ka April Highlights and 2025 AoPS Online Class Information
jlacosta0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.
WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.
Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!
Prealgebra 1
Sunday, Apr 13 - Aug 10
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Introduction to Algebra A
Monday, Apr 7 - Jul 28
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Introduction to Counting & Probability
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Introduction to Number Theory
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Introduction to Algebra B
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Introduction to Geometry
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Intermediate: Grades 8-12
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Introduction to Programming with Python
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Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:
To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.
More specifically:
For new threads:
a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.
Examples: Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿) Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"
b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.
Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".
c) Good problem statement:
Some recent really bad post was:
[quote][/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.
For answers to already existing threads:
d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve , do not answer with " is a solution" only. Either you post any kind of proof or at least something unexpected (like " is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.
e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.
To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!
Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).
The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
Let be a prime. We say that a sequence of integers is a -pod if for each , there is an such that whenever , divides the sum
Prove that if both sequences and are -pods, then the sequence is a -pod.
Let be a positive integer. A Nordic square is an board containing all the integers from to so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a valley. An uphill path is a sequence of one or more cells such that:
(i) the first cell in the sequence is a valley,
(ii) each subsequent cell in the sequence is adjacent to the previous cell, and
(iii) the numbers written in the cells in the sequence are in increasing order.
Find, as a function of , the smallest possible total number of uphill paths in a Nordic square.
For a positive integer , let denote the largest integer such that for any coloring of a with two colors, there exists a monochromatic subgraph of isomorphic to . Is it true that for each positive integer we can find a natural such that for any integer ,?
It is May 1st. I have been anticipating the arrival of my results displayed in the awards ceremony in the form of a digital certificate. I have unfortunately not received anything. I have heard from other sources(AoPS, and the internet), that the certificates generally arrive at the end of the month. I would like to ask the organizers, or the coordinators of the tournament, to at least give us an ETA. I would like to further elaborate on the expedition of the release of the Question Papers and the grading. The question papers would be very helpful to the people who have taken the contest, and also to other people who would like to solve them. It would also help, as people can discuss the problems that were given in the test, and know different strategies to solve a problem they have solved. In regards to the grading, it would be a crucial piece of evidence to dispute the score shown in the awards ceremony, in case the contestant is not satisfied.
In triangle with and , let be the midpoint of . Choose point on the extension of past and point on segment such that lies on . Let be on the opposite side of from such that and . Let intersect the circumcircle of again at , and let intersect the circumcircle of again at . Prove that ,, and are collinear.
Let be a point in the exterior of the circle of center and radius , and let , be the tangent segments from to the circle. On the segment consider the point such that .Let the line from parallel to intersect the segment at . If is a point on the segment other than so that , and if the incircle of the triangle has radius , then find the area of in terms of .
If you were in 8th grade, would you rather go to MOP or mc nats? I chose to study the former more and got in so was wondering if that was valid given that I'll never make mc nats.
Let be a set of integers containing the numbers and . Suppose further that any integer root of any non-zero polynomial with coefficients in also belongs to . Prove that belongs to .
Let be a positive integer. A graph on vertices is given such that the size of the largest clique in the graph is . Prove that there exists a vertex that is present in every clique of size
Prove that there exists a right angle triangle with rational sides and area if and only if and are squares of rational numbers and are in Arithmetic Progression
Okay, so I'm in sixth grade. I have been doing AMC 8 since fourth grade, but not anything else. I was wondering what other "good" math competitions there are that I am the right age for.
I'm also looking for prep tips for math competitions, because when I (mock)ace 2000-2010 AMC 8 and then get a 19 on the real thing when I was definitely able to solve everything, I feel like what I'm doing isn't really working. Anyone got any ideas? Thanks!
2025 Math and AI 4 Girls Competition: Win Up To $1,000!!!
audio-on64
N6 hours ago
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.
Converting from base to base 10, we get for a postive integer because its a divisor. Then you can simplify this to and since n or b cant be negative that means n has to be from 1 to 9 exclusive. Then casework from 2-8, and u get n=7 then b=21 and n=8 then b=40 so its 70
First, note that so it follows that must be a divisor of The only divisors of greater than are and so it follows that the possible values of are and yielding an answer of
9b+7 is divisible by b+7, so 9b+7 - 9(b+7) will still be divisible by b+7, so -56 is divisible by b+7. we first try -56-7=-49, so one b possibility could be 49. 49-7=42, 42/2 = 21, 21 is another possibility. 21-7=14, 14/2 = 7, which is less than 9, so we only have 21 and 49, giving us 70.
By the definition of a divisor, for an integer . Clearly does not work. So now we can just try every single value.
no gives which doesn't work gives which doesn't work gives which doesn't work gives which doesn't work gives which doesn't work gives which does work gives which does work
By the definition of a divisor, for an integer . Clearly does not work. So now we can just try every single value.
no gives which doesn't work gives which doesn't work gives which doesn't work gives which doesn't work gives which doesn't work gives which does work gives which does work
I was stuck on this problem for some reason I don't know why
The best way to tackle this problem is to convert everything to variables so first 17 base b = b + 7 and 97 base b = 9b+7 so we can just make a variable when multiplied by it it equals 97 base b
So first to do that we can simplify it like 9b+7 = x(b+7) for some value of x then when we multiply that out we get 9b+7=bx+7x. Because we want to solve for xb we have to subtract that on both sides to get 9b+7-bx= 7x then in any situation like this we have to factor out the b, but first we can subtract 7 on both sides.
When we do that we get b(9-x)= 7(x-1) which when we divide both sides we get
b= (7(x-1))/(9-x) we get this easy equation to solve because we know that x has to be a single digit number because anything greater than 9 won't work so we get that after guess and check x= 7 and 8 so when we plug that in we get 21 and 49, so when we add those together you get
This post has been edited 2 times. Last edited by sadas123, Feb 9, 2025, 4:44 PM
b+7 is divisible by 9b+7, but because b+7 is divisible by 9b+63, then b+7 is divisible by 56. And B+7 is greater than 7, so b=21 or 49[Click][sounds stupid but I got b+7 is divisible by 8b, and did a lot of stupid stuff to get -6+-5+-3+0+1+7+21+49=64 cause I forgot b is greater than 9, but luckily I realized this at the end.]
Find the sum of all integer bases for which is a divisor of
We need , so . Let , do casework and bounding.
If then so , doesn't work.
If then so , doesn't work.
If then so , which does work.
If then so , which does work.
If then so , doesn't work.
Our answer is .
(97)b =9b+7
(17)b=b+7
So b+7 divides 9b+7
gcd(9b+7,b+7)=b+7
gcd(-56,b+7)=b+7
So we get b+7 divides -56
Now , 56=2*2*2*7 and factors greater than 9+7 are 28 and 56 itself
So, b=21 or 49
Sum =70(answer)