Summer is a great time to explore cool problems to keep your skills sharp!  Schedule a class today!

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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies
jlacosta
May 1, 2025
0 replies
Shortest number theory you might've seen in your life
AlperenINAN   13
N 3 minutes ago by lksb
Source: Turkey JBMO TST 2025 P4
Let $p$ and $q$ be prime numbers. Prove that if $pq(p+1)(q+1)+1$ is a perfect square, then $pq + 1$ is also a perfect square.
13 replies
+2 w
AlperenINAN
May 11, 2025
lksb
3 minutes ago
n-term Sequence
MithsApprentice   15
N 34 minutes ago by Ilikeminecraft
Source: USAMO 1996, Problem 4
An $n$-term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a binary sequence of length $n$. Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.
15 replies
MithsApprentice
Oct 22, 2005
Ilikeminecraft
34 minutes ago
Drawing Triangles Against Your Clone
pieater314159   19
N an hour ago by Ilikeminecraft
Source: 2019 ELMO Shortlist C1
Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.)

Proposed by Milan Haiman
19 replies
pieater314159
Jun 27, 2019
Ilikeminecraft
an hour ago
Odd digit multiplication
JuanDelPan   12
N an hour ago by Ilikeminecraft
Source: Pan-American Girls' Mathematical Olympiad 2021, P4
Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$.

$\textit{Proposed by Pablo Serrano, Ecuador}$
12 replies
JuanDelPan
Oct 6, 2021
Ilikeminecraft
an hour ago
No more topics!
Math camp combi
ErTeeEs06   3
N May 2, 2025 by genius_007
Source: BxMO 2025 P2
Let $N\geq 2$ be a natural number. At a mathematical olympiad training camp the same $N$ courses are organised every day. Each student takes exactly one of the $N$ courses each day. At the end of the camp, every student has takes each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of $N$, the largest possible number of students at the camp?
3 replies
ErTeeEs06
Apr 26, 2025
genius_007
May 2, 2025
Math camp combi
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Source: BxMO 2025 P2
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ErTeeEs06
69 posts
#1
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Let $N\geq 2$ be a natural number. At a mathematical olympiad training camp the same $N$ courses are organised every day. Each student takes exactly one of the $N$ courses each day. At the end of the camp, every student has takes each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of $N$, the largest possible number of students at the camp?
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BR1F1SZ
578 posts
#2
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Answer: $(N-1)!$
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Davdav1232
47 posts
#3
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The answer is $(N-1)!$. For an example consider all permutations of $1, 2, ..., N-1$ for the courses and everyone takes the final course together, and for a bound note that if we cyclically shift a permutation of courses it doesn't have an intersection with itself, so out of every cycle we have at most 1 permutation and there are $(N-1)!$ such cycles.
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genius_007
270 posts
#4
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solution
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