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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]
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0 replies
jlacosta
May 1, 2025
0 replies
Quadruple Binomial Coefficient Sum
P162008   1
N 24 minutes ago by vmene
Source: Self made by my Elder brother
$\sum_{p=0}^{\infty} \sum_{r=0}^{\infty} \sum_{q=1}^{\infty} \sum_{s=0}^{p+q - 1} \frac{((-1)^{p+r+s+1})(2^{p+q-1}) \binom{p + q - s - 1}{p + q - 2s - 1}}{4^s(2p^2q + 2pqr + pq + qr)(2p + 2q + 2r + 3)}.$
1 reply
P162008
40 minutes ago
vmene
24 minutes ago
A MATHEMATICA E BONITA
P162008   0
an hour ago
Source: Self made by my Elder brother
Let $K = \sum_{i=0}^{\infty} \sum_{j=0}^{\infty}\sum_{m=0}^{\infty}\sum_{l=0}^{\infty} \frac{1}{(i+j+m+l)!}$ where $i,j,k$ and $l \in W.$

Now, consider the ratio $Z$ defined as
$Z = \frac{\sum_{r=0}^{\lfloor k \rfloor} \sum_{i=0}^{\lfloor k \rfloor} (-1)^r \binom{\lfloor k \rfloor}{r}(\lfloor k \rfloor - r)^i}{\sum_{r=0}^{\lfloor k \rfloor + 1}(-1)^r\binom{\lfloor k \rfloor + 1}{r}(\lfloor k \rfloor + 1 - r)^{\lfloor k \rfloor + 1}}.$

The summation function $S(n)$ is given by
$S(n) = \sum_{j=1}^{n} \left(\binom{n}{j} (j!)\left(\sum_{b=0}^{j} \frac{(-1)^b}{b!}\right)\right)$

Let $p$ denotes the number of points of intersection between the curves
$x^2 + y^2 - \tan(e^x) - \frac{|x|}{\sin y} = 0, (x\sin (a))^y + (y - x\cos(a))^x = |a|.$

Define $A(m)$ as
$A(m) = p\left(\sum_{k=0}^{m} \binom{2m + 1}{k} ((2m + 1) - 2k) (-1)^k\right).$

The value of $X$ is
$X = \lim_{n \to \infty} \frac{\sqrt{n}}{e^n} \text{exp} \left(\int_{0}^{\infty} \lfloor ne^{-x} \rfloor \text{dx}\right).$

And, $8Y =$ Number of subsets of $\left(1,2,3,\cdots,100\right)$ whose sum of elements is divisible by $5.$

Finally, compute the value of $\frac{1}{Z} +S(4) + 1 + e^{A(20)} + X\sqrt{8\pi} + Y.$
0 replies
P162008
an hour ago
0 replies
Ultra-hyper saddle with logarithmic weight
randomperson1021   0
3 hours ago
Fix integers \(k\ge 3\) and \(1<r<k\), a parameter \(\lambda>0\), and a real log-exponent \(\beta\in\mathbb R\). For every real \(a\) define
$$
F_{a,\beta}^{(k,r)}(x)
  \;:=\;
  \sum_{n\ge 1}
       n^{\,a}\,(\log n)^{\beta}\,e^{\lambda n^{r}}\,x^{\,n^{k}},
  \qquad 0\le x<1.
$$
Put
$$
\Lambda_{k,r,\lambda}
   \;:=\;
   \lambda\!\left(1-\frac{r}{k}\right)
   \left(\frac{\lambda r}{k}\right)^{\!\frac{r}{\,k-r\,}},
   \qquad
   \gamma=\frac{r}{k-r}.
$$
(1) Show that there exists a real constant \(c=c(k,r)\) (independent of \(\lambda\) and of \(\beta\)) such that
$$
\lim_{x\to 1^{-}}
      F_{a,\beta}^{(k,r)}(x)\,
      e^{-\Lambda_{k,r,\lambda}\,(1-x)^{-\gamma}}
      \;=\;
      \begin{cases}
          0, & a<c,\\[6pt]
          \infty, & a>c.
      \end{cases}
$$
(2) Determine this critical value \(c\) explicitly and verify that it coincides with the classical case \(r=1\), namely \(c=-\tfrac12\).

(3) Evaluate the finite, non-zero limit that occurs at the borderline \(a=c\) (your answer may depend on \(k,r,\lambda\) but not on \(\beta\)).
0 replies
randomperson1021
3 hours ago
0 replies
3rd AKhIMO for university students, P5
UzbekMathematician   1
N 5 hours ago by grupyorum
Source: AKhIMO 2025, P5
Show that for every positive integer $n$ there exist nonnegative integers $p, q$ and integers $a_1, a_2, ... , a_p, b_1, b_2, ... , b_q \ge 2$ such that $$ n=\frac{(a_1^3-1)(a_2^3-1)...(a_p^3-1)}{(b_1^3-1)(b_2^3-1)...(b_q^3-1)} $$
1 reply
UzbekMathematician
Yesterday at 2:10 PM
grupyorum
5 hours ago
No more topics!
Dimension of a Linear Space
EthanWYX2009   1
N Apr 19, 2025 by loup blanc
Source: 2024 May taca-10
Let \( V \) be a $10$-dimensional inner product space of column vectors, where for \( v = (v_1, v_2, \dots, v_{10})^T \) and \( w = (w_1, w_2, \dots, w_{10})^T \), the inner product of \( v \) and \( w \) is defined as \[\langle v, w \rangle = \sum_{i=1}^{10} v_i w_i.\]For \( u \in V \), define a linear transformation \( P_u \) on \( V \) as follows:
\[ P_u : V \to V, \quad x \mapsto x - \frac{2\langle x, u \rangle u}{\langle u, u \rangle} \]Given \( v, w \in V \) satisfying
\[ 0 < \langle v, w \rangle < \sqrt{\langle v, v \rangle \langle w, w \rangle} \]let \( Q = P_v \circ P_w \). Then the dimension of the linear space formed by all linear transformations \( P : V \to V \) satisfying \( P \circ Q = Q \circ P \) is $\underline{\quad\quad}.$
1 reply
EthanWYX2009
Apr 19, 2025
loup blanc
Apr 19, 2025
Dimension of a Linear Space
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G H BBookmark kLocked kLocked NReply
Source: 2024 May taca-10
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EthanWYX2009
872 posts
#1 • 1 Y
Y by Euler_Gauss
Let \( V \) be a $10$-dimensional inner product space of column vectors, where for \( v = (v_1, v_2, \dots, v_{10})^T \) and \( w = (w_1, w_2, \dots, w_{10})^T \), the inner product of \( v \) and \( w \) is defined as \[\langle v, w \rangle = \sum_{i=1}^{10} v_i w_i.\]For \( u \in V \), define a linear transformation \( P_u \) on \( V \) as follows:
\[ P_u : V \to V, \quad x \mapsto x - \frac{2\langle x, u \rangle u}{\langle u, u \rangle} \]Given \( v, w \in V \) satisfying
\[ 0 < \langle v, w \rangle < \sqrt{\langle v, v \rangle \langle w, w \rangle} \]let \( Q = P_v \circ P_w \). Then the dimension of the linear space formed by all linear transformations \( P : V \to V \) satisfying \( P \circ Q = Q \circ P \) is $\underline{\quad\quad}.$
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loup blanc
3601 posts
#2
Y by
$P_u$ is the orthogonal symmetry with respect to $u^{\perp}$.
Let $E=span(v,w),F=E^{\perp}$; then $Q_{| F}=id$ and $Q_{|E}$ is a rotation with angle $\theta\notin \pi\mathbb{Z}$.
Then $Q$ is diagonalizable over $\mathbb{C}$, with the eigenvalues $1$ ($8$ times) and $e^{\pm i\theta}$.
Then the commutant of $Q$ has dimension $8^2+2=66$.
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