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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 11:16 PM
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]
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0 replies
jlacosta
Yesterday at 11:16 PM
0 replies
Values of x
Ecrin_eren   0
4 minutes ago
Given 0 ≤ x < 2π, what is the difference between the largest and the smallest of the values of x
that satisfy the equation 5cosx + 2sin2x = 4 in radians?
0 replies
Ecrin_eren
4 minutes ago
0 replies
Maximum value
Ecrin_eren   3
N 18 minutes ago by Royal_mhyasd
a,b,c are positive real numbers such that
(a+b)^2 (a+c)^2=16abc
What is the maximum value of a+b+c
3 replies
Ecrin_eren
5 hours ago
Royal_mhyasd
18 minutes ago
Floor and exact value
Ecrin_eren   1
N 33 minutes ago by Soupboy0
The exact value of a real number a is denoted by [a] and
the fractional value {a}.
For example; [3.7]= 3 and {3, 7} = 0.7
For a positive real number x,
Given the equality of [x]{x} = 2023, what can
[X^2]-[x]^2 be?
1 reply
Ecrin_eren
an hour ago
Soupboy0
33 minutes ago
2025 CMIMC team p7, rephrased
scannose   8
N 43 minutes ago by Soupboy0
In the expansion of $(x^2 + x + 1)^{2024}$, find the number of terms with coefficient divisible by $3$.
8 replies
scannose
Apr 18, 2025
Soupboy0
43 minutes ago
No more topics!
Nested Permutations
P_Groudon   4
N Apr 24, 2025 by P_Groudon
Let $S = \{1, 2, 3, 4, 5\}$ and let $\sigma_1 : S \to S$ and $\sigma_2 : S \to S$ be permutations of $S$. Suppose there exists a permutation $\tau : S \to S$ such that $\sigma_1(\tau(s)) = \tau(\sigma_2(s))$ for all $s$ in $S$.

If $N$ is the number of possible pairs of permutations $(\sigma_1, \sigma_2)$, find the remainder when $N$ is divided by 1000.
4 replies
P_Groudon
Apr 24, 2025
P_Groudon
Apr 24, 2025
Nested Permutations
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P_Groudon
871 posts
#1 • 1 Y
Y by Sedro
Let $S = \{1, 2, 3, 4, 5\}$ and let $\sigma_1 : S \to S$ and $\sigma_2 : S \to S$ be permutations of $S$. Suppose there exists a permutation $\tau : S \to S$ such that $\sigma_1(\tau(s)) = \tau(\sigma_2(s))$ for all $s$ in $S$.

If $N$ is the number of possible pairs of permutations $(\sigma_1, \sigma_2)$, find the remainder when $N$ is divided by 1000.
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P_Groudon
871 posts
#2
Y by
This is inspired by high level mathematics, but can be solved without the need for it and cannot be trivialized by using it. I'd put this around 10 on AIME.
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Sedro
5845 posts
#3
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Sketch?
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P_Groudon
871 posts
#4 • 1 Y
Y by Sedro
Correct, $\tau$ exists $\iff$ $\sigma_1$ and $\sigma_2$ have matching cycle lengths.

The problem came from conjugates in group theory.
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P_Groudon
871 posts
#5 • 1 Y
Y by Sedro
Here's a sketch of the proof:

($\Rightarrow$): Let $\sigma_2$ have a cycle $a_1 \to a_2 \to \cdots \to a_{\ell} \to a_1$. Then, taking $s = a_i$, we have $\sigma_1(\tau(a_i)) = \tau(a_{i+1})$ (where we set $a_{\ell + 1} = a_1$).

So $\sigma_1$ has $\tau(a_1) \to \tau(a_2) \to \cdots \to \tau(a_{\ell}) \to \tau(a_1)$. We can't yet say this is a cycle, since there could be repeats, but there cannot be repeats. $\tau$ as a permutation has to map distinct inputs to distinct outputs, so $a_1, a_2, \cdots, a_{\ell}$ being distinct means $\tau(a_1), \tau(a_2), \cdots, \tau(a_{\ell})$ are distinct. So we have a cycle of the same length.

So cycles of length $\ell$ in $\sigma_2$ become cycles of length $\ell$ in $\sigma_1$ under $\tau$.

Also, as a permutation, we can't have something like 2 separate 2-cycles mapping to the same 2-cycle; they have to map to separate 2-cycles. This shows one direction.

($\Leftarrow$): From the work in the other direction, it becomes obvious how we want to construct $\tau$. If we match cycles of the same length in $\sigma_2$ and $\sigma_1$, suppose $\sigma_2$ has $a_1, \cdots, a_{\ell}$ and $\sigma_1$ has $b_1, \cdots, b_{\ell}$. Set $\tau(a_i) = b_i$ and repeat for all other cycles. It's easy to show this works.
This post has been edited 1 time. Last edited by P_Groudon, Apr 24, 2025, 4:38 PM
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