[CASH PRIZES] IndyINTEGIRLS Spring Math Competition
by Indy_Integirls, May 11, 2025, 2:36 AM

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!
Important Information
Eligibility: This competition is open to all woman-identifying and non-binary students in middle
or elementary school students capable of solving problems at the AMC 8 to AMC 10 level
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.
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!
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!
L
Jane street swag package? USA(J)MO
by arfekete, May 7, 2025, 4:34 PM
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.
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.
L
Question about AMC 10
by MathNerdRabbit103, May 2, 2025, 2:53 AM
Hi,
Can anybody predict a good score that I can get on the AMC 10 this November by only being good at counting and probability, number theory, and algebra? I know some geometry because I took it in school though, but it isn’t competition math so it probably doesn’t count.
Thanks.
Can anybody predict a good score that I can get on the AMC 10 this November by only being good at counting and probability, number theory, and algebra? I know some geometry because I took it in school though, but it isn’t competition math so it probably doesn’t count.
Thanks.
usamOOK geometry
by KevinYang2.71, Mar 21, 2025, 12:00 PM
Let
be the orthocenter of acute triangle
, let
be the foot of the altitude from
to
, and let
be the reflection of
across
. Suppose that the circumcircle of triangle
intersects line
at two distinct points
and
. Prove that
is the midpoint of
.














what the yap
by KevinYang2.71, Mar 20, 2025, 12:00 PM
Alice the architect and Bob the builder play a game. First, Alice chooses two points
and
in the plane and a subset
of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair
of cities, they are connected with a road along the line segment
if and only if the following condition holds:
Note:
is directly similar to
if there exists a sequence of rotations, translations, and dilations sending
to
,
to
, and
to
.





For every city
distinct from
and
, there exists
such




that
is directly similar to either
or
.
Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy.


Note:








Prove a polynomial has a nonreal root
by KevinYang2.71, Mar 20, 2025, 12:00 PM
Let
and
be positive integers with
. Let
be a polynomial of degree
with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers
such that the polynomial
divides
, the product
is zero. Prove that
has a nonreal root.










Stanford Math Tournament (SMT) 2025
by stanford-math-tournament, Feb 1, 2025, 3:16 AM





We are excited to announce that registration is now open for Stanford Math Tournament (SMT) 2025!
This year, we will welcome 800 competitors from across the nation to participate in person on Stanford’s campus. The tournament will be held April 11-12, 2025, and registration is open to all high-school students from the United States. This year, we are extending registration to high school teams (strongly preferred), established local mathematical organizations, and individuals; please refer to our website for specific policies. Whether you’re an experienced math wizard, a puzzle hunt enthusiast, or someone looking to meet new friends, SMT has something to offer everyone!
Register here today! We’ll be accepting applications until March 2, 2025.
For those unable to travel, in middle school, or not from the United States, we encourage you to instead register for SMT 2025 Online, which will be held on April 13, 2025. Registration for SMT 2025 Online will open mid-February.
For more information visit our website! Please email us at stanford.math.tournament@gmail.com with any questions or reply to this thread below. We can’t wait to meet you all in April!
prime spam
by fruitmonster97, Nov 7, 2024, 4:54 PM
What is the sum of the digits of the smallest prime that can be written as a sum of
distinct primes?



This post has been edited 1 time. Last edited by fruitmonster97, Nov 7, 2024, 8:21 PM
1. Algebra and sigma algebra on a set
by adityaguharoy, Feb 28, 2018, 3:28 PM
Definitions of algebra on a set and sigma algebra on a set
Let
be an infinite set. And let
be the collection of all subsets
of
such that either
or
is finite. Prove that
is an algebra on
, but not a sigma algebra on
.
Proof
A related exercise :
Let
be an uncountable set
be the collection of all subsets
of
such that either
is countable or
is countable.
Is
a
algebra on
?
Answer
Hint to sketch
Definition (of Algebra on a set )
Let
be an arbitrary set. Then a collection
of subsets of
is called an algebra on
if and only if all the following are true :

for each
the set 
for every finite sequence
of elements each
the union 
Definition (of sigma algebra on a set )
Let
be an arbitrary set. Then a collection
of subsets of
is called an
algebra on
if and only if all the following are true :

for each
the set 
for every infinite sequence
of elements each
the union 
Note that : since
, so we conclude that every
algebra on a set
is also an algebra on
.
Let













Definition (of sigma algebra on a set )
Let














Note that : since




Let









Proof
Note that given a set
then
is infinite if and only if there is a one one function from
to
.
Thus, since given that
is an infinite set so, there must be a one one function from
to
. Let
be such a function.
Then, note that since each of the sets
(
) is finite so they all belong to
.
But, if
be the union
then
is an infinite set and also
is an infinite set. Thus,
is not an element of
.
So, (as per definitions),
is not a
algebra on
.
However, since finite union of finite sets is finite and by de Moivre’s law, we get that,
has to be an algebra on
.
This completes the proof.




Thus, since given that




Then, note that since each of the sets



But, if






So, (as per definitions),



However, since finite union of finite sets is finite and by de Moivre’s law, we get that,


This completes the proof.
A related exercise :
Let

a set
is called to be uncountable if and only if there is no one one function from
to 
. Let 







Is



Answer
Yes.
Hint to sketch
Countable union of countable sets is countable. Show this ! and then use this to conclude the result.
This post has been edited 2 times. Last edited by adityaguharoy, Feb 28, 2018, 4:16 PM
The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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