.
,
.
and 
to be the remainder when 
is divided by 
.
and 
is odd for 
.
or an equivalent formulation. To prove this construction works, we need to show that
is always an integer between 
and 
inclusive.
is divided by 
.
,
since it telescopes.
,![\[\frac{r_{i+1} n - r_i}{2^i} > \frac{r_{i+1} n}{2^i} - 1 \geq \frac{n}{2^i} - 1 \geq \frac{n}{2^k} - 1.\]](http://latex.artofproblemsolving.com/4/2/0/420d26b6099ca264846885e7eef23b2c9c7b2dd0.png)
Thus, each digit is lower bounded by 
,
fails.
,
such that 
has no nonreal roots. We can also assume 
case is trivial as the constant term must be nonzero, so fix 
.
.
.
which has the 
for all integers 
.
be the coefficient of the 
term of 
for all 
,
for which 
.
and 
.
and 
.
is the coefficient of the 
term in 
and 
.
,
.
,
has 
distinct real roots as well, along with two consecutive nonleading coefficients equal to zero, but one degree lower, by the Power Rule. By doing this repeatedly, we eventually end up with a polynomial where the two consecutive coefficients have degree 
,
to be the set of points strictly outside the circle with diameter 
.
is always acute. Suppose two roads 
and 
cross. Then, 
is a convex quadrilateral. Since 
.
is not acute but there does not exist an 
such that 
are connected by some path. This is trivially true for 
,
are also connected.
is obtuse, we have 
.
,
,
is not mentioned. Do not reward points if the induction argument is incomplete or incorrect.
be the center of 
.
be the intersection of 
and the line through 
,
.
be the foot of the altitude from 
and 
,
is a midline and thus 
.
is a midline of 
,
and thus 
.
(by definition), by either HL congruence or the Pythagorean theorem, we must have 
.
.
and 
to attempt to identify 
.
or 
,
.
.
is divisible by 
,
be a prime power dividing 
.
.
,
if 
and 
otherwise, so for each term, either both the numerator are divisible by 
,
,
,
.
has an inverse modulo 
.![\[\binom ni \equiv \pm 1 \binom{\lfloor n/p \rfloor}{\lfloor i/p \rfloor} \pmod{p^a}.\]](http://latex.artofproblemsolving.com/0/1/c/01c00beb6efc07ed028fd0273f262b8bfb965d33.png)
modulo 
.
,![\[\sum_{i=0}^n \binom ni^k \equiv \sum_{i=0}^n (\pm 1)^k \equiv p \equiv 0 \pmod p.\]](http://latex.artofproblemsolving.com/2/e/c/2ececb6afa8a10e116056cdc57a6c948298d0c29.png)
where 
,
satisfies the induction hypothesis for the prime 
.![\[\sum_{i=0}^n \binom ni^k \equiv \sum_{i=0}^n (\pm 1)^k \binom{\lfloor n/p \rfloor}{\lfloor i/p \rfloor}^k \equiv p\sum_{i=0}^d \binom di^k \pmod {p^a}\]](http://latex.artofproblemsolving.com/6/c/9/6c9a4b6aac62b1befef1511451c9fa30993d2cf7.png)
By the induction hypothesis, 
divides the inner binomial sum, so since we are multiplying it by 
.
.
is incorrect, reward up to 
between the set of people except Evan 
if the total score of the cupcakes in that bubble with respect to 
.
denotes the set of neighbors of 
people.
.
from the graph, noting that no person in 
unless of your age can be written as 
.
Y by
Find all continuous functions 
such that 
for all 
.
such that 
for all 
.G
Topic
First Poster
Last Poster
Equivalent definition for C^1 functions
Ciobi_ 2
N
by Alphaamss
Source: Romania NMO 2025 11.3
Prove that, for a function 
, the following 
statements are equivalent:
a)
is differentiable, with continuous first derivative.
b) For any
and for any two sequences 
, convergent to 
, such that 
for any positive integer 
, the sequence 
is convergent.
, the following 
statements are equivalent:a)
is differentiable, with continuous first derivative.b) For any
and for any two sequences 
, convergent to 
, such that 
for any positive integer 
, the sequence 
is convergent.
2 replies
2025 USAMO Rubric
plang2008 18
N
by mathprodigy2011
1. Let 
and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
, the digits in the base-
representation of 
are all greater than .
2. 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.
3. 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:
[center]For every city
distinct from 
and 
, there exists 
such[/center]
[center]that
is directly similar to either 
or 
.[/center]
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:
is directly similar to 
if there exists a sequence of rotations, translations, and dilations sending 
to 
, 
to 
, and 
to 
.
4. 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 
.
5. Determine, with proof, all positive integers such that ![\[\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k\]](//latex.artofproblemsolving.com/3/c/5/3c5984d0003c1ca2df2e68841dd9e93ed6c2d27c.png)
is an integer for every positive integer .
6. Let
and be positive integers with 
. There are cupcakes of different flavors arranged around a circle and people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person , it is possible to partition the circle of cupcakes into groups of consecutive cupcakes so that the sum of 's scores of the cupcakes in each group is at least 
. Prove that it is possible to distribute the cupcakes to the people so that each person receives cupcakes of total score at least with respect to .
and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
, the digits in the base-
representation of 
are all greater than .2. 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.3. 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:[center]For every city
distinct from 
and 
, there exists 
such[/center][center]that
is directly similar to either 
or 
.[/center]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:
is directly similar to 
if there exists a sequence of rotations, translations, and dilations sending 
to 
, 
to 
, and 
to 
.4. 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 
.5. Determine, with proof, all positive integers
such that ![\[\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k\]](http://latex.artofproblemsolving.com/3/c/5/3c5984d0003c1ca2df2e68841dd9e93ed6c2d27c.png)
is an integer for every positive integer .6. Let
and be positive integers with 
. There are cupcakes of different flavors arranged around a circle and people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person , it is possible to partition the circle of cupcakes into groups of consecutive cupcakes so that the sum of 's scores of the cupcakes in each group is at least 
. Prove that it is possible to distribute the cupcakes to the people so that each person receives cupcakes of total score at least with respect to .18 replies
A Confused Canadian
sximoz 3
N
by Gavin_Deng
I always wanted to apply for the AMC, and this year, I think I might have a chance. A friend did AMC 8, and she came back telling me I should do it too. I was really enthusiastic, and wanted to apply.
I do not have prior experience with the AMC, and I live in Alberta, Canada. Through my research, I learned that applications must be submitted via an International Group Leader. However, I am uncertain about who they are and what steps I need to take in order to apply. If you have any information about the application, I would greatly appreciate your help.
Additionally, I would be grateful for any advice on how to best prepare for the AMC, particularly the AMC 8, as this may be one of my last opportunities to participate before moving on to the AMC 10. Specifically, I am interested in understanding the scoring system, the format of the contest, and whether it is possible to participate online from my location.
If you have any further info or tips, I would sincerely appreciate your assistance.
Thanks you very much,
sximoz
I do not have prior experience with the AMC, and I live in Alberta, Canada. Through my research, I learned that applications must be submitted via an International Group Leader. However, I am uncertain about who they are and what steps I need to take in order to apply. If you have any information about the application, I would greatly appreciate your help.
Additionally, I would be grateful for any advice on how to best prepare for the AMC, particularly the AMC 8, as this may be one of my last opportunities to participate before moving on to the AMC 10. Specifically, I am interested in understanding the scoring system, the format of the contest, and whether it is possible to participate online from my location.
If you have any further info or tips, I would sincerely appreciate your assistance.
Thanks you very much,
sximoz
3 replies
Question about USAMO, self esteem, and college
xHypotenuse 20
N
by qwerty123456asdfgzxcvb
Hello everyone. I know this question may sound ridiculous/neagtive but I really want to know how the rest of the community thinks on this issue. Please excuse this yap session and feel free to ignore this post if it doesn't make sense, I don't think I really have a sane mind these days and something has gotten into my head.
I want your advice on what I should do in this situation. It has been my dream to make usamo since ~second semester of 9th grade and I started grinding from that time on. Last year, I qualified for the aime and got a 5. This year I really wanted to qualify for the olympiad and studied really hard. I spent my entire summer working on counting and probability, the subject I suck at the most. And yet, on amc 12, I fumbled hard. I usually mocked ~120-130s on amc 10s but on amc 12 this year, I got really mediocre scores ~100. So I had no chance of making usamo.
So during winter of 2024-2025 I kinda gave up on aime studying and I was like "hey, if I can't get into usamo, maybe ill qualify for usapho." Since I was pretty good at physics at that time. So I spended my winter hard grinding for f=ma and guess what? The test had stupid and ridiculous questions and I only got an 11. What really sucks is that even with the stupid amount of cheaters in f=ma, if I changed all of my "D" guesses to "C," then I would have qualified. Since I solved 10 actually and guessed the rest. Absolutely unfair that only 1 of my guesses were correct.
And also since I didn't study for aime, I ended up being super rusty and so I only got a 7. Solved 9 tho. (I usually can consistently solve 10+ on aimes).
And now here's my senior year and ofc I want to apply to a prestigious college. But it feels stupid that I don't have any usamo or usapho titles like the people I know do. I think I will have good essays primarily due to a varied amount of life experiences but like, I don't feel like I will contribute much to the college without being some prestigious olympiad qualifier. So this led to me having a self esteem issue.
This also led me to the question: should I study one last year so that I can get into usamo in my senior year, or is there no point? Since like, colleges don't care about whatever the hell you do in your senior year, and also, it seems just 'weird' to be grinding math contests while the rest of the people from my school are playing around, etc. So this time around I've really been having an internal crisis between my self esteem (since getting into usamo will raise my self esteem a lot) and college/senior choices.
I know this may seem like a dumb question to some and you are free to completely ignore the post. That's fine. I just really want advice for what I should do in this situation and it would really help bring my life quality up
Thanks,
hypotenuse
I want your advice on what I should do in this situation. It has been my dream to make usamo since ~second semester of 9th grade and I started grinding from that time on. Last year, I qualified for the aime and got a 5. This year I really wanted to qualify for the olympiad and studied really hard. I spent my entire summer working on counting and probability, the subject I suck at the most. And yet, on amc 12, I fumbled hard. I usually mocked ~120-130s on amc 10s but on amc 12 this year, I got really mediocre scores ~100. So I had no chance of making usamo.
So during winter of 2024-2025 I kinda gave up on aime studying and I was like "hey, if I can't get into usamo, maybe ill qualify for usapho." Since I was pretty good at physics at that time. So I spended my winter hard grinding for f=ma and guess what? The test had stupid and ridiculous questions and I only got an 11. What really sucks is that even with the stupid amount of cheaters in f=ma, if I changed all of my "D" guesses to "C," then I would have qualified. Since I solved 10 actually and guessed the rest. Absolutely unfair that only 1 of my guesses were correct.
And also since I didn't study for aime, I ended up being super rusty and so I only got a 7. Solved 9 tho. (I usually can consistently solve 10+ on aimes).
And now here's my senior year and ofc I want to apply to a prestigious college. But it feels stupid that I don't have any usamo or usapho titles like the people I know do. I think I will have good essays primarily due to a varied amount of life experiences but like, I don't feel like I will contribute much to the college without being some prestigious olympiad qualifier. So this led to me having a self esteem issue.
This also led me to the question: should I study one last year so that I can get into usamo in my senior year, or is there no point? Since like, colleges don't care about whatever the hell you do in your senior year, and also, it seems just 'weird' to be grinding math contests while the rest of the people from my school are playing around, etc. So this time around I've really been having an internal crisis between my self esteem (since getting into usamo will raise my self esteem a lot) and college/senior choices.
I know this may seem like a dumb question to some and you are free to completely ignore the post. That's fine. I just really want advice for what I should do in this situation and it would really help bring my life quality up
Thanks,
hypotenuse
20 replies
Orange MOP Opportunity
blueprimes 17
N
by MathRook7817
Hello AoPS,
A reputable source that is of a certain credibility has communicated me about details of Orange MOP, a new pathway to qualify for MOP. In particular, 3 rounds of a 10-problem proof-style examination, covering a variety of mathematical topics that requires proofs will be held from September 27, 2025 12:00 AM - November 8, 2025 11:59 PM EST. Each round will occur biweekly on a Saturday starting from September 27 as described above. The deadline for late submissions will be November 20, 2025 11:59 PM EST.
Solutions can be either handwritten or typed digitally with
. If you are sending solutions digitally through physical scan, please make sure your handwriting is eligible. Inability to discern hand-written solutions may warrant point deductions.
As for rules, digital resources and computational intelligence systems are allowed. Textbooks, reference handouts, and calculators are also a freedom provided by the MAA.
The link is said to be posted on the MAA website during the summer, and invites aspiring math students of all grade levels to participate. As for scoring, solutions will be graded on a
-point scale, and solutions will be graded in terms of both elegance and correctness.
As for qualification for further examinations, the Orange MOP examination passes both the AIME and USAJMO/USAMO requirement thresholds, and the top 5 scorers will receive the benefits and prestige of participating at the national level in the MOP program, and possibly the USA TST and the USA IMO team.
I implore you to consider this rare oppourtunity.
Warm wishes.
A reputable source that is of a certain credibility has communicated me about details of Orange MOP, a new pathway to qualify for MOP. In particular, 3 rounds of a 10-problem proof-style examination, covering a variety of mathematical topics that requires proofs will be held from September 27, 2025 12:00 AM - November 8, 2025 11:59 PM EST. Each round will occur biweekly on a Saturday starting from September 27 as described above. The deadline for late submissions will be November 20, 2025 11:59 PM EST.
Solutions can be either handwritten or typed digitally with
. If you are sending solutions digitally through physical scan, please make sure your handwriting is eligible. Inability to discern hand-written solutions may warrant point deductions.As for rules, digital resources and computational intelligence systems are allowed. Textbooks, reference handouts, and calculators are also a freedom provided by the MAA.
The link is said to be posted on the MAA website during the summer, and invites aspiring math students of all grade levels to participate. As for scoring, solutions will be graded on a
-point scale, and solutions will be graded in terms of both elegance and correctness.As for qualification for further examinations, the Orange MOP examination passes both the AIME and USAJMO/USAMO requirement thresholds, and the top 5 scorers will receive the benefits and prestige of participating at the national level in the MOP program, and possibly the USA TST and the USA IMO team.
I implore you to consider this rare oppourtunity.
Warm wishes.
17 replies
LMT Spring 2025 and Girls' LMT 2025
vrondoS 24
N
by AtlantisII
The Lexington High School Math Team is proud to announce LMT Spring 2025 and our inaugural Girls’ LMT 2025! LMT is a competition for middle school students interested in math. Students can participate individually, or on teams of 4-6 members. This announcement contains information for BOTH competitions.
LMT Spring 2025 will take place from 8:30 AM-5:00 PM on Saturday, May 3rd at Lexington High School, 251 Waltham St., Lexington, MA 02421.
The competition will include two individual rounds, a Team Round, and a Guts Round, with a break for lunch and mini-events. A detailed schedule is available at https://lhsmath.org/LMT/Schedule.
There is a $15 fee per participant, paid on the day of the competition. Pizza will be provided for lunch, at no additional cost.
Register for LMT at https://lhsmath.org/LMT/Registration/Home.
Girls’ LMT 2025 will be held ONLINE on MathDash from 11:00 AM-4:15 PM EST on Saturday, April 19th, 2025. Participation is open to middle school students who identify as female or non-binary. The competition will include an individual round and a team round with a break for lunch and mini-events. It is free to participate.
Register for GLMT at https://www.lhsmath.org/LMT/Girls_LMT.
More information is available on our website: https://lhsmath.org/LMT/Home. Email lmt.lhsmath@gmail.com with any questions.
LMT Spring 2025 will take place from 8:30 AM-5:00 PM on Saturday, May 3rd at Lexington High School, 251 Waltham St., Lexington, MA 02421.
The competition will include two individual rounds, a Team Round, and a Guts Round, with a break for lunch and mini-events. A detailed schedule is available at https://lhsmath.org/LMT/Schedule.
There is a $15 fee per participant, paid on the day of the competition. Pizza will be provided for lunch, at no additional cost.
Register for LMT at https://lhsmath.org/LMT/Registration/Home.
Girls’ LMT 2025 will be held ONLINE on MathDash from 11:00 AM-4:15 PM EST on Saturday, April 19th, 2025. Participation is open to middle school students who identify as female or non-binary. The competition will include an individual round and a team round with a break for lunch and mini-events. It is free to participate.
Register for GLMT at https://www.lhsmath.org/LMT/Girls_LMT.
More information is available on our website: https://lhsmath.org/LMT/Home. Email lmt.lhsmath@gmail.com with any questions.
24 replies
MOP Cutoffs Out?
Mathandski 29
N
by Mathandski
MAA has just emailed a press release announcing the formula they will be using this year to come up with the MOP cutoff that applies to you! Here's the process:
1. Multiply your age by
, let be the result.
2. Calculate
, where 
is the Euler's totient theorem, which calculates the number of integers less than relatively prime to .
3. Multiply your result by again because why not, let the result be .
4. Define the Fibonacci sequence
for 
. Let 
be the remainder 
leaves when you divide it by 
.
5. Let
be your predicted USA(J)MO score.
6. You will be invited if your score is at least![$\lfloor \frac{x + \sqrt[r]{r^2} + r \ln(r)}{r} \rfloor$](//latex.artofproblemsolving.com/c/d/3/cd3f7d917b0ec944ec3017e1f737c538e1edd10e.png)
.
7. Note that there may be additional age restrictions for non-high schoolers.
See here for MAA's original news message.
.
.
.
Edit (4/2/2025): This was an April Fool's post.
1. Multiply your age by
, let be the result.2. Calculate
, where 
is the Euler's totient theorem, which calculates the number of integers less than relatively prime to .3. Multiply your result by
again because why not, let the result be .4. Define the Fibonacci sequence
for 
. Let 
be the remainder 
leaves when you divide it by 
.5. Let
be your predicted USA(J)MO score.6. You will be invited if your score is at least
![$\lfloor \frac{x + \sqrt[r]{r^2} + r \ln(r)}{r} \rfloor$](http://latex.artofproblemsolving.com/c/d/3/cd3f7d917b0ec944ec3017e1f737c538e1edd10e.png)
.7. Note that there may be additional age restrictions for non-high schoolers.
See here for MAA's original news message.
.
.
.
Edit (4/2/2025): This was an April Fool's post.
29 replies
2025 Math and AI 4 Girls Competition: Win Up To $1,000!!!
audio-on 19
N
by WhitePhoenix
Join the 2025 Math and AI 4 Girls Competition for a chance to win up to $1,000!
Hey Everyone, I'm pleased to announce the dates for the 2025 MA4G Competition are set!
Applications will open on March 22nd, 2025, and they will close on April 26th, 2025 (@ 11:59pm PST).
Applicants will have one month to fill out an application with prizes for the top 50 contestants & cash prizes for the top 20 contestants (including $1,000 for the winner!). More details below!
Eligibility:
The competition is free to enter, and open to middle school female students living in the US (5th-8th grade).
Award recipients are selected based on their aptitude, activities and aspirations in STEM.
Event dates:
Applications will open on March 22nd, 2025, and they will close on April 26th, 2025 (by 11:59pm PST)
Winners will be announced on June 28, 2025 during an online award ceremony.
Application requirements:
Complete a 12 question problem set on math and computer science/AI related topics
Write 2 short essays
Prizes:
1st place: $1,000 Cash prize
2nd place: $500 Cash prize
3rd place: $300 Cash prize
4th-10th: $100 Cash prize each
11th-20th: $50 Cash prize each
Top 50 contestants: Over $50 worth of gadgets and stationary
Many thanks to our current and past sponsors and partners: Hudson River Trading, MATHCOUNTS, Hewlett Packard Enterprise, Automation Anywhere, JP Morgan Chase, D.E. Shaw, and AI4ALL.
Math and AI 4 Girls is a nonprofit organization aiming to encourage young girls to develop an interest in math and AI by taking part in STEM competitions and activities at an early age. The organization will be hosting an inaugural Math and AI 4 Girls competition to identify talent and encourage long-term planning of academic and career goals in STEM.
Contact:
mathandAI4girls@yahoo.com
For more information on the competition:
https://www.mathandai4girls.org/math-and-ai-4-girls-competition
More information on how to register will be posted on the website. If you have any questions, please ask here!
Hey Everyone, I'm pleased to announce the dates for the 2025 MA4G Competition are set!
Applications will open on March 22nd, 2025, and they will close on April 26th, 2025 (@ 11:59pm PST).
Applicants will have one month to fill out an application with prizes for the top 50 contestants & cash prizes for the top 20 contestants (including $1,000 for the winner!). More details below!
Eligibility:
The competition is free to enter, and open to middle school female students living in the US (5th-8th grade).
Award recipients are selected based on their aptitude, activities and aspirations in STEM.
Event dates:
Applications will open on March 22nd, 2025, and they will close on April 26th, 2025 (by 11:59pm PST)
Winners will be announced on June 28, 2025 during an online award ceremony.
Application requirements:
Complete a 12 question problem set on math and computer science/AI related topics
Write 2 short essays
Prizes:
1st place: $1,000 Cash prize
2nd place: $500 Cash prize
3rd place: $300 Cash prize
4th-10th: $100 Cash prize each
11th-20th: $50 Cash prize each
Top 50 contestants: Over $50 worth of gadgets and stationary
Many thanks to our current and past sponsors and partners: Hudson River Trading, MATHCOUNTS, Hewlett Packard Enterprise, Automation Anywhere, JP Morgan Chase, D.E. Shaw, and AI4ALL.
Math and AI 4 Girls is a nonprofit organization aiming to encourage young girls to develop an interest in math and AI by taking part in STEM competitions and activities at an early age. The organization will be hosting an inaugural Math and AI 4 Girls competition to identify talent and encourage long-term planning of academic and career goals in STEM.
Contact:
mathandAI4girls@yahoo.com
For more information on the competition:
https://www.mathandai4girls.org/math-and-ai-4-girls-competition
More information on how to register will be posted on the website. If you have any questions, please ask here!
19 replies
INTEGIRLS Spring Competition on 4/20!!!
integirls.bayarea 1
N
by Inaaya
[center]IMAGE[/center]
[br]
[center]INTEGIRLS Bay Area Spring Competition![/center]
Hi everyone! INTEGIRLS Bay Area is excited to invite you to participate in our eighth biannual, free, virtual math competition. The event is open to all girls or non-binary individuals comfortable with being grouped with girls in middle or high school and will take place on Sunday, April 20th from 9 AM - 1:00 PM (PST).
If you're excited to dive into a day of math, make new friends, and win fun prizes, then we encourage you to sign up here!
**Note that the Bay Area chapter of INTEGIRLS writes their own problems, so you can participate in another INTEGIRLS chapter's Spring Competition as well :thumbup:
------
[center]Competition Information[/center]
WHO All middle school and high school students who identify as female or non-binary are invited to join our competition! You can sign up with teams of up to 4 people, or choose to be paired with other students at random.
WHAT Our competition will feature individual, team and tiebreaker rounds with problems written by our amazingly talented team, fun games and a social room to meet new people! There will be separate rounds for middle and high school students as well as exciting prizes for our participants.
WHEN The competition will take place on Sunday, April 20th from 9 AM to 1:00 PM (PST).
WHERE We will host the competition over Zoom, so students from all over the world may attend!
WHY Explore exciting math problems, make friends, and most of all, have fun! Through our competition, we hope to inspire a passion for math in more students, and by bringing together girls who love math together, we aim to create a community of future female mathematicians. Math is an amazing subject full of hidden puzzles and strategies, and together, we seek to create an event full of joy where girls bond over the beauty of the subject.
HOW Register for the competition now here!
CONTACT Feel free to email us at bayarea@integirls.org with any questions! Join our community on Discord, and follow us on Instagram at @integirls.bayarea :laugh:
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[center]INTEGIRLS Bay Area Spring Competition![/center]
Hi everyone! INTEGIRLS Bay Area is excited to invite you to participate in our eighth biannual, free, virtual math competition. The event is open to all girls or non-binary individuals comfortable with being grouped with girls in middle or high school and will take place on Sunday, April 20th from 9 AM - 1:00 PM (PST).
If you're excited to dive into a day of math, make new friends, and win fun prizes, then we encourage you to sign up here!
**Note that the Bay Area chapter of INTEGIRLS writes their own problems, so you can participate in another INTEGIRLS chapter's Spring Competition as well :thumbup:
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[center]Competition Information[/center]
WHO All middle school and high school students who identify as female or non-binary are invited to join our competition! You can sign up with teams of up to 4 people, or choose to be paired with other students at random.
WHAT Our competition will feature individual, team and tiebreaker rounds with problems written by our amazingly talented team, fun games and a social room to meet new people! There will be separate rounds for middle and high school students as well as exciting prizes for our participants.
WHEN The competition will take place on Sunday, April 20th from 9 AM to 1:00 PM (PST).
WHERE We will host the competition over Zoom, so students from all over the world may attend!
WHY Explore exciting math problems, make friends, and most of all, have fun! Through our competition, we hope to inspire a passion for math in more students, and by bringing together girls who love math together, we aim to create a community of future female mathematicians. Math is an amazing subject full of hidden puzzles and strategies, and together, we seek to create an event full of joy where girls bond over the beauty of the subject.
HOW Register for the competition now here!
CONTACT Feel free to email us at bayarea@integirls.org with any questions! Join our community on Discord, and follow us on Instagram at @integirls.bayarea :laugh:
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1 reply
The Empty Set Exists
Archimedes15 37
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by lpieleanu
Source: 2021 AIME II P6
For any finite set 
, let 
denote the number of elements in . FInd the number of ordered pairs 
such that and are (not necessarily distinct) subsets of 
that satisfy
, let 
denote the number of elements in . FInd the number of ordered pairs 
such that and are (not necessarily distinct) subsets of 
that satisfy
37 replies
Gheorghe Țițeica 2025 Grade 11 P1
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Source: Gheorghe Țițeica 2025
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and 
(where
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Y by TheBlackPuzzle913, Mathzeus1024
Very beautiful!
is good.
Assume there exists
such that 
, say 
. (*)
The hypothesis can be rewritten as
, 
, where 
, 
.
As is continuous, 
is continuous as well, and by (*) we get that either is constant (which easily gives 
, false) or there exist 
such that ![$[T_1, T_2] \subset \text{Im}\ g$](//latex.artofproblemsolving.com/2/9/a/29a2de24e01e099c7b1a75dd334150af6d19c328.png)
. Notice that any number in 
is a period of (again, by (*)). We will now prove that is constant.
For doing so, we will show that
such that
![\[
r = x_r \alpha_r + y_r \beta_r
\]](//latex.artofproblemsolving.com/7/0/b/70b956bc6f09d1a268bbd515fd9c7fa176d61865.png)
This could be rewritten as (dividing by, say, 
; if 
we divide by 
, and if 
as well then 
and this is of no interest) ![\[ \exists \gamma_r \in \mathbb{Q}, \exists x_r, y_r \in \text{Im}\ g \text{ such that } r = x_r+ y_r \gamma_r, \]](//latex.artofproblemsolving.com/e/9/1/e912eb2b13a626851965ab9ec3b901bc78633aaf.png)
Let ![$y_r \in [T_1, T_2] \subset \text{Im}\ g$](//latex.artofproblemsolving.com/5/e/8/5e84588e05602de7db728858227979321dd6e187.png)
be arbitrary.
It suffices to find a
such that
![\[
T_1 \leq r - y_r \gamma_r \le T_2
\]](//latex.artofproblemsolving.com/7/9/4/794e1d47d8ac955b775745adeda91c936b1188fc.png)
But this can be rewritten as
, which is obviously true as 
is dense in 
.
Let
be arbitrary with 
. By the above, there exist 
such that
![\[
y - x = x_0 \alpha_0 + y_0 \beta_0
\]](//latex.artofproblemsolving.com/b/b/9/bb9966cdb11ef786e7bde6ada0126b8d55d0f27d.png)
Thus
, and so is constant.
is good.Assume there exists
such that 
, say 
. (*)The hypothesis can be rewritten as
, 
, where 
, 
.As
is continuous, 
is continuous as well, and by (*) we get that either is constant (which easily gives 
, false) or there exist 
such that ![$[T_1, T_2] \subset \text{Im}\ g$](http://latex.artofproblemsolving.com/2/9/a/29a2de24e01e099c7b1a75dd334150af6d19c328.png)
. Notice that any number in 
is a period of (again, by (*)). We will now prove that is constant.For doing so, we will show that
such that![\[
r = x_r \alpha_r + y_r \beta_r
\]](http://latex.artofproblemsolving.com/7/0/b/70b956bc6f09d1a268bbd515fd9c7fa176d61865.png)
This could be rewritten as (dividing by, say, 
; if 
we divide by 
, and if 
as well then 
and this is of no interest) ![\[ \exists \gamma_r \in \mathbb{Q}, \exists x_r, y_r \in \text{Im}\ g \text{ such that } r = x_r+ y_r \gamma_r, \]](http://latex.artofproblemsolving.com/e/9/1/e912eb2b13a626851965ab9ec3b901bc78633aaf.png)
Let ![$y_r \in [T_1, T_2] \subset \text{Im}\ g$](http://latex.artofproblemsolving.com/5/e/8/5e84588e05602de7db728858227979321dd6e187.png)
be arbitrary.It suffices to find a
such that![\[
T_1 \leq r - y_r \gamma_r \le T_2
\]](http://latex.artofproblemsolving.com/7/9/4/794e1d47d8ac955b775745adeda91c936b1188fc.png)
But this can be rewritten as
, which is obviously true as 
is dense in 
.Let
be arbitrary with 
. By the above, there exist 
such that![\[
y - x = x_0 \alpha_0 + y_0 \beta_0
\]](http://latex.artofproblemsolving.com/b/b/9/bb9966cdb11ef786e7bde6ada0126b8d55d0f27d.png)
Thus
, and so is constant.This post has been edited 4 times. Last edited by RobertRogo, Mar 30, 2025, 1:02 PM
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N Quick Reply
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