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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 AMC 10/12 A&B Coming up Soon!
jlacosta   0
Nov 1, 2024
There is still time to train for the November 6th and November 12th AMC 10A/12A and AMC 10B/12B, respectively! Enroll in our weekend seminars to be held on November 2nd and 3rd (listed below) and you will learn problem strategies, test taking techniques, and be able to take a full practice test! Note that the “B” seminars will have different material from the “A” seminars which were held in October.

[list][*]Special AMC 10 Problem Seminar B
[*]Special AMC 12 Problem Seminar B[/list]
For those who want to take a free practice test before the AMC 10/12 competitions, you can simulate a real competition experience by following this link. As you assess your performance on these exams, be sure to gather data!

[list][*]Which problems did you get right?
[list][*]Was the topic a strength (e.g. number theory, geometry, counting/probability, algebra)?
[*]How did you prepare?
[*]What was your confidence level?[/list]
[*]Which problems did you get wrong?
[list][list][*]Did you make an arithmetic error?
[*]Did you misread the problem?
[*]Did you have the foundational knowledge for the problem?
[*]Which topics require more fluency through practice (e.g. number theory, geometry, counting/probability, algebra)?
[*]Did you run out of time?[/list][/list]
Once you have analyzed the results with the above questions, you will have a plan of attack for future contests! BEST OF LUCK to all competitors at this year’s AMC 10 and AMC 12!

Did you know that the day after both the AMC 10A/12A and AMC 10B/12B you can join a free math jam where our AoPS team will go over the most interesting problems? Find the schedule below under “Mark your calendars”.

Mark your calendars for these upcoming free math jams!
[list][*]November 20th: Amherst College Info Session, 7:30 pm ET: Matt McGann, Dean of Admission and Financial Aid at Amherst College, and Nathan Pflueger, math professor at Amherst College, will host an info session exploring both Amherst College specifically and liberal arts colleges generally. Topics include opportunities in math, the admission process, and financial aid for both US and international students.
[*]November 7th: 2024 AMC 10/12 A Discussion, Thursday, 7:30 pm ET:
[*]AoPS instructors will discuss problems from the AMC 10/12 A, administered November 6. We will discuss some of the most interesting problems from each test!
[*]November 13th: 2024 AMC 10/12 B Discussion, Wednesday, 7:30 pm ET:
[*]AoPS instructors will discuss problems from the AMC 10/12 B, administered November 12. We will discuss some of the most interesting problems from each test![/list]
AoPS Spring classes are open for enrollment. Get a jump on the New Year and enroll in our math, contest prep, coding, and science classes today! Need help finding the right plan for your goals? Check out our recommendations page!

Don’t forget: Highlight your AoPS Education on LinkedIn!
Many of you are beginning to build your education and achievements history on LinkedIn. Now, you can showcase your courses from Art of Problem Solving (AoPS) directly on your LinkedIn profile!

Whether you've taken our classes at AoPS Online or AoPS Academies or reached the top echelons of our competition training with our Worldwide Online Olympiad Training (WOOT) program, you can now add your AoPS experience to the education section on your LinkedIn profile.

Don't miss this opportunity to stand out and connect with fellow problem-solvers in the professional world and be sure to follow us at: https://www.linkedin.com/school/art-of-problem-solving/mycompany/ Check out our job postings, too, if you are interested in either full-time, part-time, or internship opportunities!

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
Nov 1, 2024
0 replies
Logical guessing game!
Mathdreams   22
N an hour ago by JH_K2IMO
Source: 2021 Fall AMC10B P10
Fourty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat?

$\textbf{(A) }27\qquad\textbf{(B) }37\qquad\textbf{(C) }47\qquad\textbf{(D) }57\qquad\textbf{(E) }67$
22 replies
Mathdreams
Nov 17, 2021
JH_K2IMO
an hour ago
Possibility of USAMO?
MathXplorer10   4
N an hour ago by MathXplorer10
Hi guys!


I got a 118.5 on the 12B test this year. I am wondering if it is possible to make USAMO (what do you think the cutoffs would be this year?)

For some background, I got 121.5/127.5 on the 10s last year, and got a 7 on AIME with no extra prep. Is it possible to go from a 7 to a 10 (or whatever I need to get on AIME)?

Thank you!
4 replies
MathXplorer10
4 hours ago
MathXplorer10
an hour ago
10a vs 10b
golden_star_123   111
N an hour ago by happyfish0922
Post the difference between your 10a and 10b score!
111 replies
golden_star_123
Wednesday at 6:24 PM
happyfish0922
an hour ago
What do next?
FuturePanda   2
N an hour ago by Tem8
Hi everyone,

I think I got an 81 and 102 for 12A and 10B, sillying way too much on both. I read all of the AOPS books, and I know most of the theorems for the AMC’s I just don’t know which ones to apply to solve the problems. Additionally, I suck at trig, complex, and logarithms. What should I do to improve?

For example, should I be grinding past AIME’s?
I plan on reading most of the Awesomemath books for L3
2 replies
FuturePanda
3 hours ago
Tem8
an hour ago
No more topics!
complex quadrilateral
StressedPineapple   15
N Yesterday at 1:39 AM by EaZ_Shadow
Source: 2024 AMC 12B #12
Suppose $z$ is a complex number with positive imaginary part, with real part greater than $1$, and with $|z| = 2$. In the complex plane, the four points $0$, $z$, $z^{2}$, and $z^{3}$ are the vertices of a quadrilateral with area $15$. What is the imaginary part of $z$?

$\textbf{(A)}~\displaystyle\frac{3}{4}\qquad\textbf{(B)}~1\qquad\textbf{(C)}~\displaystyle\frac{4}{3}\qquad\textbf{(D)}~\displaystyle\frac{3}{2}\qquad\textbf{(E)}~\displaystyle\frac{5}{3}$
15 replies
StressedPineapple
Wednesday at 5:20 PM
EaZ_Shadow
Yesterday at 1:39 AM
complex quadrilateral
G H J
Source: 2024 AMC 12B #12
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StressedPineapple
3 posts
#1
Y by
Suppose $z$ is a complex number with positive imaginary part, with real part greater than $1$, and with $|z| = 2$. In the complex plane, the four points $0$, $z$, $z^{2}$, and $z^{3}$ are the vertices of a quadrilateral with area $15$. What is the imaginary part of $z$?

$\textbf{(A)}~\displaystyle\frac{3}{4}\qquad\textbf{(B)}~1\qquad\textbf{(C)}~\displaystyle\frac{4}{3}\qquad\textbf{(D)}~\displaystyle\frac{3}{2}\qquad\textbf{(E)}~\displaystyle\frac{5}{3}$
This post has been edited 1 time. Last edited by jlacosta, Wednesday at 6:08 PM
Reason: adjusted to official wording
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HonestCat
947 posts
#2
Y by
StressedPineapple wrote:
Let $z$ be a complex number with real part greater than $1$ and $|z| = 2$. In the complex plane, the four points $0$, $z$, $z^{2}$, and $z^{3}$ are the vertices of a quadrilateral with area $15$. What is the imaginary part of $z$?

$\textbf{(A)}~\displaystyle\frac{3}{4}\qquad\textbf{(B)}~1\qquad\textbf{(C)}~\displaystyle\frac{4}{3}\qquad\textbf{(D)}~\displaystyle\frac{3}{2}\qquad\textbf{(E)}~\displaystyle\frac{5}{3}$

3/2
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Bluesoul
756 posts
#3 • 1 Y
Y by Sedro
$\frac{1}{2}(8\sin(\theta)+32\sin(\theta))=15, \sin(\theta)=\frac{3}{4}$, the imaginary part is $2\sin(\theta)=\frac{3}{2}$
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samrocksnature
8781 posts
#4 • 1 Y
Y by aidensharp
NOOOOOOOOOO
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Michw08
48 posts
#5
Y by
I couldn't get this during the test and then headsolved it while I was going home :oops_sign:
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OronSH
1623 posts
#6 • 2 Y
Y by David_He, ivyshine13
best geometry problem on the test ngl
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plang2008
271 posts
#7
Y by
i skipped this at first becuase i thought imaginary part 3/2 implied real part < 1 oops
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mathprodigy2011
57 posts
#8
Y by
I skipped at first but question is trivial once you draw a coordinate plane
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Shreyasharma
589 posts
#9
Y by
This was super creative, probably my favorite problem on the test.
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Sedro
5739 posts
#10 • 1 Y
Y by sami1618
I loved this problem. Let $z = 2e^{i\theta}$. Then, the area of the quadrilateral is $20\sin\theta$ and thus $\sin \theta = \tfrac{3}{4}$. We have $\text{Im}(z) = 2\sin\theta = \tfrac{3}{2}$ for answer choice D.
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LostDreams
124 posts
#11 • 1 Y
Y by Jack_w
This was one of the problems of AMC
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john0512
4142 posts
#12 • 4 Y
Y by pog, Jack_w, ivyshine13, centslordm
By spiral similarity, $0,z^2,z^3$ has four times the area of $0,z,z^2$, so $0,z,z^2$ has area $3$. Now $\frac{1}{2}(2)(4)\sin\theta=3$ so $\sin\theta=\frac{3}{4}$, so the answer is $2\cdot \frac{3}{4}=\frac{3}{2}$.

So clean and conceptual. why can't more amc problems be like this
this is why for computational i main college comps and arml now
This post has been edited 1 time. Last edited by john0512, Wednesday at 7:24 PM
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apotosaurus
75 posts
#13
Y by
I did not cook...
Due to the given conditions, we can express this as the sum of the areas of triangles $0,z,z^2$ and $0,z^2,z^3$. By Complex Shoelace, this is\[\frac i4 \left(\begin{vmatrix} z & \frac 4z & 1 \\ z^2 & \frac{16}{z^2} & 1 \\ 0 & 0 & 1 \end{vmatrix} + \begin{vmatrix} z^2 & \frac {16}{z^2} & 1 \\ z^3 & \frac{64}{z^3} & 1 \\ 0 & 0 & 1 \end{vmatrix}\right) =\frac i4 \left( \frac{16}{z}-4z+\frac{64}{z}-16z\right) = \frac i4 \left(20\bar z - 20z\right) = 10 \mathrm{Im}(z),\]so the answer is $\boxed{\frac 32}$.
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clarkculus
126 posts
#14 • 1 Y
Y by centslordm
NOOOOO I FORGOT TO MULTIPLY BY TWO
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EaZ_Shadow
279 posts
#15
Y by
Can someone explain to me the sol step by step
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EaZ_Shadow
279 posts
#16
Y by
Nvm nvm nvm I understand now
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N Quick Reply
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