ka March Highlights and 2025 AoPS Online Class Information
jlacosta0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.
Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!
Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.
Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
Our full course list for upcoming classes is below:
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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:
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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.
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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.
There are people at a round table. Some of them are knights who always tell the truth, and the rest are liars who always tell lies. Each of those sitting said the phrase: “among the people sitting clockwise from where I sit there are as many knights as among the people seated counterclockwise from where I sit”. For what could this happen?
Source: Bundeswettbewerb Mathematik 2025, Round 1 - Problem 4
For integers we consider a rectangular frame, consisting of the boundary squares of a rectangle.
Renate and Erhard play the following game on this frame, with Renate to start the game. In a move, a player colours a rectangular area consisting of a single or several white squares. If there are any more white squares, they have to form a connected region. The player who moves last wins the game.
Determine all pairs for which Renate has a winning strategy.
Let be distinct real numbers such that Find the value of
Let be distinct real numbers such that Find the value of
Let be distinct real numbers such that Find the value of
Source: Bundeswettbewerb Mathematik 2025, Round 1 - Problem 3
Let be a semicircle with diameter and midpoint . Let be a point on different from and .
The circle touches in a point , the segment in a point , and additionally the segment . The circle touches in a point and additionally the segments and .
Source: Bundeswettbewerb Mathematik 2025, Round 1 - Problem 2
For each integer we consider the last digit different from zero in the decimal expansion of . The infinite sequence of these digits starts with . Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.
Fridolin just can't get enough from jumping on the number line
Tintarn0
an hour ago
Source: Bundeswettbewerb Mathematik 2025, Round 1 - Problem 1
Fridolin the frog jumps on the number line: He starts at , then jumps in some order on each of the numbers exactly once and finally returns with his last jump to . Can the total distance he travelled with these jumps be a) , b) ?
1. before posting another problem please try your best to provide the solution to the previous solution because we don't want a backlog of many problems
2.one is welcome to send functional equations involving calculus (mainly basic real analysis type of proofs) as long it is of the form
Observe that , since . So, is a power of . Let . Note that both are prime, but one of , must be divisible by , so one of them must be . Of course, , so .
We have . Note that , so . means that is even, let . We have .
So, . Note for , for , by simply inducting. So, we have an obvious contradiction by size and we are reduced to a minimal case check which is left as an excercise to the reader.
The equation modulo gives implying is a power of . At least one of , and must be a multiple of , so it must be one of or . It thus follows that . Now Zsigmondy finishes as it implies . This gives us our only solution of .
Observe that the left hand side is divisible by and so for some Clearly, we have that is a prime. Hence, if then However, we already know that 3 divides one of It also can't divide as this would imply Hence, The rest is trivial by Zsigmondy
This post has been edited 1 time. Last edited by Ilikeminecraft, Mar 7, 2025, 9:28 PM
Using modulo , we find , i.e., for a suitable . Suppose . As is a prime, we must have is a prime. At the same time, is also a prime, which is possible only when . Thus, and . Using modulo 3, we find that is even, set and factorize Clearly . So, . Since cannot be simultaneously divisible by 3, we must have and . The cases are easy, is the only solution. Let . Then, using modulo 8, we find is even. Moreover, using modulo 5, we find . So, where are both even, which doesn't have any solutions.
Since divides the RHS, it's clear that is a power of . Note that this means , so divides one of , from which we obviously have . By Zsigmondy (or just noting that is even and factoring) we can find that , so must hold, and it clearly works.
edit (even easier way to show ): If , then neither of are , so and , so divides the RHS, which is absurd as .
This post has been edited 2 times. Last edited by megarnie, Mar 7, 2025, 10:11 PM
Okay so inmediately gives that is a power of but also if then and thus which can't happen so , now by Zsigmondy we have for a contradiction so if then by checking so contradiction and if then and thus we are done .
Sketch of my in-contest solution:
1. Take mod to get for some
2. Take mod and by orders we get which means
3. Then by LTE we get or
4. Realize that since is a power of then one of is a multiple of which obviously implies
5. If then by mod we have which gives by orders.
I kinda forgot what I did afterwards. If I can get my submission/scratch work from my proctor I will complete the solution.
This post has been edited 4 times. Last edited by PEKKA, Mar 8, 2025, 7:25 PM
If I am not mistaken, heavy theorems can be very easily avoided?
The right-hand side is divisible by , so divides and hence . In particular, for some . But then and if , then is divisible by and hence not prime, contradiction. Hence .
The equation now reads . By mod we get that is even. With we factor and the greatest common divisor of the factors on the right is . Since and , we get and . Work only with the former -- if , then and no solution for . If , then by mod we get is even, so , factor and by mod , contradiction. Therefore, the only solution is .