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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:
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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
math olympiad question - please help
senboy   0
2 hours ago
I got stumped on this math olympiad question



Some perfect $5^\text{th}$ powers of positive integers have all distinct digits, and some do not. For example, $5^5 = 3125$ has all distinct digits, while $6^5 = 7776$ does not. Show that the maximum number of perfect $5^\text{th}$ powers of positive integers with distinct digits is $89$.

0 replies
senboy
2 hours ago
0 replies
Find min and max
lgx57   0
2 hours ago
Source: Own
$x_1,x_2, \cdots ,x_n\ge 0$,$\displaystyle\sum_{i=1}^n x_i=m$. $k_1,k_2,\cdots,k_n >0$. Find min and max of
$$\sum_{i=1}^n(k_i\prod_{j=1}^i x_j)$$
0 replies
lgx57
2 hours ago
0 replies
Find min
lgx57   0
2 hours ago
Source: Own
$a,b>0$, $a^4+ab+b^4=60$. Find min of
$$4a^2-ab+4b^2$$
$a,b>0$, $a^4-ab+b^4=60$. Find min of
$$4a^2-ab+4b^2$$
0 replies
lgx57
2 hours ago
0 replies
III Lusophon Mathematical Olympiad 2013 - Problem 5
DavidAndrade   2
N 3 hours ago by KTYC
Find all the numbers of $5$ non-zero digits such that deleting consecutively the digit of the left, in each step, we obtain a divisor of the previous number.
2 replies
DavidAndrade
Aug 12, 2013
KTYC
3 hours ago
Mathcount strategies anyone?
Glowworm   6
N 3 hours ago by Glowworm
Does anyone know good MATHCOUNTS strategies for a higher nationals score? Any tips would be appreciated!
6 replies
Glowworm
Apr 9, 2025
Glowworm
3 hours ago
Maximum number of terms in the sequence
orl   11
N 4 hours ago by navier3072
Source: IMO LongList, Vietnam 1, IMO 1977, Day 1, Problem 2
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
11 replies
orl
Nov 12, 2005
navier3072
4 hours ago
USAMO 2003 Problem 1
MithsApprentice   68
N 4 hours ago by Mamadi
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
68 replies
MithsApprentice
Sep 27, 2005
Mamadi
4 hours ago
Number Theory
Vmathwizard3.14   2
N 4 hours ago by aidan0626
Prove for any n an integer that n^5 -n is divisible by 30.

Please answer the question
2 replies
Vmathwizard3.14
4 hours ago
aidan0626
4 hours ago
Centroid, altitudes and medians, and concyclic points
BR1F1SZ   3
N 4 hours ago by EeEeRUT
Source: Austria National MO Part 1 Problem 2
Let $\triangle{ABC}$ be an acute triangle with $BC > AC$. Let $S$ be the centroid of triangle $ABC$ and let $F$ be the foot of the perpendicular from $C$ to side $AB$. The median $CS$ intersects the circumcircle $\gamma$ of triangle $\triangle{ABC}$ at a second point $P$. Let $M$ be the point where $CS$ intersects $AB$. The line $SF$ intersects the circle $\gamma$ at a point $Q$, such that $F$ lies between $S$ and $Q$. Prove that the points $M,P,Q$ and $F$ lie on a circle.

(Karl Czakler)
3 replies
BR1F1SZ
Monday at 9:45 PM
EeEeRUT
4 hours ago
IMO Genre Predictions
ohiorizzler1434   60
N 4 hours ago by Yiyj
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
60 replies
ohiorizzler1434
May 3, 2025
Yiyj
4 hours ago
square root problem
kjhgyuio   5
N 4 hours ago by Solar Plexsus
........
5 replies
kjhgyuio
May 3, 2025
Solar Plexsus
4 hours ago
Diodes and usamons
v_Enhance   47
N 5 hours ago by EeEeRUT
Source: USA December TST for the 56th IMO, by Linus Hamilton
A physicist encounters $2015$ atoms called usamons. Each usamon either has one electron or zero electrons, and the physicist can't tell the difference. The physicist's only tool is a diode. The physicist may connect the diode from any usamon $A$ to any other usamon $B$. (This connection is directed.) When she does so, if usamon $A$ has an electron and usamon $B$ does not, then the electron jumps from $A$ to $B$. In any other case, nothing happens. In addition, the physicist cannot tell whether an electron jumps during any given step. The physicist's goal is to isolate two usamons that she is sure are currently in the same state. Is there any series of diode usage that makes this possible?

Proposed by Linus Hamilton
47 replies
v_Enhance
Dec 17, 2014
EeEeRUT
5 hours ago
9 AMC 10 Prep
bluedino24   28
N 5 hours ago by Craftybutterfly
I'm in 7th grade and thought it would be good to start preparing for the AMC 10. I'm not extremely good at math though.

What are some important topics I should study? Please comment below. Thanks! :D
28 replies
bluedino24
May 2, 2025
Craftybutterfly
5 hours ago
3-var inequality
sqing   1
N 5 hours ago by sqing
Source: Own
Let $ a,b\geq  0 ,a^3-ab+b^3=1  $. Prove that
$$  \frac{1}{2}\geq     \frac{a}{a^2+3 }+ \frac{b}{b^2+3}   \geq  \frac{1}{4}$$$$  \frac{1}{2}\geq     \frac{a}{a^3+3 }+ \frac{b}{b^3+3}   \geq  \frac{1}{4}$$$$  \frac{1}{2}\geq \frac{a}{a^2+ab+2}+ \frac{b}{b^2+ ab+2}  \geq  \frac{1}{3}$$$$  \frac{1}{2}\geq \frac{a}{a^3+ab+2}+ \frac{b}{b^3+ ab+2}  \geq  \frac{1}{3}$$Let $ a,b\geq  0 ,a^3+ab+b^3=3  $. Prove that
$$  \frac{1}{2}\geq     \frac{a}{a^2+3 }+ \frac{b}{b^2+3}   \geq  \frac{1}{4}(\frac{1}{\sqrt[3]{3}}+\sqrt[3]{3}-1)$$$$  \frac{1}{2}\geq     \frac{a}{a^3+3 }+ \frac{b}{b^3+3}   \geq  \frac{1}{2\sqrt[3]{9}}$$$$  \frac{1}{2}\geq \frac{a}{a^2+ab+2}+ \frac{b}{b^2+ ab+2}  \geq  \frac{4\sqrt[3]{3}+3\sqrt[3]{9}-6}{17}$$$$  \frac{1}{2}\geq \frac{a}{a^3+ab+2}+ \frac{b}{b^3+ ab+2}  \geq  \frac{\sqrt[3]{3}}{5}$$
1 reply
sqing
5 hours ago
sqing
5 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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Soupboy0
376 posts
#62 • 1 Y
Y by legospartan
WARNING: WHAT IM ABOUT TO GO THROUGH INVOLVES A LENGTHY COORDBASH. IF ONE IS SENSITIVE TO SUCH CONTENT, I WOULD STRONGLY RECOMMEND TO USE THE 1434 GEO LEMMA INSTEAD. ONE NOT FAMILLIAR WITH THE 1434 GEO LEMMA CAN READ MY EARLIER SOLUTION. PLEASE DO NOT THAT THE 2 FORMULAS USED IN THIS TEXT ARE USEFUL FOR COMPETITIONS LIKE THE AMC14 AND THE AMY.THE AREA MUST FIRST BE FOUND IN TERMS OF THE ARBITRARY VARIABLE $T$. NOTE THAT $T^4 \equiv 1\pmod 4$, AND THE AREA OF THE TRIANGLE CAN BE FOUND USING 1434 GEO LEMMA OR THE METHOD IM ABOUT TO DESCRIBE. FIRST, DEFINE THE SKIBIDI POINT $S(P) = (x, y)$ OF A TRIANGLE WITH A POINT $P$ AT CARTESIAN COORDINATES $(x, y)$ TRANSFORMED TO $(x-y+5, x+y)$, AND THEN THE DISTANCE FROM THE ORIGIN IS FOUND. NOW USE THE chicken jockey theorem TO FIND THAT THE AREA OF THE TRIANGLE IN TERMS OF THE ARBITRARY VALUE $T$ IS $\frac{1}{3}(36-T^3)(T^3+1)$. NOW, WE USE THE SKIBIDI POINT OF OUR TRIANGLE AND LET THE VERTEX OPPOSITE THE SIDE OF LENGTH $15$ REST AT $A=(0, 0)$ SO THAT THE VERTEX OPPOSITE THE SIDE OF LENGTH $14$ IS LOCATED AT $B=(5, 12)$, AND THE LAST VERTEX IS AT $C=(0, 14)$. NOW, WE USE THE chicken jockey lemma WHICH STATES THAT $\text{JACK BLACK'S AGE} + [S(C) \cdot S(B)]^2 - ([S(B)]^2-1])([S(C)]^2-1) - 5S(A)  = T$. THIS ULTIMATELY EVALUATES TO $T=3$, SO PLUGGING $T=3$ INTO OUR EARILER FORMULA YEALDS $\frac{1}{3}(9)(28)=\fbox{84}$. HOWEVER, THE ULTIMATE FAULT OF THE chicken jockey lemma LIES IN THE FACT THAT IF $S(P) = 5$ IN ANY TRIANGLE, THEN WE MUST MULTIPLY BY $\frac{T^5-T-1}{12}$ BECAUSE OF LOST AREA, AND IN OUR TRIANGLE, $A=(0, 0)$ AND $S(A) = (0-0+5, 0+0) = (5, 0) = 5$. THEREFORE, THE ANSWER IS $84 \cdot \frac{243-3-1}{14} = \frac{84 \cdot 239}{14} = 6 \cdot 239 = 1434$. THEREFORE, THIS PROVES THAT JACK BLACK IS STEVE.
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Yiyj1
1266 posts
#63
Y by
What is 1434*9+1434+9
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Yrock
1289 posts
#64
Y by
Using Heron's formula:
$s=\frac{13+14+15}{2}=21$
$A=\sqrt{21\cdot(21-13)\cdot(21-14)\cdot(21-15)}=\sqrt{21\cdot8\cdot7\cdot6}=\sqrt{7\cdot7\cdot3\cdot3\cdot8\cdot2}=\sqrt{(7\cdot3\cdot4)^2}=7\cdot3\cdot4 = \boxed{1434}$.

hehehe
This post has been edited 1 time. Last edited by Yrock, Apr 10, 2025, 3:29 AM
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Soupboy0
376 posts
#65
Y by
Yrock wrote:
Using Heron's formula:
$s=\frac{13+14+15}{2}=21$
$A=\sqrt{21\cdot(21-13)\cdot(21-14)\cdot(21-15)}=\sqrt{21\cdot8\cdot7\cdot6}=\sqrt{7\cdot7\cdot3\cdot3\cdot8\cdot2}=\sqrt{(7\cdot3\cdot4)^2}=7\cdot3\cdot4 = \boxed{1434}$.

hehehe

makes sense although you forgot to use the 1434 geo lemma
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Yrock
1289 posts
#66
Y by
Soupboy0 wrote:
Yrock wrote:
Using Heron's formula:
$s=\frac{13+14+15}{2}=21$
$A=\sqrt{21\cdot(21-13)\cdot(21-14)\cdot(21-15)}=\sqrt{21\cdot8\cdot7\cdot6}=\sqrt{7\cdot7\cdot3\cdot3\cdot8\cdot2}=\sqrt{(7\cdot3\cdot4)^2}=7\cdot3\cdot4 = \boxed{1434}$.

hehehe

makes sense although you forgot to use the 1434 geo lemma

The fact that every number is equal to 1434 is the 1434 trivial lemma.
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maxamc
568 posts
#68
Y by
EaZ_Shadow wrote:
Triangle has base 14, height of 12, so the answer is 84.$\phantom{01}$

EaZ_Shadow is a genius and clearly should win a fields medal for this correct statement.
Attachments:
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HacheB2031
393 posts
#69
Y by
HacheB2031 wrote:
the area is $2$
proof: look at the diagram
[asy]
unitsize(1cm);
pair A = (0,0);
pair B = (2,0);
pair C = (0,2);
draw(A--B--C--cycle);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NW);
label("$14$",A--C,W);
label("$15$",B--C,NE);
label("$13$",A--B,S);
label("$2$",(1,1),2SW);
[/asy]
the diagram is true and stuff so yeah the area is $2$ because the diagram said so

erm actually my diagram is better and proves the area is $3$ and stuff by like the induction hypothesis and the really weird lemma and the proof of the theorem and the because i said so strategy and like a really complicated proof and i bolded this i italiciziaeds edsdedsd this i strikethroughed this i underlined this WHY IS THIS CAPITAL AND STUFF ITS IMPORTANT so yeah my argument is true
[asy]
unitsize(1cm);
pair A = (0,0);
pair B = (2,0);
pair C = (0,3);
draw(A--B--C--cycle);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NW);
label("$14$",A--C,W);
label("$15$",B--C,NE);
label("$13$",A--B,S);
label("$3$",(1,1.5),2SW);
[/asy]
This post has been edited 1 time. Last edited by HacheB2031, Apr 10, 2025, 4:56 AM
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mdk2013
638 posts
#70
Y by
i said this last time and i say it now: if the side lengths are 13, 14, 15 by the pywagorean teorem we get that the mass of the sun is equal to the mass of jupiter * 0.5647
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HacheB2031
393 posts
#71 • 1 Y
Y by legospartan
The 1434 Theorem: Let $f:\mathbb R\to\mathbb R$ be any real valued function. Then \[f(x)=1434.\]
"Because I Said So" Theorem: Let $1$ and $2$ be real numbers such that $1=1$ and $2=2.$ Then \[1=2.\]
Proof: If this wasn't true, then we would die. So, it must be true. Q.E.D.

Proof of 1434: By the "Because I Said So" theorem, the theorem is true. Q.E.D.

Corollary: the area of the triangle is $1434$
This post has been edited 3 times. Last edited by HacheB2031, Apr 10, 2025, 5:02 AM
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jkim0656
990 posts
#72
Y by
side len 13, 14, 15?
seems like a job for the jkim formula!
$\dfrac{\dfrac{13}{14}}{15+13+14}$
= $\dfrac{13}{588}$~
that is our totally correct answer
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whwlqkd
99 posts
#73
Y by
The triangle has length $13cm,14cm,15mm$
So we get $sqrt{14.25\times 0.25\times 1.25 \times 12.75}=\frac{3}{16} \sqrt{1615}$.
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kjhgyuio
59 posts
#74
Y by
MathRook7817 wrote:
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

7
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fruitmonster97
2491 posts
#75 • 2 Y
Y by legospartan, ChaitraliKA
We use the well known area=2*perimeter theorem to get $2(13+14+15)=84$ - oh darn, you want a wrong answer

We can split it into two right triangles. the area of a right triangle is simply the sum of the largest and smallest sides times two. This yields 2(9+15)+2(5+13)=84 - whoops mb

We use bird formula. Birds are cool. the number 6 is also cool. being the middle child is cool. Thus 6 times the middle side 14 equals 84 - oh wow im bad at this.

xooks xonks area=1434-iltg by 1434 geo lemma. basic computation yields i=2(13+14)/3=18. l=15. t=14-13=1. g is the shortest side made from splitting into two pythag triples, or 5. Our answer is 1434-(15)(18)(1)(5)=84 - oops sry

lamelo ball is 6' 7''. he plays like hes short, so this triangle wants to be double 6' 7'', or 2(6)(7)=84 - well this is awkward

the height is 12. thus, ans is 15*14-13*12+15+14+13-12=84 - ok i give up
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Aaronjudgeisgoat
897 posts
#76
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fruitmonster97 wrote:
We use the well known area=2*perimeter theorem to get $2(13+14+15)=84$ - oh darn, you want a wrong answer

We can split it into two right triangles. the area of a right triangle is simply the sum of the largest and smallest sides times two. This yields 2(9+15)+2(5+13)=84 - whoops mb

We use bird formula. Birds are cool. the number 6 is also cool. being the middle child is cool. Thus 6 times the middle side 14 equals 84 - oh wow im bad at this.

xooks xonks area=1434-iltg by 1434 geo lemma. basic computation yields i=2(13+14)/3=18. l=15. t=14-13=1. g is the shortest side made from splitting into two pythag triples, or 5. Our answer is 1434-(15)(18)(1)(5)=84 - oops sry

lamelo ball is 6' 7''. he plays like hes short, so this triangle wants to be double 6' 7'', or 2(6)(7)=84 - well this is awkward

the height is 12. thus, ans is 15*14-13*12+15+14+13-12=84 - ok i give up

larry bird ahh moment (iykyk)
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MathRook7817
677 posts
#77 • 1 Y
Y by ZMB038
The (in)correct answer is:

Area = 3/(1/13+1/14+1/15)
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