Yeah… this for the thinkers, the dreamers, the ones who see past the numbers.
This for the ones who don’t just memorize—nah, they understand.
Richard Rusczyk… let’s talk.

[Verse 1] (calm, poetic flow)
Started with a vision, just a kid with a dream,
Saw the flaws in the system, had to switch up the scheme.
Math ain't just numbers, it's the way that you think,
Logic in my blood, put that proof in the ink.

They ain't teach it right, so he built his own lane,
Took the problem, solved it—yeah, he changed the whole game.
Not about the grind, it's about the finesse,
Art of Problem Solving, every lesson a test.

[Chorus] (aggressive, layered vocals)
I do numbers, I don’t do luck! (Nah!)
We see patterns, we construct! (Yeah!)
This ain’t school, this is war, sharpen minds, then we up!
You want shortcuts? Ain’t no crutch!

[Verse 2] (faster, controlled intensity)
Young mathlete, he was top-tier,
Did the contests, made it crystal clear.
But he saw through the system, the cracks in the code,
They was teaching for tests, not the long road.

So he flipped it, re-wrote the playbook,
Math with a purpose, take a good look.
AMC, AIME, yeah, he built beasts,
Taught the next gen how to never face defeat.

[Bridge] (slower, eerie melody in background)
You don’t get better by guessing.
You don’t get stronger by stressing.
You see the proof, you trust the process.
Master the game, now that’s a lesson.

[Verse 3] (powerful, cinematic build-up)
They say math is cold, but the truth too raw,
Either solve the problem or the problem solve y’all.
Ain’t about memorizing, gotta see past the page,
Every theorem that you master just unlocks a new stage.

Now we global, AoPS worldwide,
Young minds trained up, watch the numbers divide.
Not just numbers—it’s a way of life,
Rusczyk showed us how to sharpen the knife.

[Outro] (quiet, dramatic pause, deep voice)
They was stuck in the box… so he built his own.
Now we legends.

( Beat drops...)

[Verse 1]
Yo, let me tell you ‘bout a legend in the game,
Richard Rusczyk, yeah, you know the name.
Math OG, built AoPS from the ground,
Turned number theory into beats that astound.

From the Ivy halls, he took a different lane,
Math competitions, yeah, he changed the game.
Wrote the books, made the classes elite,
Now every mathlete got skills on repeat.

[Chorus]
Richard Rusczyk, making math real slick,
Taught the world how to think real quick.
If you wanna flex, better solve that trick,
AoPS squad, yeah, we stay too sick!

[Verse 2]
AMC, AIME, yeah, he cracked the code,
Training young minds, put ‘em on beast mode.
Proofs so clean, like a calculus dream,
Made problem-solving the ultimate scheme.

No rote learning, nah, he ain’t with that,
Logic, intuition—keep your brain on track.
Geometry kings, combinatorics pros,
Thanks to Richard, yeah, the knowledge just flows.

[Bridge]
Mathletes worldwide, put your hands up high,
If you reppin’ AoPS, let ‘em know why.
From the forums to the books, yeah, we rise,
Solvin’ every problem ‘til we touch the skies!

[Outro]
So next time you’re stuck, just think like Rich,
Break it down, flip the script, make the switch.
From a contest kid to a math elite,
Rusczyk’s legacy can’t be beat!

Math never sounded this fresh!

[quote=sprint p27] What is the greatest value of such that divides . [/quote]
p-adic valuation ftw

[quote=sprint p28]Find the first positive digit after the decimal point in the rational number [/quote]
complete the square

[quote=sprint p29]How many order pairs of positive integers and such that [/quote]
funny sfft

and finally, what you've all been waiting for:
[quote=target p8]There exists two positive reals and such that and form an equilateral triangle. Find the area of this triangle.[/quote]
trig ftw

apparently sprint p20 = target p8 in difficulty

Every day, I will try to post a new problem for you all to solve! If you want to post a daily problem, you can! :)

Please hide solutions and answers, hints are fine though! :)

The first problem is:
[quote=March 21st Problem]Alice flips a fair coin until she gets 2 heads in a row, or a tail and then a head. What is the probability that she stopped after 2 heads in a row? Express your answer as a common fraction.[/quote]

In the spirit of parmenides51, I guess.

p1. We call a time on a hour digital clock nice if the sum of the minutes digits is equal to the hour. For example, , and are nice times. How many nice times occur during the course of one day? (We do not consider times of the form .)

p2. Along Stanford’s University Avenue are palm trees which are either red, green, or blue. Let the positive integers , , be the number of red, green, and blue palm trees respectively. Given that
compute .

p3. integers are each selected uniformly at random from the range to inclusive and put into a set . Each integer is selected independently of the others. What is the expected value of the minimum element of ?

p4. Cornelius chooses three complex numbers uniformly at random from the complex unit circle. Given that real parts of and are , compute the expected value of the real part of .

p5. A computer virus starts off infecting a single device. Every second an infected computer has a chance to stay infected and not do anything else, a chance to infect a new computer, and a chance to infect two new computers. Otherwise (a chance), the virus gets exterminated, but other copies of it on other computers are unaffected. Compute the probability that a single infected computer produces an infinite chain of infections.

p6. In the language of Blah, there is a unique word for every integer between and inclusive. A team of students has an unordered list of these words, but do not know what integer each word corresponds to. However, the team is given access to a machine that, given two, not necessarily distinct, words in Blah, outputs the word in Blah corresponding to the sum modulo of their corresponding integers. What is the minimum such that the team can narrow down the possible translations of "" to a list of Blah words, using the machine as many times as they want?

p7. Compute

p8. What is the area that is swept out by a regular hexagon of side length as it rotates about its center?

p9. Let be the the area enclosed by the relation
Let be the area enclosed by the relation
Compute the limit of as for .

p10. Let be the set of positive integers which are the product of an even number of distinct primes, including . Let be the set of positive integers which are the product of an odd number of distinct primes. Compute

p11. Define the Fibonacci sequence by , , and for . Compute

p12. Let , , , and be points in the plane with integer coordinates such that no three of them are collinear, and where the distances , , , , , and are all integers. Compute the smallest possible length of a side of a convex quadrilateral formed by such points.

p13. Suppose the real roots of are (roots
may be repeated). Compute

p14. A teacher stands at and has some students, where there is exactly one student at each integer position in the following triangle:
[center]IMAGE[/center]
Here, the circle denotes the teacher at and the triangle extends until and includes the column . A teacher can see a student if there is no student in the direct line of sight between the teacher and the position . Compute the number of students the teacher can see (assume that each student has no width—that is, each student is a point).

p15. Suppose we have a right triangle where is the right angle and lengths . Suppose we have points , , and on , , and respectively with . What is the minimum possible length of ?

With are all real positive number that: . Prove that:

If satisfies for all pairs except finitely many, prove or disprove that must hold for all .

**Problem:**

Define as the sum of the digits of a natural number . Also, define
The smallest natural number such that
is .

The MAA may not be perfect, but in my view, their handling of the 2024–2025 competition cycle was truly commendable. Not a single leak of the AMC 8/10/12, AIME I/II, or USA(J)MO appeared on AoPS. This is an impressive feat in an environment where information travels fast and far.

Other supposed leaks remain nothing more than rumors; they are unverifiable, anecdotal, and almost certainly unfounded.

Let’s be realistic: the MAA did everything reasonably possible to protect the integrity of these contests. Leaks exist in many corners of life. Take government (look: https://www.washingtonpost.com/national-security/2025/03/24/trump-leak-signal-jeffrey-goldberg-atlantic/), corporations, and yes, even standardized testing. But here’s the difference: in those other domains, people don’t constantly talk about it. When someone scores a 1600 on the SAT or gets a 5 on the AP Calculus exam, no one jumps to accuse them of cheating. There's a presumption of merit and legitimacy.


Sadly, the same can’t be said for AMC competitions. Ironically, it's not cheating that undermines the credibility of the AMC; it’s the constant speculation about it. Even when there's no clear evidence, the persistent noise about leaks creates doubt. And that doubt sticks. It cheapens success.

You might feel justified in complaining about alleged cheating because you didn’t qualify this year. But what happens next year, when you make USA(J)MO? If the narrative continues to be “oh, everyone just cheats anyway,” your own hard-earned success will be met with skepticism. Not because you cheated—but because you've helped build a culture where perception overrides reality.

I’ve heard plenty of damaging and unfounded accusations just being an active member in my local math community. Teachers claiming a student with a B+ in Algebra II must have cheated to make AIME. Parents whispering that a “Chinese teacher” who runs group classes must be leaking problems because several of their students got DHR on the AMC 8. These accusations are unfair, baseless, and rooted in misunderstanding. School math is not competition math, and success often comes from years of focused training, not shortcuts.

So let’s stop. The more we talk about unverified cheating, the more we hurt ourselves and our peers. It’s time we protect the prestige of these contests—for the sake of those who’ve earned it, and for those who will in the future.

If you’re frustrated by your results, let that energy drive your preparation—not blame. Focus on growth, not gossip.

Hey everyone!

I’m back with another practice test. Sorry this one took a while to pump out since I have been busy lately.

Post your score/distribution, favorite problems, and thoughts on the difficulty of the test down below. Hope you enjoy!


Practice AMC 10A

1. Find the sum of the infinite geometric series 1/2 + 7/36 + 49/648 + …
A - 18/11, B - 9/22, C - 9/11, D - 18/7, E - 9/14

2. What is the first digit after the decimal point in the square root of 420?
A - 1, B - 2, C - 3, D - 4, E - 5

3. Caden’s calculator is broken and two of the digits are swapped for some reason. When he entered in 9 + 10, he got 21. What is the sum of the two digits that got swapped?
A - 2, B - 3, C - 4, D - 5, E - 6

4. Two circles with radiuses 47 and 96 intersect at two points A and B. Let P be the point 82% of the way from A to B. A line is drawn through P that intersects both circles twice. Let the four intersection points, from left to right be W, X, Y, and Z. Find (PW/PX)*(PY/PZ).
A - 50/5863, B - 47/96, C - 1, D - 96/47, E - 5863/50

5. Two dice are rolled, and the two numbers shown are a and b. How many possible values of ab are there?
A - 17, B - 18, C - 19, D - 20, E - 21

6. What is the largest positive integer that cannot be expressed in the form 6a + 9b + 4c + 20d, where a, b, c, and d are positive integers?
A - 29, B - 38, C - 43, D - 76, E - 82

7. What is the absolute difference of the probabilities of getting at least 6/10 on a 10-question true or false test and at least 3/5 on a 5-question true or false test?
A - 0, B - 1/504, C - 1/252, D - 1/126, E - 1/63

8. How many arrangements of the letters in the word “ginger” are there such that the two vowels have an even number of letters (remember 0 is even) between them (including the original “ginger”)?
A - 72, B - 108, C - 144, D - 216, E - 432

9. After opening his final exam, Jason does not know how to solve a single question. So he decides to pull out his phone and search up the answers. Doing this, Jason has a success rate of anywhere from 94-100% for any given question he uses his phone on. However, if the teacher sees his phone at any point during the test, then Jason gets a 0.5 multiplier on his final test score, as well as he must finish the rest of the test questions without his phone. (Assume Jason uses his phone on every question he does until he finishes the test or gets caught.) Every question is a 5-choice multiple choice question. Jason has a 90% chance of not being caught with his phone. What is the expected value of Jason’s test score, rounded to the nearest tenth of a percent?
A - 89.9%, B - 90.0%, C - 90.1%, D - 90.2%, E - 90.3%

10. A criminal is caught by a police officer. Due to a lack of cooperation, the officer calls in a second officer so they can start the arrest smoothly. Officer 1 takes 26:18 to arrest a criminal, and officer 2 takes 13:09 to arrest a criminal. With these two police officers working together, how long should the arrest take?
A - 4:23, B - 5:26, C - 8:46, D - 17:32, E - 19:44

11. Suppose that on the coordinate grid, the x-axis represents economic freedom, and the y-axis represents social freedom, where -1 <= x, y <= 1 and a higher number for either coordinate represents more freedom along that particular axis. Accordingly, the points (0, 0), (1, 1), (-1, 1), (-1, -1), and (1, -1) represent democracy, anarchy, socialism, communism, and fascism, respectively. A country is classified as whichever point it is closest to. Suppose a theoretical new country is selected by picking a random point within the square bounded by anarchy, socialism, communism, and fascism as its vertices. What is the probability that it is fascist?
A - 1 - (1/4)pi, B - 1/5, C - (1/16)pi, D - 1/4, E - 1/8

12. Statistics show that people in Memphis who eat at KFC n days a week have a (1/10)(n+2) chance of liking kool-aid, and the number of people who eat at KFC n days a week is directly proportional to 8 - n (Note that n can only be an integer from 0 to 7, inclusive). A random person in Memphis is selected. Find the probability that they like kool-aid.
A - 13/30, B - 17/30, C - 19/30, D - 23/30, E - 29/30

13. A southern plantation has a length of 50 meters and a width of 60 meters. On the plantation, there is 1 kg of cotton per square meter waiting to be picked. The master of the plantation initially calls over 25 cotton pickers, each picking cotton at a rate of 5 kg per hour starting at 9 AM. However, he wants all of the cotton to be picked by 9 PM, and realizes that he needs to speed up the process. At 12 PM, the master then encourages his pickers to work faster by whipping them, in which they then all speed up to 6 kg per hour. At 1 PM, the master calls in 15 more pickers which pick at 5 kg per hour. Unfortunately, at 3 PM, the clouds drift away and the hot sun starts beating down, which slows every picker down by 2 kg per hour. At 4 PM, the clouds return, and all pickers return to picking at 5 kg per hour. At 5 PM, the master calls in 30 more pickers, which again pick at 5 kg per hour. At 6 PM, he calls in 30 more pickers. At 7 PM, he whips all the pickers again, speeding them up to 6 kg per hour. But at 8 PM, n pickers suddenly crash out and stop working due to fatigue, and the rest all slow back down to 5 kg per hour because they are tired. The master does not have any more pickers, so if too many of them drop out, he is screwed and will have to go overtime. Find the maximum value of n such that all of the cotton can still be picked on time, done no later than 9 PM.
A - 51, B - 52, C - 53, D - 54, E - 55

14. Find the number of positive integers n less than 69 such that the average of all the squares from 1^2 to n^2, inclusive, is an integer.
A - 11, B - 12, C - 23, D - 24, E - 48

15. Find the number of ordered pairs (a, b) of integers such that (a - b)^2 = 625 - 2ab.
A - 6, B - 10, C - 12, D - 16, E - 20

16. What is the 420th digit after the decimal point in the decimal expansion of 1/13?
A - 4, B - 5, C - 6, D - 7, E - 8

17. Two congruent towers stand near each other. Both take the shape of a right rectangular prism. A plane that cuts both towers into two pieces passes through the vertical axes of symmetry of both towers and does not cross the floor or roof of either tower. Let the point that the plane crosses the axis of symmetry of the first tower be A, and the point that the plane crosses the axis of symmetry of the second tower be B. A is 81% of the way from the floor to the roof of the first tower, and B is 69% of the way from the floor to the roof of the second tower. What percent of the total mass of both towers combined is above the plane?
A - 19%, B - 25%, C - 50%, D - 75%, E - 81%

18. What is the greatest number of positive integer factors an integer from 1 to 100 can have?
A - 10, B - 12, C - 14, D - 15, E - 16

19. On an analog clock, the minute hand makes one full revolution every hour, and the hour hand makes one full revolution every 12 hours. Both hands move at a constant rate. During which of the following time periods does the minute hand pass the hour hand?
A - 7:35 - 7:36, B - 7:36 - 7:37, C - 7:37 - 7:38, D - 7:38 - 7:39, E - 7:39 - 7:40

20. Find the smallest positive integer that is a leg in three different Pythagorean triples.
A - 12, B - 14, C - 15, D - 20, E - 21

21. How many axes of symmetry does the graph of (x^2)(y^2) = 69 have?
A - 2, B - 3, C - 4, D - 5, E - 6

22. Real numbers a, b, and c are chosen uniformly and at random from 0 to 3. Find the probability that a + b + c is less than 2.
A - 4/81, B - 8/81, C - 4/27, D - 8/27, E - 2/3

23. Let f(n) be the sum of the positive integer divisors of n. Find the sum of the digits of the smallest odd positive integer n such that f(n) is greater than 2n.
A - 15, B - 18, C - 21, D - 24, E - 27

24. Find the last three digits of 24^10.
A - 376, B - 576, C - 626, D - 876, E - 926

25. A basketball has a diameter of 9 inches, and the hoop has a diameter of 18 inches. Peter decides to pick up the basketball and make a throw. Given that Peter has a 1/4 chance of accidentally hitting the backboard and missing the shot, but if he doesn’t, he is guaranteed that the frontmost point of the basketball will be within 18 inches of the center of the hoop at the moment when a great circle of the basketball crosses the plane containing the rim. No part of the ball will extend behind the backboard at any point during the throw, and the rim is attached directly to the backboard. What is the probability that Peter makes a green FN?
A - 3/128, B - 3/64, C - 3/32, D - 3/16, E - 3/8

Let be a set of integers with the following properties:
[list]
[*] .
[*] If and , then .
[*] If for some , is composite, then all positive divisors of are in .
[/list]
Prove that contains all positive integers.

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!

edgar has three bank accounts, each with an integer amount of dollars in it. He is only allowed to transfer money from one account to another if, by doing so, the latter ends up with double the money it had previously. Prove that edgar can always transfer all of his money into two accounts. Will he always be able to transfer all of his money into a single account?

Let and Prove that

Very hard math problem

Find all k such that \frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a}\le \frac{k}{abc}+3-k
holds for a,b,c>0:a+b+c=3.

The Integral equation with k and u continuous in the unit square and unit interval can have only the trivial solution. Prove this in detail. Here k(x,y)=sin(xy)

Let A, B are two square matrices with same size.
p(x) and q(x) are real coefficients polynomial
prove:

Given -

Find-

Evaluate

It’s seems that there are suspicions about AoPS selling Beast Academy to another company. Is that true? Becuase AoPS online and Beast Academy are connected and it will affect us. I hope that AoPS decides to keep it, but if not, there really isn’t anything we can do.

Several posts have been made about this years before, but the office hours banner hasn't appeared yet even though I refreshed the page multiple times. It has been 50 minutes since when office hours should have started and no mods have responded to a question I posted 3 hours ago (relative to last edit). Can others confirm?

Summary of the problem: Pressing go down to last post button always creates a gray box overlapping last post
Page URL: any forum
Steps to reproduce:
1. Go to any topic in a forum
2. The gray box at the bottom overlaps part of the first post
Expected behavior: Should not show a gray box
Frequency: 100% of the time
Operating system(s): Linux HP EliteBook 835 G8 Notebook PC
Browser(s), including version: Chrome 133.0.6943.142 (Official Build) (64-bit) (cohort: Stable)
Additional information: It works on any other device, on my iPhone XR, a MacOS, and my iPad. Took the screenshot a month ago. The gray box still appears

I saw this on the AoPS Blogroll. closing it, the page worked normally.