersects $CA_1$ at $N$, and $CB_2$ intersects $AB_1$ at $P$. Prove that $\angle MNB=\angle AMP$.
0 replies
JustPostNorthKoreaTST
43 minutes ago
0 replies
J
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Tricolor complete graphs
JustPostNorthKoreaTST   0
an hour ago
Source: 2015 North Korea Mathematical Olympiad P3
Consider a complete graph $K_n$ on $n$ vertices, where $n \ge 3$. Each edge is colored with one of three colors, and each color is used on at least one edge. Find the minimum positive integer $k$ such that for any such edge coloring and any color $C$ chosen from the three colors, it is possible to recolor at most $k$ edges to color $C$ so that the subgraph consisting of all edges of color $C$ is connected.
0 replies
JustPostNorthKoreaTST
an hour ago
0 replies
J
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How to solve this?
Mr._Calculator   0
an hour ago
In the diagram below, $ABC$ is a triangle, where the angle bisectors of $\angle ABC$ and $\angle ACB$ intersect at point $P$. Let points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $DE$ passes through point $P$ and $\angle AED = \angle ABP$. If the areas of triangles $BDP$, $CEP$ and $BPC$ are equal to $18 \text{ cm}^2$, $36 \text{ cm}^2$ and $57 \text{ cm}^2$, respectively, then what is the area, in $\text{cm}^2$, of $ABC$?
0 replies
Mr._Calculator
an hour ago
0 replies
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Number Theory
JetFire008   1
N an hour ago by blug
Source: Elementary Number Theory by David M. Burton
Modify Euclid's proof that there are infinitely many primes by assuming the existence of a largest prime $p$ and using the integer $N=p!+1$ to arrive at a contradiction.
1 reply
JetFire008
2 hours ago
blug
an hour ago
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Theorems that really helped
SwordAxe   21
N 2 hours ago by NamelyOrange
What are some theorems that really helped you in competitions (specifically AMC 8/10/12, AIME, Mathcounts)?

Herons really helped me once
21 replies
SwordAxe
Jul 29, 2025
NamelyOrange
2 hours ago
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J
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Counting problem
lgx57   1
N Jun 6, 2025 by HAL9000sk
Calculate the number of $n$ that meet the following conditions:
1. $585 \mid n$
2.$0 \sim 7$ appears exactly once in each octal digit of $n$
1 reply
lgx57
Jun 6, 2025
HAL9000sk
Jun 6, 2025
J
Counting problem
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algebra
number theory
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lgx57
64 posts
lgx57
#1 h  Jun 6, 2025, 8:36 AM
Y by
Calculate the number of $n$ that meet the following conditions:
1. $585 \mid n$
2.$0 \sim 7$ appears exactly once in each octal digit of $n$
Z K Y
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HAL9000sk
120 posts
HAL9000sk
#2 h  Jun 6, 2025, 10:50 AM
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solution
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