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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.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
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[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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
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
A container has $40$ liters of milk. Then, $4$ liters are removed from the cont
Vulch   3
N a few seconds ago by evt917
A container has $40$ liters of milk. Then, $4$ liters are removed from the container and replaced with $4$ liters of water. This process of replacing $4$ liters of the liquid in the container with an equal volume of water is continued repeatedly. The smallest number of times of doing this process, after which the volume of milk in the container becomes less than that of water, is
3 replies
Vulch
Yesterday at 10:11 AM
evt917
a few seconds ago
9 middle school olympiads forum ?
kjhgyuio   6
N an hour ago by kjhgyuio
There is a high school olympiads forum,so I am thinking why is there no middle school olympiads forum Should i create a middle school olympiads forum ?
here is the link if you are interested ->https://artofproblemsolving.com/community/c4318171_middle_school_olympiads
6 replies
kjhgyuio
3 hours ago
kjhgyuio
an hour ago
MATHCOUNTS
ILOVECATS127   33
N 2 hours ago by bluedino24
Hi,

I am looking to get on my school MATHCOUNTS team next year in 7th grade, and I had a question: Where do the school round questions come from? (Sprint, Chapter, Team, Countdown)
33 replies
ILOVECATS127
May 7, 2025
bluedino24
2 hours ago
BossLu for CDR
Math-lover1   0
2 hours ago
Ben lu is gonna make CDR this year's nationals

Mark my words
0 replies
Math-lover1
2 hours ago
0 replies
A strong inequality problem
hn111009   0
2 hours ago
Source: Somewhere
Let $a,b,c$ be the positive number satisfied $a^2+b^2+c^2=3.$ Find the minimum of $$P=\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}+\dfrac{3abc}{2(ab+bc+ca)}.$$
0 replies
hn111009
2 hours ago
0 replies
help me please,thanks
tnhan.129   0
2 hours ago
find f: R+ -> R such that:
f(x)/x + f(y)/y = (1/x + 1/y).f(sqrt(xy))
0 replies
tnhan.129
2 hours ago
0 replies
Easy divisibility
a_507_bc   2
N 2 hours ago by TUAN2k8
Source: ARO Regional stage 2023 9.4~10.4
Let $a, b, c$ be positive integers such that no number divides some other number. If $ab-b+1 \mid abc+1$, prove that $c \geq b$.
2 replies
1 viewing
a_507_bc
Feb 16, 2023
TUAN2k8
2 hours ago
Inspired by old results
sqing   0
2 hours ago
Source: Own
Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2 =3$. Prove that$$\frac{9}{5}>(a-b)(b-c)(2a-1)(2c-1)\geq -16$$
0 replies
sqing
2 hours ago
0 replies
integer functional equation
ABCDE   149
N 2 hours ago by ezpotd
Source: 2015 IMO Shortlist A2
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\]holds for all $x,y\in\mathbb{Z}$.
149 replies
ABCDE
Jul 7, 2016
ezpotd
2 hours ago
A geometry problem involving 2 circles
Ujiandsd   0
2 hours ago
Source: L
Point M is the midpoint of side BC of triangle ABC. The length of the radius of the outer circle of triangle ABM, triangle ACM
is 5 and 7 respectively find the distance between the center of their outer circles
0 replies
Ujiandsd
2 hours ago
0 replies
Inequality, inequality, inequality...
Assassino9931   10
N 2 hours ago by sqing
Source: Al-Khwarizmi Junior International Olympiad 2025 P6
Let $a,b,c$ be real numbers such that \[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]Find the smallest possible value of $a^2 + b^2 + c^2$.

Binh Luan and Nhan Xet, Vietnam
10 replies
Assassino9931
Yesterday at 9:38 AM
sqing
2 hours ago
Grid with rooks
a_507_bc   3
N 2 hours ago by TUAN2k8
Source: ARO Regional stage 2022 9.3
Given is a positive integer $n$. There are $2n$ mutually non-attacking rooks placed on a grid $2n \times 2n$. The grid is splitted into two connected parts, symmetric with respect to the center of the grid. What is the largest number of rooks that could lie in the same part?
3 replies
a_507_bc
Feb 16, 2023
TUAN2k8
2 hours ago
IMO Shortlist 2013, Number Theory #3
lyukhson   47
N 2 hours ago by cursed_tangent1434
Source: IMO Shortlist 2013, Number Theory #3
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
47 replies
lyukhson
Jul 10, 2014
cursed_tangent1434
2 hours ago
Darboux cubic
srirampanchapakesan   1
N 3 hours ago by srirampanchapakesan
Source: Own
Let P be a point on the Darboux cubic (or the McCay Cubic ) of triangle ABC.

P1P2P3 is the circumcevian or pedal triangle of P wrt ABC.

Prove that P also lie on the Darboux cubic ( or the McCay Cubic) of P1P2P3 .
1 reply
srirampanchapakesan
May 7, 2025
srirampanchapakesan
3 hours ago
k Wrong Answers Only Pt.2
MathRook7817   72
N Apr 10, 2025 by MathRook7817
Problem: What is the area of a triangle with side lengths 13,14, and 15?
WRONG ANSWERS ONLY!

other one got locked for some reason
72 replies
MathRook7817
Apr 9, 2025
MathRook7817
Apr 10, 2025
Wrong Answers Only Pt.2
G H J
G H BBookmark kLocked kLocked NReply
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MathRook7817
692 posts
#1 • 2 Y
Y by Exponent11, jkim0656
Problem: What is the area of a triangle with side lengths 13,14, and 15?
WRONG ANSWERS ONLY!

other one got locked for some reason
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Soupboy0
393 posts
#3
Y by
$\sqrt{13^2+14^2-15^2} = 80$
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unicornlover9763
38 posts
#4
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(13+14)/15=1.8
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Mathdreams
1472 posts
#5
Y by
$\sqrt(21 * 8 * 7 * 6) = 1$
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BossLu99
1386 posts
#7
Y by
like 6-7
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Soupboy0
393 posts
#8
Y by
Using the 1434 Geo Lemma, we find the area of the triangle to be $\frac{[\binom{13}{2}-\binom{13}{4}+\binom{14}{2}-\binom{14}{4}+\binom{15}{2}-\binom{15}{4}][14-15+13]}{13^2+15^2-\frac{14}{2}}(-1)^{\frac{1}{2}(13+14+15)}$


edit: this totally isn't the correct answer
This post has been edited 1 time. Last edited by Soupboy0, Apr 9, 2025, 8:49 PM
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Lankou
1399 posts
#9
Y by
$13^{14^{15}}\approx 10^{10^{17.24}}$$
This post has been edited 1 time. Last edited by Lankou, Apr 9, 2025, 6:20 PM
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komodraqon
149 posts
#10
Y by
13+14+15 = 131415
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maxamc
574 posts
#11
Y by
the inradius is $\frac{13 \cdot 14 \cdot 15}{4 Area}$. it is also $\frac{Area}{\frac{13+14+15}{2}},$ solving gives $Area=\frac{21\sqrt{130}}{2}$.
This post has been edited 3 times. Last edited by maxamc, Apr 9, 2025, 6:50 PM
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iwastedmyusername
143 posts
#12 • 9 Y
Y by komodraqon, donut_bear, Exponent11, MathRook7817, huajun78, aidan0626, legospartan, Rice_Farmer, EthanNg6
We want to find the area of a triangle with side lengths 13, 14, and 15. Now this might seem quite difficult, but there's a neat trick to find the area of this specific triangle. First, split the side with length 14 into two sides of lengths 5 and 9. Now, we want to connect the point connecting the sides with lengths 13 and 15 to the center of the splitting (idk). Hmmm, its not looking promising. Then I realized. What rhymes with five? Hive. Beehive. A beehive is shaped like a bunch of hexagons. Hexagons have 6 sides. So I found that the number 6 has something to do with this. Because its a triangle and triangles have 3 sides, and multiplication is the third operation most students learn in elementary school, I know it has to be 6 times something. What could 6 possibly be multiplied by? Then I realized that I haven't used the 9 yet. So I searched up on google what happened in the year 9. According to Gemini AI which is a very credible source, there was a devastating defeat for the Roman army led by Publiius Quinticlius Varus. Wait. P, Q, V. How many letters are in between Q and V? Let's see, Q, R, S, T, U. There are 4 letters. Because it's a triangle and it has 1 less side than a square, we want to find the square root. So 2. 2 * 6 = 12. Wait a minute. 5-12-13 and 9-12-15 are both right triangles. So I just need to find the area of two right triangles. This is fairly simple. The area of a right triangle is the product of its two legs, divided by 2. So the area of 30+54=84 right? No actually. You see, when you consider the transformations of a function, f(x+b) is actually a shift b to the LEFT of f(x). f(x-b) is a shift to the RIGHT of f(x). Wait a minute. There are two right triangles. And shifting to the right is adding NEGATIVE b. So we have to multiply 84 by -1. So the answer is -84. Q.E.D.
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ChickensEatGrass
45 posts
#14
Y by
$13^2+14^2=15^2$ using the Pythagorean $- 140$ Theorem, so $13 \cdot 7=91$.
But, we mustn’t forget to add the $140$ back to make up for lost aura—which I lose every day sob sob sob. Therefore, our final answer is $231$.
This post has been edited 3 times. Last edited by ChickensEatGrass, Apr 9, 2025, 7:23 PM
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ilikemath247365
253 posts
#15
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Remember that $13 + 14 + 15 = 42$. Now, the area is 4 times the perimeter so $4 * 42 = 168$.
This post has been edited 1 time. Last edited by ilikemath247365, Apr 9, 2025, 7:26 PM
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Runner1600
12 posts
#16
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We know that $13=1+1+1+1+1+1+1+1+1+1+1+1+1$, right and $14=1+1+1+1+1+1+1+1+1+1+1+1+1+1$ and $15=1+1+1+1+1+1+1+1+1+1+1+1+1+1+1$. So according to the 5th Symphony by Beethoven, you have to plug in any form of adding 1s into the quadratic formula. So $a=1+1+1+1+1+1+1+1+1+1+1+1+1, b=1+1+1+1+1+1+1+1+1+1+1+1+1+1 c=1+1+1+1+1+1+1+1+1+1+1+1+1+1+1$. Plugging in, we get

$\frac{-(1+1+1+1+1+1+1+1+1+1+1+1+1+1) \pm \sqrt (1+1+1+1+1+1+1+1+1+1+1+1+1+1)^{2}-4(1+1+1+1+1+1+1+1+1+1+1+1+1)(1+1+1+1+1+1+1+1+1+1+1+1+1+1+1)} {2(1+1+1+1+1+1+1+1+1+1+1+1+1)}$

= $\frac{-14 \pm \sqrt (196-780)} {26}$
=$\frac{-7 \pm \sqrt -274} {13}$
So our two solutions are $\frac{-7-\sqrt-274} {13}$ and $\frac{-7+\sqrt-274} {13}$
This post has been edited 2 times. Last edited by Runner1600, Apr 9, 2025, 7:45 PM
Reason: Sillied, forgot about i
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Runner1600
12 posts
#17
Y by
These are the two different areas
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Existing_Human1
212 posts
#18 • 1 Y
Y by EthanNg6
Runner1600 wrote:
We know that $13=1+1+1+1+1+1+1+1+1+1+1+1+1$, right and $14=1+1+1+1+1+1+1+1+1+1+1+1+1+1$ and $15=1+1+1+1+1+1+1+1+1+1+1+1+1+1+1$. So according to the 5th Symphony by Beethoven, you have to plug in any form of adding 1s into the quadratic formula. So $a=1+1+1+1+1+1+1+1+1+1+1+1+1, b=1+1+1+1+1+1+1+1+1+1+1+1+1+1 c=1+1+1+1+1+1+1+1+1+1+1+1+1+1+1$. Plugging in, we get

$\frac{-(1+1+1+1+1+1+1+1+1+1+1+1+1+1) \pm \sqrt (1+1+1+1+1+1+1+1+1+1+1+1+1+1)^{2}-4(1+1+1+1+1+1+1+1+1+1+1+1+1)(1+1+1+1+1+1+1+1+1+1+1+1+1+1+1)} {2(1+1+1+1+1+1+1+1+1+1+1+1+1)}$

= $\frac{-14 \pm \sqrt (196-780)} {26}$ =$\frac{-7 \pm -\sqrt 274} {13}$
So our two solutions are $\frac{-7-\sqrt274} {13}$ and $\frac{-7+\sqrt274} {13}$

Wait, I just realized re-listening to Beethoven's 5th Symphony:

Lyrics:
Da da da da
duhn duhn duhn duhn
My name is Beethoven
And what you should do
Is, whenever you have to find the area of a 13-14-15 triangle, plug the numbers into the quadratic formula
Da da da da
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=
a