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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 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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Inspired by Zhejiang 2025
sqing   1
N an hour ago by WallyWalrus
Source: Own
Let $ x,y,z $ be reals such that $ 5x^2+6y^2+6z^2-8yz\leq 5. $ Prove that$$ x+y+z\leq \sqrt{6}$$
1 reply
1 viewing
sqing
4 hours ago
WallyWalrus
an hour ago
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incircle excenter midpoints
danepale   9
N an hour ago by Want-to-study-in-NTU-MATH
Source: Middle European Mathematical Olympiad T-6
Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$. Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$. Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively.

Prove that the points $B, C, N,$ and $L$ are concyclic.
9 replies
danepale
Sep 21, 2014
Want-to-study-in-NTU-MATH
an hour ago
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geometry
gggzul   0
2 hours ago
Let $ABC$ be a triangle with $\angle ACB=90^{\circ}$. $D$ is the midpoint of $AC$. Let the angle bisector of $\angle ACB$ cut $BD$ at $P$ and $G$ be the centroid of $ABC$. $(CPG)$ meets $BC$ at $Q\ne C$ and $R$ is the projection of $Q$ onto $AB$. Prove that $R, G, P, A$ lie on a common circle.
0 replies
gggzul
2 hours ago
0 replies
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Maximum Area of Triangle ABC
steven_zhang123   0
2 hours ago
Let the coordinates of point \( A \) be \( (0,3) \). Points \( B \) and \( C \) are two moving points on the circle \( O \): \( x^2+y^2=25 \), satisfying \( \angle BAC=90^\circ \). Find the maximum area of \( \triangle ABC \).
0 replies
steven_zhang123
2 hours ago
0 replies
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IMO 2016 Problem 4
termas   56
N 2 hours ago by sansgankrsngupta
Source: IMO 2016 (day 2)
A set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$is fragrant?
56 replies
termas
Jul 12, 2016
sansgankrsngupta
2 hours ago
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Interesting inequalities
sqing   3
N 2 hours ago by sqing
Source: Own
Let $a,b,c \geq 0 $ and $ abc+2(ab+bc+ca) =32.$ Show that
$$ka+b+c\geq 8\sqrt k-2k$$Where $0<k\leq 4. $
$$ka+b+c\geq 8 $$Where $ k\geq 4. $
$$a+b+c\geq 6$$$$2a+b+c\geq 8\sqrt 2-4$$
3 replies
sqing
May 15, 2025
sqing
2 hours ago
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Every popular person is the best friend of a popular person?
yunxiu   8
N 3 hours ago by HHGB
Source: 2012 European Girls’ Mathematical Olympiad P6
There are infinitely many people registered on the social network Mugbook. Some pairs of (different) users are registered as friends, but each person has only finitely many friends. Every user has at least one friend. (Friendship is symmetric; that is, if $A$ is a friend of $B$, then $B$ is a friend of $A$.)
Each person is required to designate one of their friends as their best friend. If $A$ designates $B$ as her best friend, then (unfortunately) it does not follow that $B$ necessarily designates $A$ as her best friend. Someone designated as a best friend is called a $1$-best friend. More generally, if $n> 1$ is a positive integer, then a user is an $n$-best friend provided that they have been designated the best friend of someone who is an $(n-1)$-best friend. Someone who is a $k$-best friend for every positive integer $k$ is called popular.
(a) Prove that every popular person is the best friend of a popular person.
(b) Show that if people can have infinitely many friends, then it is possible that a popular person is not the best friend of a popular person.

Romania (Dan Schwarz)
8 replies
yunxiu
Apr 13, 2012
HHGB
3 hours ago
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2021 EGMO P2: f(xf(x)+y) = f(y) + x^2 for rational x, y
anser   80
N 3 hours ago by math-olympiad-clown
Source: 2021 EGMO P2
Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that the equation
\[f(xf(x)+y) = f(y) + x^2\]holds for all rational numbers $x$ and $y$.

Here, $\mathbb{Q}$ denotes the set of rational numbers.
80 replies
anser
Apr 13, 2021
math-olympiad-clown
3 hours ago
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D1033 : A problem of probability for dominoes 3*1
Dattier   1
N 3 hours ago by Dattier
Source: les dattes à Dattier
Let $G$ a grid of 9*9, we choose a little square in $G$ of this grid three times, we can choose three times the same.

What the probability of cover with 3*1 dominoes this grid removed by theses little squares (one, two or three) ?
1 reply
Dattier
May 15, 2025
Dattier
3 hours ago
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2010 Japan MO Finals
parkjungmin   4
N 3 hours ago by parkjungmin
Is there anyone who can solve question problem 5?
4 replies
parkjungmin
May 15, 2025
parkjungmin
3 hours ago
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9 How many squares do you have memorized
LXC007   27
N Today at 4:53 AM by whwlqkd
How many squares have you memorized. I have 1-20
27 replies
LXC007
Yesterday at 3:44 PM
whwlqkd
Today at 4:53 AM
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2000th Post!
PikaPika999   7
N Today at 4:25 AM by elizhang101412
1. How many ways can you arrange the letters in the word ALGEBRA such that no two identical letters are adjacent?

2. Find the smallest positive integer n such that $n^2+n+41$ is not a prime number.

3. You have 4 red tiles, 3 blue tiles, and 2 green tiles. How many ways can you arrange them in a row such that no two tiles of the same color are adjacent?

4. You flip a fair coin repeatedly until you either get 3 tails or 4 heads. What is the expected number of flips before stopping?

5. Let $A(2,3)$ and $B(8,7)$ be two points in the coordinate plane. A circle is drawn such that $\overline{AB}$ is a diameter.

(a). Find the equation of the circle in the form $(x+a)^2+(y+b)^2=r^2$

(b). A line passes through the point P(6,2) and is tangent to the circle. Find the equation of this tangent line.

hopefully these problems weren't too easy lol

also,
Please tell me if any of these problems have any flaws! (also please put your answers in hide tags or quote tags)
7 replies
PikaPika999
Today at 1:53 AM
elizhang101412
Today at 4:25 AM
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Challenge: Make every number to 100 using 4 fours
CJB19   126
N Today at 4:15 AM by e_is_2.7182818
I've seen this attempted a lot but I want to see if the AoPS community can actually do it. Using ONLY 4 fours and math operations, make as many numbers as you can. Try to go in order. I'll start:
$$(4-4)*4*4=0$$$$4-4+4/4=1$$$$4/4+4/4=2$$$$(4+4+4)/4=3$$$$4+(4-4)*4=4$$$$4+4^{4-4}=5$$$$4!/4+4-4=6$$$$4+4-4/4=7$$$$4+4+4-4=8$$
126 replies
CJB19
May 15, 2025
e_is_2.7182818
Today at 4:15 AM
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9 Am I going insane?
PenguFish   26
N Today at 4:08 AM by nmlikesmath
I feel stupid honestly, and fyi, I did go over every question. Focus is counting with symmetry
26 replies
PenguFish
Mar 26, 2025
nmlikesmath
Today at 4:08 AM
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Can someone explain this one
hawa   10
N Apr 29, 2025 by VivaanKam
Suppose n is the largest integer obtained by solving the following inequality:

3+9+18+30+...+n
n < 2021.
10 replies
hawa
Apr 29, 2025
VivaanKam
Apr 29, 2025
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Can someone explain this one
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Reveal topic tags
inequalities
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hawa
2 posts
hawa
#1 h  Apr 29, 2025, 1:36 AM
Y by
Suppose n is the largest integer obtained by solving the following inequality:

3+9+18+30+...+n
n < 2021.
Z K Y
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FredyCrooger
64 posts
FredyCrooger
#2 h  Apr 29, 2025, 1:45 AM
Y by
So first you find the change in the difference between the numbers and assuming the question is asking to find the largest $n$, you can add them up and continue from there.
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johnnie.walker
2 posts
johnnie.walker
#3 h  Apr 29, 2025, 1:38 PM
Y by
Hint
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fossasor
606 posts
fossasor
#4 h  Apr 29, 2025, 1:48 PM
Y by
Hint #2
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snake2020
4510 posts
snake2020
#5 h  Apr 29, 2025, 1:51 PM
Y by
Hint #3
This post has been edited 1 time. Last edited by snake2020, Apr 29, 2025, 1:51 PM
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snake2020
4510 posts
snake2020
#6 h  Apr 29, 2025, 2:08 PM
Y by
Solution
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Andrew2019
2319 posts
Andrew2019
#7 h  Apr 29, 2025, 5:32 PM
Y by
snake2020 wrote:
Solution

believe this is incorrect, it says $n$ is less that $2021$, not the sum
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VivaanKam
167 posts
VivaanKam
#8 h  Apr 29, 2025, 8:01 PM
Y by
The general term of the sequence is $a_n=\dfrac{3n(n-1))}{2}$
This post has been edited 1 time. Last edited by VivaanKam, Apr 29, 2025, 8:10 PM
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VivaanKam
167 posts
VivaanKam
#9 h  Apr 29, 2025, 8:04 PM
Y by
But I don't get the problem. Are you saying that $3+9+18+30+\dots +n<2021$ and $n<2021$?
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VivaanKam
167 posts
VivaanKam
#10 h  Apr 29, 2025, 8:05 PM
Y by
nvm I figured it out.
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VivaanKam
167 posts
VivaanKam
#11 h  Apr 29, 2025, 8:23 PM
Y by
Solution
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