Polygon formed by the edges of an infinite chessboard
by AlperenINAN, Mar 18, 2025, 6:27 AM
Let
be a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers
represent the number of unit squares that have exactly
edges on the boundary of
respectively. Find the largest real number
such that the inequality
holds for each polygon constructed with these conditions.






Nice FE as the First Day Finale
by swynca, Mar 18, 2025, 6:24 AM
Interesting inequality
by sqing, Mar 18, 2025, 4:16 AM
Let
Prove that









This post has been edited 2 times. Last edited by sqing, 5 hours ago
Inspired by my own results
by sqing, Mar 17, 2025, 8:32 AM
Let
be reals such that
Show that
Let
and
Show that






Problem 2830
by sqing, Mar 17, 2025, 8:15 AM
This post has been edited 1 time. Last edited by sqing, Yesterday at 8:24 AM
hard problem
by Noname23, Mar 16, 2025, 4:57 PM
problem
Let
and
. Prove that




This post has been edited 1 time. Last edited by Noname23, Sunday at 5:34 PM
Natural function and cubelike expression
by sarjinius, Mar 9, 2025, 3:42 PM
Let
be the set of positive integers. Find all functions
such that for all
,
is a perfect cube.



![\[m^2f(m) + n^2f(n) + 3mn(m + n)\]](http://latex.artofproblemsolving.com/0/f/f/0ff783fa217b7be4bda5006e417f1471ea3a6fca.png)
Roots, bounding and other delusions
by anantmudgal09, Mar 7, 2021, 10:33 AM
Let
be the set of all polynomials with real coefficients. Find all functions
satisfying the following conditions:
Proposed by Anant Mudgal, Sutanay Bhattacharya, Pulkit Sinha
![$\mathbb{R}[x]$](http://latex.artofproblemsolving.com/a/8/8/a88b9f4858016b1771635c611d44b56161a08009.png)
![$f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$](http://latex.artofproblemsolving.com/0/3/3/033be3ab043bd47e537f01791c813739ad5c7da5.png)
maps the zero polynomial to itself,
- for any non-zero polynomial
,
, and
- for any two polynomials
, the polynomials
and
have the same set of real roots.
Proposed by Anant Mudgal, Sutanay Bhattacharya, Pulkit Sinha
This post has been edited 1 time. Last edited by anantmudgal09, Mar 7, 2021, 5:20 PM
Reason: I had to do this after Ankoganit suggested it. Missed opportunity.
Reason: I had to do this after Ankoganit suggested it. Missed opportunity.
square geometry bisect $\angle ESB$
by GorgonMathDota, Nov 8, 2020, 12:52 AM
Let
be a square of center
and let
be the symmetric of the point
with respect to point
. Let
be the intersection of
and
, and let
be the intersection of
and
. Show that
is the angle bisector of
.













This post has been edited 2 times. Last edited by GorgonMathDota, Nov 8, 2020, 1:10 AM
Pieces // Class of 2022 // Final Update On This Blog [Part 1/3]
by shiningsunnyday, Mar 29, 2018, 2:09 PM
Rejected from Harvard, Princeton, Yale, MIT, UCLA, UCSD.
Your spots are the most coveted of any US college.
Yet you took a chance in me when nobody else did.
This means the world to me.
Destiny calls from the West.
Committed Stanford Class of 2022!
Pieces from memory
Harvard
New kid
Betrayal
Isolation
Garbage
Class clown
A bit too far
Awakening
Hustle begins
Lonelier the higher you get
Hello, anyone there?
Overnight sensation
Back when I was nothing.
Where were you guys again?
When I had nothing and stood bluffing.
Where were you back then?
Crazy they say my aspirations then,
Change they ask of me to fit in there,
Inspirational they say my aspirations are now,
Now people change to be more like me here.
Cause one's success comes at others' costs,
With scarcity there is competition, that's the natural law,
Which filters our behaviors through selfish thoughts,
Scarcity makes us products o' the system of us all.
Your spots are the most coveted of any US college.
Yet you took a chance in me when nobody else did.
This means the world to me.
Destiny calls from the West.
Committed Stanford Class of 2022!
Pieces from memory
Harvard
"Hey dad, what's the best university on the planet?"
"That would be Harvard."
"I'll get into Harvard!"
"Son, I'll be in tears of joy if you attend my alma mater, USC."
"Go to a UC public school, in-state tuition saves your dad and I a lot of money."
"But I want to prove the other kids at school wrong! I want to get in Harvard!"
"Eat your rice and sleep early. You have counseling session tomorrow."
"Students, today we're writing about our dreams and aspirations in a letter to your future self. Be sure to complete it before you're dismissed to go to lunch."
Dear me,
...
I want to get into a top school like Harvard, the school all the parents want their kids to go to. I will work as hard as I can, no matter what it takes to prove others wrong.
...
*Taking pictures*
"Michael! Stand right there, let me take some photos of you by the John Harvard statue!"
"This is an once-in-a-lifetime tour opportunity!"
"But I'll be back here anyways for college."
"I want to get in to Harvard."
"Lol ok."
"I'm serious. I will change completely in high school. I will stuff my schedule up with AP classes, SAT prep, extracurriculars and do everything perfectly to get into Harvard."
"Why do you want to get into Harvard?"
"I just want to prove people wrong. They all think I'm weird and a clown and a perv. They don't understand me. Once I get in, I'll change back. Party every day."
"That's the funniest thing I've ever heard. Good luck with that."
"That would be Harvard."
"I'll get into Harvard!"
"Son, I'll be in tears of joy if you attend my alma mater, USC."
"Go to a UC public school, in-state tuition saves your dad and I a lot of money."
"But I want to prove the other kids at school wrong! I want to get in Harvard!"
"Eat your rice and sleep early. You have counseling session tomorrow."
"Students, today we're writing about our dreams and aspirations in a letter to your future self. Be sure to complete it before you're dismissed to go to lunch."
Dear me,
...
I want to get into a top school like Harvard, the school all the parents want their kids to go to. I will work as hard as I can, no matter what it takes to prove others wrong.
...
*Taking pictures*
"Michael! Stand right there, let me take some photos of you by the John Harvard statue!"
"This is an once-in-a-lifetime tour opportunity!"
"But I'll be back here anyways for college."
"I want to get in to Harvard."
"Lol ok."
"I'm serious. I will change completely in high school. I will stuff my schedule up with AP classes, SAT prep, extracurriculars and do everything perfectly to get into Harvard."
"Why do you want to get into Harvard?"
"I just want to prove people wrong. They all think I'm weird and a clown and a perv. They don't understand me. Once I get in, I'll change back. Party every day."
"That's the funniest thing I've ever heard. Good luck with that."
New kid
"Who can complete the blank in the sentence? The way the turtle moved was very (slow, fast, cool)."
*Shyly raises hand*
"Michael?"
"Cool?"
*Class laughs*
*Shyly raises hand*
"Michael?"
"Cool?"
*Class laughs*
Betrayal
"Sorry Michael, I don't want to be friends with you anymore. I'm now hanging out with **** and their group."
Isolation
"Ms. ***, why doesn't Michael ever talk?"
"Why is there a [participation quota] checklist on Michael's desk?"
"Son, I'm sending you to a counseling group. There're two other kids who're younger than you. Hope you can get along with them. I also booked sessions with your homeroom teacher after school."
"Why is there a [participation quota] checklist on Michael's desk?"
"Son, I'm sending you to a counseling group. There're two other kids who're younger than you. Hope you can get along with them. I also booked sessions with your homeroom teacher after school."
Garbage
"Hey [crush]! I know that you ___."
"You're such a stalker."
"Sorry, what's a stalker?"
"Go screw yourself."
*Blocks me*
My english was still quite bad then, and had to search up the meanings of "stalk" and "screw."
"You're such a stalker."
"Sorry, what's a stalker?"
"Go screw yourself."
*Blocks me*
My english was still quite bad then, and had to search up the meanings of "stalk" and "screw."
Class clown
"Michael, how are you so talkative now!?"
"Michael needs to learn to better use his humor. The video he made for last unit's project wasn't the most appropriate, even though his group was by far the most popular."
"Let's forget this incident. Mr. *******'s wife had to run back home to get you a white shirt cause you were going to go on stage as the only one with a black shirt."
"Michael, you NEED to become an actor!"
"You're a natural born actor."
"Michael's hilarious."
"Michael's such a weirdo."
"And the award for most vocal improvement goes to... Michael Sun!"
"Michael needs to learn to better use his humor. The video he made for last unit's project wasn't the most appropriate, even though his group was by far the most popular."
"Let's forget this incident. Mr. *******'s wife had to run back home to get you a white shirt cause you were going to go on stage as the only one with a black shirt."
"Michael, you NEED to become an actor!"
"You're a natural born actor."
"Michael's hilarious."
"Michael's such a weirdo."
"And the award for most vocal improvement goes to... Michael Sun!"
A bit too far
"Michael's perverted."
"They hate you."
"They hate you."
Awakening
"Hey Michael, just saw your ex ***** at the restaurant with your best friend. It's ok man."
"Son, your best friend's mom told me he invited everyone to his birthday party except you, cause he says you're becoming too anti-social. What has gotten into you?"
"Michael is a perfectionist in all his work, but I'm worried for his health if he continues to be this tense in high school, where the workload will be a lot heavier."
"[Chinese guards] You can't stay on campus every day past 6; we're in charge of locking the doors."
On middle school graduation day, I skipped the graduation party, went home, and began reading Brave New World. The whole summer, I locked myself in my bedroom and read the whole summer.
"Son, your best friend's mom told me he invited everyone to his birthday party except you, cause he says you're becoming too anti-social. What has gotten into you?"
"Michael is a perfectionist in all his work, but I'm worried for his health if he continues to be this tense in high school, where the workload will be a lot heavier."
"[Chinese guards] You can't stay on campus every day past 6; we're in charge of locking the doors."
On middle school graduation day, I skipped the graduation party, went home, and began reading Brave New World. The whole summer, I locked myself in my bedroom and read the whole summer.
Hustle begins
"I got a 4.0 this semester!"
"Let's ignore Michael. He's so arrogant."
"All he cares about is grades."
"Son, *****'s dad and I were just talking. *****, your childhood friend, even thinks you've changed. Look at how your old friends are talking about you."
"Let's ignore Michael. He's so arrogant."
"All he cares about is grades."
"Son, *****'s dad and I were just talking. *****, your childhood friend, even thinks you've changed. Look at how your old friends are talking about you."
Lonelier the higher you get
"Michael wants to be left alone. Let's go somewhere else."
"Son, it's your problem nobody gets you."
"Son, it's your problem nobody gets you."
Hello, anyone there?
"Michael's just crazy."
"Your mind is only on math."
"You can't just ditch your grades!"
"Based on your attendance record you may not receive credit for this course."
"You can't be irresponsible. Remember, all the contests you do is extracurricular activities, which comes after curricular activities."
"Your mind is only on math."
"You can't just ditch your grades!"
"Based on your attendance record you may not receive credit for this course."
"You can't be irresponsible. Remember, all the contests you do is extracurricular activities, which comes after curricular activities."
Overnight sensation
"Michael you're my role model. I want to be just like you."
"Thank you for inspiring all of us to pursue our goals."
"The legend."
"I want to be the next Michael Sun."
"How are you so good at everything."
"How can my child be like you?"
"You're my son's inspiration."
"Michael don't forget our friendship in the future, when you get rich and famous."
“Michael my friend says she wants to marry you!”
"Michael I'm following you from now on."
"Thank you for inspiring all of us to pursue our goals."
"The legend."
"I want to be the next Michael Sun."
"How are you so good at everything."
"How can my child be like you?"
"You're my son's inspiration."
"Michael don't forget our friendship in the future, when you get rich and famous."
“Michael my friend says she wants to marry you!”
"Michael I'm following you from now on."
Personal Note from Stanford Admission Officer wrote:
"Your passion for math and learning is contagious. I really enjoyed your personal essay."
Back when I was nothing.
Where were you guys again?
When I had nothing and stood bluffing.
Where were you back then?
Crazy they say my aspirations then,
Change they ask of me to fit in there,
Inspirational they say my aspirations are now,
Now people change to be more like me here.
Cause one's success comes at others' costs,
With scarcity there is competition, that's the natural law,
Which filters our behaviors through selfish thoughts,
Scarcity makes us products o' the system of us all.
This post has been edited 13 times. Last edited by shiningsunnyday, Apr 1, 2018, 6:13 AM
The ones who are crazy enough to think they can change the world are the ones who do.
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