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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies
jlacosta
Mar 2, 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
MAA messed up the order(n)
skipiano   199
N 10 minutes ago by littlefox_amc
Source: 2017 USAJMO #1/USAMO #1
Prove that there are infinitely many distinct pairs $(a, b)$ of relatively prime integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.
199 replies
skipiano
Apr 19, 2017
littlefox_amc
10 minutes ago
Looks Like Mount Inequality Erupted :(
jasonhu4   158
N 12 minutes ago by alexanderhamilton124
Source: 2017 USAMO #6
Find the minimum possible value of \[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}\]given that $a$, $b$, $c$, $d$ are nonnegative real numbers such that $a+b+c+d=4$.

Proposed by Titu Andreescu
158 replies
jasonhu4
Apr 20, 2017
alexanderhamilton124
12 minutes ago
Functional Equations
kootrapali   106
N 20 minutes ago by scannose
Source: 2019 USAJMO 2, by Ankan
Let $\mathbb{Z}$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f \colon \mathbb{Z}\rightarrow \mathbb{Z}$ and $g \colon \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying
\[ f(g(x))=x+a \quad\text{and}\quad g(f(x))=x+b \]for all integers $x$.

Proposed by Ankan Bhattacharya
106 replies
kootrapali
Apr 17, 2019
scannose
20 minutes ago
Inequality
anhduy98   5
N 33 minutes ago by JK1603JK
Source: Own
Given three real numbers $   a,b,c\ge 0   $ satisfying $:   a+b+c=3   $.Prove that:
$$\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}+\sqrt{c^2-ca+a^2}\ge 3+\frac{a^2+b^2+c^2-3abc}{3}.$$
5 replies
anhduy98
Oct 28, 2024
JK1603JK
33 minutes ago
Nice and easy FE on R+
sttsmet   22
N 43 minutes ago by jasperE3
Source: EMC 2024 Problem 4, Seniors
Find all functions $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x+yf(x)) = xf(1+y)$
for all x, y positive reals.
22 replies
sttsmet
Dec 23, 2024
jasperE3
43 minutes ago
p^k divides term of sequence
KevinYang2.71   33
N an hour ago by jcoons91
Source: USAJMO 2024/3
Let $a(n)$ be the sequence defined by $a(1)=2$ and $a(n+1)=(a(n))^{n+1}-1$ for each integer $n\geq 1$. Suppose that $p>2$ is a prime and $k$ is a positive integer. Prove that some term of the sequence $a(n)$ is divisible by $p^k$.

Proposed by John Berman
33 replies
1 viewing
KevinYang2.71
Mar 20, 2024
jcoons91
an hour ago
Existence of AP of interesting integers
DVDthe1st   33
N an hour ago by tchange7575
Source: 2018 China TST Day 1 Q2
A number $n$ is interesting if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
33 replies
1 viewing
DVDthe1st
Jan 2, 2018
tchange7575
an hour ago
A diophantine equation
crazyfehmy   13
N an hour ago by Primeniyazidayi
Source: Turkey Junior National Olympiad 2012 P1
Let $x, y$ be integers and $p$ be a prime for which

\[ x^2-3xy+p^2y^2=12p \]
Find all triples $(x,y,p)$.
13 replies
crazyfehmy
Dec 12, 2012
Primeniyazidayi
an hour ago
nf(f(n)) = f(n)^2, f : N->N
Zhero   19
N an hour ago by HamstPan38825
Source: ELMO Shortlist 2010, A1; also ELMO #4
Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$.

Carl Lian and Brian Hamrick.
19 replies
Zhero
Jul 5, 2012
HamstPan38825
an hour ago
RMM 2019 Problem 2
math90   77
N 2 hours ago by ihatemath123
Source: RMM 2019
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.

Jakob Jurij Snoj, Slovenia
77 replies
math90
Feb 23, 2019
ihatemath123
2 hours ago
Two circles concur on a line
math154   59
N 2 hours ago by Mathandski
Source: ELMO Shortlist 2012, G1; also ELMO #1
In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$.

Ray Li.
59 replies
math154
Jul 2, 2012
Mathandski
2 hours ago
functional equation
Anni   7
N 2 hours ago by HamstPan38825
Source: albanian TST 2008 bmo
Find all functions $f: \mathbb R \to \mathbb R$ such that
\[ f(x+f(y))=y+f(x+1),\]for all $x,y \in \mathbb R$.
7 replies
Anni
May 24, 2009
HamstPan38825
2 hours ago
"pseudo-Fibonnaci" sequence
pohoatza   11
N 2 hours ago by asdf334
Source: IMO Shortlist 2006, Algebra 3
The sequence $c_{0}, c_{1}, . . . , c_{n}, . . .$ is defined by $c_{0}= 1, c_{1}= 0$, and $c_{n+2}= c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\textstyle\sum_{j \in J}{c_{j}}$, $y=\textstyle\sum_{j \in J}{c_{j-1}}$. Prove that there exist real numbers $\alpha$, $\beta$, and $M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality \[m < \alpha x+\beta y < M\] if and only if $(x, y) \in S$.

Remark: A sum over the elements of the empty set is assumed to be $0$.
11 replies
pohoatza
Jun 28, 2007
asdf334
2 hours ago
IMO ShortList 2001, algebra problem 4
orl   35
N 2 hours ago by HamstPan38825
Source: IMO ShortList 2001, algebra problem 4
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfying \[
f(xy)(f(x) - f(y)) = (x-y)f(x)f(y)
\] for all $x,y$.
35 replies
orl
Sep 30, 2004
HamstPan38825
2 hours ago
Mock AMC 1/12
Lord.of.AMC   28
N Jan 16, 2021 by firebolt360
Hello everyone! My friend and I created a mock AMC 10, and it is almost ready. Naturally, we would need some proofreaders for the test, as we might have made some errors. Also, we are planning to also make a mock AMC 12, with some harder problems (in particular, #25 and #23, as well as some others; I will tell you in detail when you become a helper) in addition to around 12 of the same problems on the AMC 10. We would like some helpers to help us finish that. If you are interested, post on this thread (so we can keep an active list). (Note: If you help with the test, you will not be allowed to participate.)

However, if you wish to take the test, please choose only ONE contest (10 or 12) to take as there will be problems that are replicas. We will decide testing dates a little later when we are done with the test.

So, Happy New Year everyone, and thanks in advance for participating!

Helpers [4]

$\uparrow$
Copy this list.

EDIT: From now on, don't pm me saying which test you want to take. Say that when you submit solutions (e.g. put your first sentence saying "This is for the AMC 12 test" or something like that).

EDIT2: There is no longer any limit on the number of helpers. However, in order to be a helper, you can not be a participant, as I mentioned before.
28 replies
Lord.of.AMC
Jan 5, 2012
firebolt360
Jan 16, 2021
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Lord.of.AMC
1210 posts
#1 • 10 Y
Y by MathTwo, rdj5933mile5, dft, NewAlbionAcademy, Adventure10, and 5 other users
Hello everyone! My friend and I created a mock AMC 10, and it is almost ready. Naturally, we would need some proofreaders for the test, as we might have made some errors. Also, we are planning to also make a mock AMC 12, with some harder problems (in particular, #25 and #23, as well as some others; I will tell you in detail when you become a helper) in addition to around 12 of the same problems on the AMC 10. We would like some helpers to help us finish that. If you are interested, post on this thread (so we can keep an active list). (Note: If you help with the test, you will not be allowed to participate.)

However, if you wish to take the test, please choose only ONE contest (10 or 12) to take as there will be problems that are replicas. We will decide testing dates a little later when we are done with the test.

So, Happy New Year everyone, and thanks in advance for participating!

Helpers [4]

$\uparrow$
Copy this list.

EDIT: From now on, don't pm me saying which test you want to take. Say that when you submit solutions (e.g. put your first sentence saying "This is for the AMC 12 test" or something like that).

EDIT2: There is no longer any limit on the number of helpers. However, in order to be a helper, you can not be a participant, as I mentioned before.
This post has been edited 3 times. Last edited by Lord.of.AMC, Jan 9, 2012, 10:38 PM
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Eagle_luvMath
15 posts
#2 • 2 Y
Y by Lord.of.AMC, Adventure10
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mathwizarddude
1976 posts
#3 • 2 Y
Y by Adventure10, Mango247
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engarde
250 posts
#4 • 1 Y
Y by Adventure10
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esque
299 posts
#5 • 2 Y
Y by Adventure10, Mango247
When will the test be released for participants?
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Lord.of.AMC
1210 posts
#6 • 2 Y
Y by Adventure10, Mango247
They will be released soon; at least a week before the A date.

Here are the rules for the competition (i.e. the first page of the AMC 10 test).
Attachments:
mock amc 112 rules.pdf (36kb)
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exmath89
2572 posts
#7 • 2 Y
Y by Adventure10, Mango247
@Lord.of.AMC, will the AMC 10 test be released soon?

Thanks.
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Lord.of.AMC
1210 posts
#8 • 1 Y
Y by Adventure10
[edit: deleted]
This post has been edited 1 time. Last edited by Lord.of.AMC, Feb 27, 2013, 2:57 AM
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jeff10
1117 posts
#9 • 3 Y
Y by osmosis92, Adventure10, Mango247
I want to sign up too!
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Lord.of.AMC
1210 posts
#10 • 10 Y
Y by mymathboy, bulldog23, exmath89, NewAlbionAcademy, greatwhiteshark98, MathLearner01, Adventure10, Mango247, and 2 other users
Sorry, sorry, never mind my previous post! We will be releasing the AMC 10 test after all. Here it is!

It would be best to take this test before the B date. However, you may take it at any time you want, and just submit to me for grading, and I'll give you a score!
Attachments:
mock amc 10 1211.pdf (127kb)
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Lord.of.AMC
1210 posts
#11 • 2 Y
Y by Adventure10, Mango247
Change the answer choices of #19 to:

(A) 243 (B) 342 (C) 486 (D) 684 (E) 1026
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michael_moore
18 posts
#12 • 1 Y
Y by Adventure10
this was awesome, thank you so much for going to the trouble of making this mock amc, you guys should make more!!!!
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MathTwo
541 posts
#13 • 1 Y
Y by Adventure10
michael_moore wrote:
this was awesome, thank you so much for going to the trouble of making this mock amc, you guys should make more!!!!

I'm glad you liked the test (I was one of the test writers too). However, due to the fact that the AMC 10/12 B is in 3 days, there won't be enough time to write another test, so that will probably have to wait until next year.
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niraekjs
1861 posts
#14 • 2 Y
Y by Adventure10, Mango247
Awesome test, thanks!
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Lord.of.AMC
1210 posts
#15 • 1 Y
Y by Adventure10
There was an error in the grading. Everyone I have graded up to now, please resubmit your answer to #14. I have realized that my answer to #14 was wrong, and I was grading everyone against that wrong answer. Sorry for any inconvenience this may cause!
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Lord.of.AMC
1210 posts
#16 • 2 Y
Y by Adventure10, Mango247
Hey everyone,

The AMC 10B is now over, so I think it would be a good idea to end the official period of this mock contest here. However, you may take this test as many times as you want (as it'll remain up), and check your answers against the answer key posted below.

Answer Key

Just so you'll know, here are the statistics.

# submissions = 11
Mean score = 108
Std. Dev. = 17.493
Max. score = 136.5
Median score = 108
Mode score = 96

Top 4 scorers:
1. SteinsChaos with a 136.5
2. NewAlbionAcademy with a 132
3. exmath89 with a 124.5
4. esque with a 115.5

Congratulations to all who took this test!!!
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AplayerDanny
257 posts
#17 • 2 Y
Y by Adventure10, Mango247
How do you solve Question 25?
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tenniskidperson3
2376 posts
#18 • 2 Y
Y by Adventure10, Mango247
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Edit: Fixed a huge error
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newchie123
412 posts
#19 • 1 Y
Y by Adventure10
Solution for 16 and 17 please
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tenniskidperson3
2376 posts
#20 • 1 Y
Y by Adventure10
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Willingo
149 posts
#21 • 1 Y
Y by Adventure10
sol for 8 and 13 please :D
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tenniskidperson3
2376 posts
#22 • 2 Y
Y by Adventure10, Mango247
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Willingo
149 posts
#23 • 2 Y
Y by Adventure10, Mango247
Solutions for 8 , 13 and 15 please :D
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tenniskidperson3
2376 posts
#24 • 2 Y
Y by Adventure10, Mango247
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Willingo
149 posts
#25 • 2 Y
Y by Adventure10, Mango247
How did you conclude M was circumcenter, and why is the big triangle two times PTC the formed triangle(extendeD)

Also for the question where you used vietas, i understand how you got the m-n to the fourth, but why did you add the part after with like 4mn and etc, in the beginning.
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tenniskidperson3
2376 posts
#26 • 2 Y
Y by Adventure10, Mango247
We know that $M$ is the circumcenter because if we draw the circumcircle, since $\angle ACB$ is right, we know that $AB$ is a diameter. Thus the circumcenter is halfway between the endpoints of this diameter, at the midpoint of $A$ and $B$, which is $M$.

We saw that $AP$ is half of $AC$ because triangle $APN$ is similar to triangle $ACM$ with ratio $\frac{1}{2}$. Hence $CP$ is also half of $AC$, just the other half. We showed that $\triangle PTC\sim\triangle ABC$ by AA similarity, so to find the ratio we compare two corresponding sides. In this case we look at $PC$ and $AC$. The ratio of these sides is $\frac{1}{2}$ so the ratio of the triangles is $\frac{1}{2}$.

What I did for the Vieta question is I tried to write $-m^4-n^4$ as a combination of $m+n$ and $mn$. So I used $-(m+n)^4$ to knock out the $-m^4-n^4$, but that leaves me with some extra terms that we must add back in to cancel. These terms are exactly $4m^3n+6m^2n^2+4mn^3$. If we add these to $-(m+n)^4$ they cancel to give $-m^4-n^4$. Then we calculate it from there.

I hope this helps.
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droid347
2679 posts
#27 • 1 Y
Y by Adventure10
Are there any solutions for the last 5 problems?
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gamjawon
3496 posts
#28 • 2 Y
Y by Adventure10, Mango247
Thanks for the test again!
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firebolt360
903 posts
#29
Y by
Huge revive anyone have solutions for 19-23?
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