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Docked 4 points Help
sadas123   8
N 2 hours ago by bjump
In school we had this beginners like middle school contest, but we had to right down our solution kind of like usajmo except no proofs. It was also graded out of 7 but I got 4 Points docked for this question. what was my problem??? But I kind of had to rush the solution on this question because there was another problem before this that was like 1000x times harder.

Question:The solutions to the equation x^3-13x^2+ax−48=0 are all positive whole numbers. What is $a$?


Solution: We can see that we can use Vieta's formulas to find that the product of the roots is $48$, and the sum of the roots is $13$. So we need to find a combination of integers that multiply to $48$ and add up to $13$. Let's call the roots of the equation p, q, and r. From Vieta's, we get that $p+q+r=-13$ and $pqr = -48$. Looking at the factors of $48$, which is $2^4*3$, we try to split the numbers in a way that gives us the correct sum and product. Trying 3, -2, and -8, we see that they add up to $-13$ and multiply to $-48$, so they work. That means the roots of the polynomial are -3, -2, and -8, and the factorization is $(x-3)(x-2)(x-8)$. Multiplying it out, we get $x^3-13x^2+46x-48$, so we find that a = 46.
8 replies
sadas123
Yesterday at 4:06 PM
bjump
2 hours ago
Cyclic Quad
worthawholebean   131
N 3 hours ago by mathwiz_1207
Source: USAMO 2008 Problem 2
Let $ ABC$ be an acute, scalene triangle, and let $ M$, $ N$, and $ P$ be the midpoints of $ \overline{BC}$, $ \overline{CA}$, and $ \overline{AB}$, respectively. Let the perpendicular bisectors of $ \overline{AB}$ and $ \overline{AC}$ intersect ray $ AM$ in points $ D$ and $ E$ respectively, and let lines $ BD$ and $ CE$ intersect in point $ F$, inside of triangle $ ABC$. Prove that points $ A$, $ N$, $ F$, and $ P$ all lie on one circle.
131 replies
worthawholebean
May 1, 2008
mathwiz_1207
3 hours ago
Jane street swag package? USA(J)MO
arfekete   40
N 4 hours ago by Pengu14
Hey! People are starting to get their swag packages from Jane Street for qualifying for USA(J)MO, and after some initial discussion on what we got, people are getting different things. Out of curiosity, I was wondering how they decide who gets what.
Please enter the following info:

- USAMO or USAJMO
- Grade
- Score
- Award/Medal/HM
- MOP (yes or no, if yes then color)
- List of items you got in your package

I will reply with my info as an example.
40 replies
arfekete
May 7, 2025
Pengu14
4 hours ago
OTIS or MathWOOT 2
math_on_top   4
N 4 hours ago by Pengu14
Hey AoPS community I took MathWOOT 1 this year and scored an 8 on AIME (last year I got a 6). My goal is to make it to MOP next year through USAMO. It's gonna be a lot of work, but do you think that I should do MathWOOT 2 or OTIS? Personally, I felt that MathWOOT 1 taught me a lot but was more focused on computational - not sure how to split computation vs olympiad prep. So, for those who can address this question:

(1) How much compuational vs olympiad
(2) OTIS or MathWOOT 2 and why
4 replies
math_on_top
Yesterday at 9:56 PM
Pengu14
4 hours ago
An FE. Who woulda thunk it?
nikenissan   118
N 5 hours ago by maromex
Source: 2021 USAJMO Problem 1
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for positive integers $a$ and $b,$ \[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]
118 replies
nikenissan
Apr 15, 2021
maromex
5 hours ago
[CASH PRIZES] IndyINTEGIRLS Spring Math Competition
Indy_Integirls   21
N 6 hours ago by Penguin117
[center]IMAGE

Greetings, AoPS! IndyINTEGIRLS will be hosting a virtual math competition on May 25,
2024 from 12 PM to 3 PM EST.
Join other woman-identifying and/or non-binary "STEMinists" in solving problems, socializing, playing games, winning prizes, and more! If you are interested in competing, please register here![/center]

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[center]Important Information[/center]

Eligibility: This competition is open to all woman-identifying and non-binary students in middle and high school. Non-Indiana residents and international students are welcome as well!

Format: There will be a middle school and high school division. In each separate division, there will be an individual round and a team round, where students are grouped into teams of 3-4 and collaboratively solve a set of difficult problems. There will also be a buzzer/countdown/Kahoot-style round, where students from both divisions are grouped together to compete in a MATHCOUNTS-style countdown round! There will be prizes for the top competitors in each division.

Problem Difficulty: Our amazing team of problem writers is working hard to ensure that there will be problems for problem-solvers of all levels! The middle school problems will range from MATHCOUNTS school round to AMC 10 level, while the high school problems will be for more advanced problem-solvers. The team round problems will cover various difficulty levels and are meant to be more difficult, while the countdown/buzzer/Kahoot round questions will be similar to MATHCOUNTS state to MATHCOUNTS Nationals countdown round in difficulty.

Platform: This contest will be held virtually through Zoom. All competitors are required to have their cameras turned on at all times unless they have a reason for otherwise. Proctors and volunteers will be monitoring students at all times to prevent cheating and to create a fair environment for all students.

Prizes: At this moment, prizes are TBD, and more information will be provided and attached to this post as the competition date approaches. Rest assured, IndyINTEGIRLS has historically given out very generous cash prizes, and we intend on maintaining this generosity into our Spring Competition.

Contact & Connect With Us: Follow us on Instagram @indy.integirls, join our Discord, follow us on TikTok @indy.integirls, and email us at indy@integirls.org.

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[center]Help Us Out

Please help us in sharing the news of this competition! Our amazing team of officers has worked very hard to provide this educational opportunity to as many students as possible, and we would appreciate it if you could help us spread the word!
21 replies
Indy_Integirls
May 11, 2025
Penguin117
6 hours ago
Erecting Rectangles
franchester   103
N Yesterday at 9:10 PM by torch
Source: 2021 USAMO Problem 1/2021 USAJMO Problem 2
Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that \[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\]Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.
103 replies
franchester
Apr 15, 2021
torch
Yesterday at 9:10 PM
9 best high school math competitions hosted by a college/university
ethan2011   21
N Yesterday at 7:38 PM by linjiah
I only included college-hosted comps since MAA comps are very differently formatted, and IMO would easily beat the rest on quality since mathematicians around the world give questions, and so many problems are shortlisted, so IMO does release the IMO shortlist for people to practice. I also did not include the not as prestigious ones(like BRUMO, CUBRMC, and others), since most comps with very high quality questions are more prestigious(I did include other if you really think those questions are really good).
21 replies
1 viewing
ethan2011
Apr 12, 2025
linjiah
Yesterday at 7:38 PM
d_k-eja Vu
ihatemath123   50
N Yesterday at 5:57 PM by MathematicalArceus
Source: 2024 USAMO Problem 1
Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n!$ in increasing order as $1 = d_1 < d_2 < \dots < d_k = n!$, then we have
\[ d_2 - d_1 \leq d_3 - d_2 \leq \dots \leq d_k - d_{k-1}. \]
Proposed by Luke Robitaille.
50 replies
ihatemath123
Mar 20, 2024
MathematicalArceus
Yesterday at 5:57 PM
Essentially, how to get good at olympiad math?
gulab_jamun   3
N Yesterday at 5:29 PM by sus_rbo
Ok, so I'm posting this as an anynonymous user cuz I don't want to get flamed by anyone I know for my goals but I really do want to improve on my math skill.

Basically, I'm alright at computational math (10 AIME, dhr stanford math meet twice) and I hope I can get good enough at olympiad math over the summer to make MOP next year (I will be entering 10th as after next year, it becomes much harder :( )) Essentially, I just want to get good at olympiad math. If someone could, please tell me how to study, like what books (currently thinking of doing EGMO) but I don't know how to get better at the other topics. Also, how would I prepare? Like would I study both proof geometry and proof number theory concurrently or just study each topic one by one?? Would I do mock jmo/amo or js prioritize olympiad problems in each topic. I have the whole summer ahead of me, and intend to dedicate it to olympiad math, so any advice would be really appreciated. Thank you!
3 replies
gulab_jamun
Yesterday at 1:53 AM
sus_rbo
Yesterday at 5:29 PM
Olympiad Combinatorics Book
Pascal96   126
N Dec 27, 2023 by zaahir
Hi everyone, I am currently writing a book on combinatorics for people preparing for national and international math competitions, especially the IMO and selection tests leading up to it. The book is intended to expose readers to a variety of ideas, techniques and problem solving strategies, ranging from the intuitive “greedy algorithms” in the first chapter to the powerful Probabilistic Method in chapter nine.
I am uploading chapter one here, and would appreciate your feedback and any suggestions. Over the coming weeks, I will be uploading the remaining chapters one at a time.
The only prerequisites are familiarity with basic graph theoretic concepts and terminology, algebraic inequalities, induction and the pigeonhole principle. Experience with invariants and the extremal principle is also helpful.
EDIT: CHAPTER 9 IS OUT! Since only 3 attachments are allowed per post, I have uploaded chapters 4, 5 and 6 in my comment below (10th on this page), and chapters 7, 8, and 9 further below (comment number 49 on this page).
NOTE: The solution to example 8 in chapter 1 is incorrect, and will be corrected in the final version of the book. For now, ignore this example.

Full book (uploaded by green_dog_7983): Dead Link
[Amir: new link]
126 replies
Pascal96
Aug 6, 2014
zaahir
Dec 27, 2023
Olympiad Combinatorics Book
G H J
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Pascal96
124 posts
#127 • 5 Y
Y by 606234, lneis1, Aimingformygoal, 554183, winniep008hfi
Here is the second handout I created for the training camp, on advanced combinatorial algorithms.
Attachments:
Advanced_Combinatorial_Algorithms.pdf (171kb)
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Pascal96
124 posts
#128 • 1 Y
Y by quirtt
DebayuRMO wrote:
Btw when can we expect the the new beginner friendly chapters that you were planning to write? I know you are really busy but at the same time I'm too excited about this extension so that I can start reading your book.
Thank You

I appreciate the excitement! I have a few rough drafts in the works, but creating the polished final versions takes time unfortunately. I don't want to commit to any particular date at this point. In the mean time, I hope the introductory problem set serves as some solid material for beginners to work through.
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MelonGirl
575 posts
#129 • 2 Y
Y by Pascal96, Mango247
not directly related to last few posts, but I pm-ed you about this sometime back, so I guess I'll post it here as well for anyone who's worked on the problem.
Quote:
On each square of a chessboard is a light which has two states-
on or off. A move consists of choosing a square and changing
the state of the bulbs in that square and in its neighboring
squares (squares that share a side with it). Show that starting
from any configuration we can make finitely many moves to
reach a point where all the bulbs are switched off

This is under the algorithms chapter (2). People have found ways to do this with linear algebra and brute force (similar to the row reduction method here.)

Does anyone know of a purely combinatorial algorithm approach to this problem?
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Pascal96
124 posts
#130
Y by
@MelonGirl no I'm actually not aware of a purely combinatorial solution. For a few of the exercises in the book, I did not know the original source (usually because I saw the problem on some handout at an olympiad training camp). This was one such problem, and I'd actually tried it and hadn't been able to solve it. I figured it would make sense as one of the later exercises in the algorithms chapters, so placed it there without realizing it required linear algebra.
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Physicsknight
642 posts
#131
Y by
I think the linalgb and matrix makes it easier to see the grid rather than a combinatorial approach. You can check the $4$ vectors $(1,1,1,0) \,(1,1,0,1)\,(1,0,1,1)\,(0,1,1,1) $ are independent over $\mathbb{F}_2. $.
Turn the the space of the states of the chessboard into a $64$ dimensional vector space over $\mathbb {F}_2$. The field with $2$ elements $0,1, $ with addition $\pmod{2} $.
The only gruesome task is to prove that $64$ vectors are not linearly independent.
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KAKAAB
23 posts
#132
Y by
Pascal96 wrote:
DebayuRMO wrote:
Btw when can we expect the the new beginner friendly chapters that you were planning to write? I know you are really busy but at the same time I'm too excited about this extension so that I can start reading your book.
Thank You

I appreciate the excitement! I have a few rough drafts in the works, but creating the polished final versions takes time unfortunately. I don't want to commit to any particular date at this point. In the mean time, I hope the introductory problem set serves as some solid material for beginners to work through.

eagerly waiting for it :D
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TheMath_boy
1235 posts
#134
Y by
the drop box link for thefull pdf isn't working
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HoRI_DA_GRe8
598 posts
#135
Y by
Pascal96 wrote:
Hi everyone, I am currently writing a book on combinatorics for people preparing for national and international math competitions, especially the IMO and selection tests leading up to it. The book is intended to expose readers to a variety of ideas, techniques and problem solving strategies, ranging from the intuitive “greedy algorithms” in the first chapter to the powerful Probabilistic Method in chapter nine.
I am uploading chapter one here, and would appreciate your feedback and any suggestions. Over the coming weeks, I will be uploading the remaining chapters one at a time.
The only prerequisites are familiarity with basic graph theoretic concepts and terminology, algebraic inequalities, induction and the pigeonhole principle. Experience with invariants and the extremal principle is also helpful.
EDIT: CHAPTER 9 IS OUT! Since only 3 attachments are allowed per post, I have uploaded chapters 4, 5 and 6 in my comment below (10th on this page), and chapters 7, 8, and 9 further below (comment number 49 on this page).
NOTE: The solution to example 8 in chapter 1 is incorrect, and will be corrected in the final version of the book. For now, ignore this example.

Full book (uploaded by green_dog_7983): Dead Link
[Amir: new link]

Is there any hints/solution key here??
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IceWolf10
1575 posts
#136
Y by
no but there's a discussion forum
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dgrozev
2463 posts
#137
Y by
MelonGirl wrote:
not directly related to last few posts, but I pm-ed you about this sometime back, so I guess I'll post it here as well for anyone who's worked on the problem.
Quote:
On each square of a chessboard is a light which has two states-
on or off. A move consists of choosing a square and changing
the state of the bulbs in that square and in its neighboring
squares (squares that share a side with it). Show that starting
from any configuration we can make finitely many moves to
reach a point where all the bulbs are switched off

This is under the algorithms chapter (2). People have found ways to do this with linear algebra and brute force (similar to the row reduction method here.)

Does anyone know of a purely combinatorial algorithm approach to this problem?
Let the cells of the chessboard be vertices of a graph $G$. Connect any two neighboring vertices (squares). So, you can change the binary state (0,1) of any vertex and all of its neighbors. It does not matter what a graph $G$ is. For any simple graph the same claim holds. It was posted already here, in this forum. Here is a solution using linear algebra. There is a T. Gallai's result saying that the vertices of any graph can be partitioned into two sets $V_1,V_2$ such that the subgraphs induced on $V_1$ and $V_2$ have all vertices of even degrees. It's possible the above problem to be proved as a corollary of Gallai's theorem. In this blog post, the converse approach is shown.
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Karmanka
3 posts
#138 • 3 Y
Y by Mango247, Mango247, Mango247
Is it a mistake in task 5 [Czech and Slovak Republics 1997] after 1st Chapter?
Shouldn't there be only (2n+1)-gon or am I missing something (pls, explain then)?
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StefanSebez
53 posts
#139 • 4 Y
Y by adorefunctionalequation, Mango247, Mango247, Mango247
Karmanka wrote:
Is it a mistake in task 5 [Czech and Slovak Republics 1997] after 1st Chapter?
Shouldn't there be only (2n+1)-gon or am I missing something (pls, explain then)?

Yes, n should be an odd integer
Also here is that problem on aops
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Helixglich
113 posts
#140 • 2 Y
Y by adorefunctionalequation, theSpider
That book is a good one. Fairly friendly to newcomers too :blush:
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Phusy
2 posts
#141
Y by
Thanks alot. I love your style. Wish all the good thing for you <3
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zaahir
6 posts
#142
Y by
When can we expect the beginner version @Pascal96 ?
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