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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.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]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
Inspired by lgx57
sqing   1
N 3 minutes ago by sqing
Source: Own
Let $ a,b>0, a^4+ab+b^4=10  $. Prove that
$$ \sqrt{10}\leq a^2+ab+b^2 \leq 6$$$$ 2\leq a^2-ab+b^2 \leq  \sqrt{10}$$$$  4\sqrt{10}\leq 4a^2+ab+4b^2 \leq18$$$$  12<4a^2-ab+4b^2 \leq14$$
1 reply
1 viewing
sqing
12 minutes ago
sqing
3 minutes ago
polonomials
Ducksohappi   1
N 5 minutes ago by top1vien
$P\in \mathbb{R}[x] $ with even-degree
Prove that there is a non-negative integer k such that
$Q_k(x)=P(x)+P(x+1)+...+P(x+k)$
has no real root
1 reply
Ducksohappi
6 hours ago
top1vien
5 minutes ago
Inspired by Bet667
sqing   3
N 38 minutes ago by sqing
Source: Own
Let $ a,b $ be a real numbers such that $a^3+kab+b^3\ge a^4+b^4.$Prove that
$$1-\sqrt{k+1} \leq  a+b\leq 1+\sqrt{k+1} $$Where $ k\geq 0. $
3 replies
sqing
an hour ago
sqing
38 minutes ago
Geometry marathon
HoRI_DA_GRe8   846
N 42 minutes ago by ItzsleepyXD
Ok so there's been no geo marathon here for more than 2 years,so lets start one,rules remain same.
1st problem.
Let $PQRS$ be a cyclic quadrilateral with $\angle PSR=90°$ and let $H$ and $K$ be the feet of altitudes from $Q$ to the lines $PR$ and $PS$,.Prove $HK$ bisects $QS$.
P.s._eeezy ,try without ss line.
846 replies
1 viewing
HoRI_DA_GRe8
Sep 5, 2021
ItzsleepyXD
42 minutes ago
Find all functions $f$ is strictly increasing : \(\mathbb{R^+}\) \(\rightarrow\)
guramuta   0
an hour ago
Find all functions $f$ is strictly increasing : \(\mathbb{R^+}\) \(\rightarrow\) \(\mathbb{R^+}\) such that:
i) $f(2x)$ \(\geq\) $2f(x)$
ii) $f(f(x)f(y)+x) = f(xf(y)) + f(x) $
0 replies
guramuta
an hour ago
0 replies
Partitioning coprime integers to arithmetic sequences
sevket12   3
N an hour ago by quacksaysduck
Source: 2025 Turkey EGMO TST P3
For a positive integer $n$, let $S_n$ be the set of positive integers that do not exceed $n$ and are coprime to $n$. Define $f(n)$ as the smallest positive integer that allows $S_n$ to be partitioned into $f(n)$ disjoint subsets, each forming an arithmetic progression.

Prove that there exist infinitely many pairs $(a, b)$ satisfying $a, b > 2025$, $a \mid b$, and $f(a) \nmid f(b)$.
3 replies
sevket12
Feb 8, 2025
quacksaysduck
an hour ago
Inspired by Bet667
sqing   3
N an hour ago by sqing
Source: Own
Let $ a,b $ be a real numbers such that $a^2+kab+b^2\ge a^3+b^3.$Prove that$$a+b\leq k+2$$Where $ k\geq 0. $
3 replies
sqing
Tuesday at 2:46 PM
sqing
an hour ago
F has at least n distinct values
nataliaonline75   0
an hour ago

Let $n$ be natural number and $S$ be the set of $n$ distinct natural numbers. Define function $f: S \times S \rightarrow N$ with $f(x,y)=\frac{xy}{(gcd(x,y))^2}$. Prove that $f$ have at least $n$ distinct values.
0 replies
nataliaonline75
an hour ago
0 replies
Junior Balkan Mathematical Olympiad 2020- P4
Lukaluce   11
N an hour ago by MR.1
Source: JBMO 2020
Find all prime numbers $p$ and $q$ such that
$$1 + \frac{p^q - q^p}{p + q}$$is a prime number.

Proposed by Dorlir Ahmeti, Albania
11 replies
Lukaluce
Sep 11, 2020
MR.1
an hour ago
Prove that lines parallel in triangle
jasperE3   5
N 2 hours ago by Thapakazi
Source: Mongolian MO 2007 Grade 11 P1
Let $M$ be the midpoint of the side $BC$ of triangle $ABC$. The bisector of the exterior angle of point $A$ intersects the side $BC$ in $D$. Let the circumcircle of triangle $ADM$ intersect the lines $AB$ and $AC$ in $E$ and $F$ respectively. If the midpoint of $EF$ is $N$, prove that $MN\parallel AD$.
5 replies
jasperE3
Apr 8, 2021
Thapakazi
2 hours ago
JBMO Shortlist 2020 N6
Lukaluce   4
N 2 hours ago by MR.1
Source: JBMO Shortlist 2020
Are there any positive integers $m$ and $n$ satisfying the equation

$m^3 = 9n^4 + 170n^2 + 289$ ?
4 replies
Lukaluce
Jul 4, 2021
MR.1
2 hours ago
Nice concyclicity involving triangle, circle center, and midpoints
Kizaruno   0
2 hours ago
Let triangle ABC be inscribed in a circle with center O. A line d intersects sides AB and AC at points E and D, respectively. Let M, N, and P be the midpoints of segments BD, CE, and DE, respectively. Let Q be the foot of the perpendicular from O to line DE. Prove that the points M, N, P, and Q lie on a circle.

0 replies
Kizaruno
2 hours ago
0 replies
non-perfect square is non-quadratic residue mod some p
SpecialBeing2017   3
N 2 hours ago by ilovemath0402
If $n$ is not a perfect square, then there exists an odd prime $p$ s.t. $n$ is a quadratic non-residue mod $p$.
3 replies
SpecialBeing2017
Apr 14, 2023
ilovemath0402
2 hours ago
Circles tangent at orthocenter
Achillys   62
N 2 hours ago by Rayvhs
Source: APMO 2018 P1
Let $H$ be the orthocenter of the triangle $ABC$. Let $M$ and $N$ be the midpoints of the sides $AB$ and $AC$, respectively. Assume that $H$ lies inside the quadrilateral $BMNC$ and that the circumcircles of triangles $BMH$ and $CNH$ are tangent to each other. The line through $H$ parallel to $BC$ intersects the circumcircles of the triangles $BMH$ and $CNH$ in the points $K$ and $L$, respectively. Let $F$ be the intersection point of $MK$ and $NL$ and let $J$ be the incenter of triangle $MHN$. Prove that $F J = F A$.
62 replies
Achillys
Jun 24, 2018
Rayvhs
2 hours ago
A Handout on Triangle Configurations in Olympiads
i3435   21
N Jul 21, 2024 by kotmhn
Source: My Own
I have finally succeeded in giving the downvote button a purpose.

For my 999th post, over the past few months, I've been slowly compiling a handout briefly covering Triangle Configurations in Olympiad Geometry, or "American" Triangle Configurations in Olympiad Geometry, meant for everyone who would like to learn about triangle geometry in an olympiad setting, and the background knowledge being EGMO.

Olympiad geometry is a subject in which there is a significant amount gained from just being familiar with the configurations. We see AoPSers on this forum talk about things like the "Ex Point", "Iran Lemma", "Humpty and Dumpty Points", "'Median-Incircle Concurrency", etc. What are these? What are the proofs of these things? What are problems that include these? How can I learn the power to insta-kill old problems by citing some "well known" lemma?

In this handout I have tried to answer these questions, by going over many of the common triangle configurations in the olympiad geometry of today. In olympiad culture, there are a lot of colloquialisms, such as all of the names above, and I have tried to use these in the handout. However, you obviously can't cite "Ex Point" as well known, and in that a secondary purpose of the handout is to act like a list of some of what I think are the best proofs of so called "well known" facts, for when they're necessary on an olympiad. I have included 100 problems in roughly increasing order of difficulty at the end, 19 of these having solutions. I hope that these solutions, and the hints that accompany 36 of the problems, show what motivation there might be to the solution of a triangle-configurational geometry problem, once one knows all of the facts about a such configuration.

I would like to give thanks to amar_04, who proofread it without any incentive. amar_04 is really pro and has seen a lot of nice problems, and without their suggestions this handout wouldn't be what it is. I would also like to give thanks to The_Turtle for letting me use this quote, to show what I'd like to try to help solve.
[quote=The_Turtle]
[quote]
everyone knows all of the configs
[/quote]
I feel attacked
[/quote]

Without further ado, here is the link. If you're wondering what the title is all about, the title is just meant to be slightly funny, nothing more. If you see any mistakes, you can put them over here, I'd like to keep all of them organized.

Thanks,
i3435
21 replies
i3435
Feb 18, 2021
kotmhn
Jul 21, 2024
A Handout on Triangle Configurations in Olympiads
G H J
G H BBookmark kLocked kLocked NReply
Source: My Own
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i3435
1350 posts
#1 • 115 Y
Y by Kamran011, JustinLee2017, L567, BaoVn, MP8148, EmilXM, fukano_2, nikenissan, Functional_equation, Eyed, Frestho, amar_04, Blossomstream, mira74, Hamroldt, Aritra12, mueller.25, A-Thought-Of-God, vsamc, Arabian_Math, aops29, Pluto1708, parmenides51, NJOY, Pitagar, ilovepizza2020, somartino, BatyrKHAN, IMOTC, CT17, Abhaysingh2003, itslumi, myh2910, Mathematicsislovely, franzliszt, UKR3IMO, Hemlock, pog, OliverA, CoolCarsOnTheRun, bissue, Inconsistent, zuss77, tenebrine, jacoporizzo, samrocksnature, tigerzhang, Bradygho, new_to_mew_too, centslordm, v4913, 554183, JAnatolGT_00, DofL, agwwtl03, OlympusHero, mathtiger6, tricky.math.spider.gold.1, Jupiter_is_BIG, EpicNumberTheory, Aryan27, FishHeadTail, Flying-Man, BVKRB-, megarnie, guptaamitu1, Bumblebee60, Kamonohashin, cadaeibf, holahello, Siddharth03, crazyeyemoody907, Mogmog8, anurag27826, CyclicISLscelesTrapezoid, sabkx, EpicBird08, MathPerson12321, BorisAngelov1, Luka13, bjump, MathJams, the_mathmagician, Om245, ESAOPS, Rounak_iitr, ApraTrip, OronSH, nguyenducmanh2705, GeoKing, cursed_tangent1434, ddami, solasky, ehuseyinyigit, MathLuis, This_deserves_a_like, Marcus_Zhang, busy-beaver, SilverBlaze_SY, ihategeo_1969, Funcshun840, ZVFrozel, akliu, shanelin-sigma, cosdealfa, MS_asdfgzxcvb, frontlinerbd, Tem8, zhaohm, Rajukian, Sedro, mpcnotnpc, Ramanujan32, cosinesine, farhad.fritl
I have finally succeeded in giving the downvote button a purpose.

For my 999th post, over the past few months, I've been slowly compiling a handout briefly covering Triangle Configurations in Olympiad Geometry, or "American" Triangle Configurations in Olympiad Geometry, meant for everyone who would like to learn about triangle geometry in an olympiad setting, and the background knowledge being EGMO.

Olympiad geometry is a subject in which there is a significant amount gained from just being familiar with the configurations. We see AoPSers on this forum talk about things like the "Ex Point", "Iran Lemma", "Humpty and Dumpty Points", "'Median-Incircle Concurrency", etc. What are these? What are the proofs of these things? What are problems that include these? How can I learn the power to insta-kill old problems by citing some "well known" lemma?

In this handout I have tried to answer these questions, by going over many of the common triangle configurations in the olympiad geometry of today. In olympiad culture, there are a lot of colloquialisms, such as all of the names above, and I have tried to use these in the handout. However, you obviously can't cite "Ex Point" as well known, and in that a secondary purpose of the handout is to act like a list of some of what I think are the best proofs of so called "well known" facts, for when they're necessary on an olympiad. I have included 100 problems in roughly increasing order of difficulty at the end, 19 of these having solutions. I hope that these solutions, and the hints that accompany 36 of the problems, show what motivation there might be to the solution of a triangle-configurational geometry problem, once one knows all of the facts about a such configuration.

I would like to give thanks to amar_04, who proofread it without any incentive. amar_04 is really pro and has seen a lot of nice problems, and without their suggestions this handout wouldn't be what it is. I would also like to give thanks to The_Turtle for letting me use this quote, to show what I'd like to try to help solve.
The_Turtle wrote:
Quote:
everyone knows all of the configs
I feel attacked

Without further ado, here is the link. If you're wondering what the title is all about, the title is just meant to be slightly funny, nothing more. If you see any mistakes, you can put them over here, I'd like to keep all of them organized.

Thanks,
i3435
Z K Y
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JustinLee2017
1703 posts
#2 • 2 Y
Y by amar_04, samrocksnature
Nice! Thank you :)
Z K Y
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L567
1184 posts
#3 • 5 Y
Y by amar_04, samrocksnature, Mango247, Mango247, Mango247
Wow, this is really good! Thanks a lot!! :-D
Z K Y
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Hamroldt
744 posts
#4 • 3 Y
Y by amar_04, MrOreoJuice, samrocksnature
Thank you ! This is going to help for the INMO !
Z K Y
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nikaaryu
55 posts
#5 • 2 Y
Y by amar_04, samrocksnature
Nice! Thanks! :)
Z K Y
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Arabian_Math
86 posts
#6 • 2 Y
Y by amar_04, samrocksnature
Wonderful! :)
Z K Y
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justin6688
840 posts
#7 • 3 Y
Y by amar_04, samrocksnature, Bradygho
Nice thanks a lot!
Z K Y
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i3435
1350 posts
#9 • 5 Y
Y by Functional_equation, amar_04, Aritra12, samrocksnature, Mango247
There's no other name. I also said that I try to use olympiad colloquialisms, but thanks for saying that the names I use are the standard.

@3below I know, I've read your original blog post on it. It's quite nice.
This post has been edited 1 time. Last edited by i3435, Feb 20, 2021, 5:56 PM
Z K Y
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parmenides51
30651 posts
#10 • 2 Y
Y by amar_04, samrocksnature
this is great
Z K Y
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MrOreoJuice
594 posts
#11 • 5 Y
Y by amar_04, samrocksnature, Mango247, Mango247, Mango247
Amazing!
Z K Y
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aops29
452 posts
#12 • 3 Y
Y by amar_04, samrocksnature, Funcshun840
i3435 wrote:
There's no other name. I also said that I try to use olympiad colloquialisms, but thanks for saying that the names I use are the standard.

I was the one who coined that name lol
Sharky -> refers to user SHARKYKESA
devil -> refers to my past Discord username.
Z K Y
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508669
1040 posts
#13 • 1 Y
Y by samrocksnature
aops29 wrote:
i3435 wrote:
There's no other name. I also said that I try to use olympiad colloquialisms, but thanks for saying that the names I use are the standard.

I was the one who coined that name lol
Sharky -> refers to user SHARKYKESA
devil -> refers to my past Discord username.

The origin does not match what you said in your blog

Very nice handout!
Z K Y
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Abhaysingh2003
222 posts
#14 • 1 Y
Y by samrocksnature
Beautiful Handout! You are great! :P .
This post has been edited 2 times. Last edited by Abhaysingh2003, Feb 23, 2021, 10:33 PM
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i3435
1350 posts
#15
Y by
Someone asked for PDF files, due to AOPS' restriction I split it into 5 separate files.
Attachments:
_Muricaaaaaaa-21-39.pdf (451kb)
_Muricaaaaaaa-1-20.pdf (374kb)
_Muricaaaaaaa-40-58.pdf (476kb)
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i3435
1350 posts
#16
Y by
Here are the other 2.
Attachments:
_Muricaaaaaaa-78-96.pdf (488kb)
_Muricaaaaaaa-59-77.pdf (340kb)
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guptaamitu1
656 posts
#18
Y by
Hi all. Lets collect the AoPS link of as many practice problems as we can here. Few reasons:
  • It is sometimes hard to a problem on AoPS. So if all links are at one place, it will save a lot of time of everyone.
  • This is neither time consuming: whenever we look at a problem on AoPS we can post it's link here too.
  • We can also particularly specify problems to which we cannot find link, so that if someone know where that particular problem is, then he can tell.
  • Lastly, we can find problems from AoPS mock contest at : https://imogeometry.blogspot.com/p/geometry-olympiads.html
I will post link of as many problems as I can in this post (I will keep editing the post). I also request i3435 to put these links in the first post of this topic too.

So lets start!

links
Problems to which I cannot find link:
This post has been edited 3 times. Last edited by guptaamitu1, Feb 11, 2022, 4:28 PM
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parmenides51
30651 posts
#19
Y by
guptaamitu1 wrote:
Hi all. Lets collect the AoPS link of as many practice problems as we can here. Few reasons:
  • It is sometimes hard to a problem on AoPS. So if all links are at one place, it will save a lot of time of everyone.
  • This is neither time consuming: whenever we look at a problem on AoPS we can post it's link here too.
  • We can also particularly specify problems to which we cannot find link, so that if someone know where that particular problem is, then he can tell.
  • Lastly, we can find problems from AoPS mock contest at : https://imogeometry.blogspot.com/p/geometry-olympiads.html
I will post link of as many problems as I can in this post (I will keep editing the post). I also request i3435 to put these links in the first post of this topic too.
a more recent list of geo mocks than the link in my webpage mentioned above, awaits you here
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crazyeyemoody907
450 posts
#20
Y by
so true- there are less configs than people think there are. in today's contests, these configs are generally avoided anyway . . .
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v4913
1650 posts
#21 • 2 Y
Y by crazyeyemoody907, SatisfiedMagma
when this handout gets featured on evan chen's new geo slang handout :love:
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MathJams
3229 posts
#22 • 18 Y
Y by GrantStar, CyclicISLscelesTrapezoid, parmenides51, The_Great_Learner, crazyeyemoody907, v_Enhance, L567, v4913, ihatemath123, superagh, EpicBird08, Upwgs_2008, pog, OronSH, GeoKing, qwerty123456asdfgzxcvb, Funcshun840, SatisfiedMagma
I've compiled a solutions manual for all the problems in this handout! I'm kind of slow at latex so these solutions are digitally handwritten but every problem has a diagram and I tried to be neat :P All of the solutions also fit on 1 page so they're on the shorter side (hopefully also simple enough). If you have any questions about any of the solutions, find a mistake, or need clarification on some handwriting pm me! I know that some of the sources are hard to find so I hope this is helpful!
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John_Mgr
69 posts
#23 • 1 Y
Y by GeoKing
Thanks a lot!!
This post has been edited 1 time. Last edited by John_Mgr, Jun 17, 2024, 4:54 AM
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kotmhn
60 posts
#24
Y by
thanks a lot !!
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