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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
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.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Inspired by old results
sqing   4
N 9 minutes ago by sqing
Source: Own
Let $ a,b>0. $ Prove that
$$\frac{(a+1)^2}{b}+\frac{(b+k)^2}{a} \geq4(k+1) $$Where $ k\geq 0. $
$$\frac{a^2}{b}+\frac{(b+1)^2}{a} \geq4$$
4 replies
1 viewing
sqing
5 hours ago
sqing
9 minutes ago
J
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Help with math problem
Glist   0
an hour ago
1. The infinite Morse sequence of zeros and ones, 011010011001..., is constructed as follows: start with 0, then at each step, append a block of the same length as the current sequence, obtained by replacing 0 with 1 and vice versa in the existing block. Is this sequence periodic?
2. On an infinite (two-way) tape, a text in Russian is written. It is known that in this text, the number of distinct 15-symbol blocks is equal to the number of distinct 16-symbol blocks. Prove that the text on the tape is periodic in both directions (i.e., bi-infinite and periodic), for example: "...мамамыларамумамамы...".
0 replies
Glist
an hour ago
0 replies
J
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Math problem
Glist   3
N an hour ago by Glist
Given six distinct points on a plane, all pairwise distances between which are different. Prove that there exists a line segment connecting two of these points which is the longest side in one triangle formed by three of the points, and the shortest side in another triangle formed by three of the points.
3 replies
Glist
Yesterday at 2:19 PM
Glist
an hour ago
J
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interesting combinatorics EGMO P5
aditya21   21
N an hour ago by endless_abyss
Source: EGMO 2015, Problem 5
Let $m, n$ be positive integers with $m > 1$. Anastasia partitions the integers $1, 2, \dots , 2m$ into $m$ pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers.
Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to $n$.
21 replies
aditya21
Apr 17, 2015
endless_abyss
an hour ago
J
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Dimension of a Linear Space
EthanWYX2009   0
5 hours ago
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 $\underline{\quad\quad}.$
0 replies
EthanWYX2009
5 hours ago
0 replies
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Matrices and Determinants
Saucepan_man02   5
N Today at 1:23 AM by Saucepan_man02
Hello

Can anyone kindly share some problems/handouts on matrices & determinants (problems like Putnam 2004 A3, which are simple to state and doesnt involve heavy theory)?

Thank you..
5 replies
Saucepan_man02
Apr 4, 2025
Saucepan_man02
Today at 1:23 AM
J
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Jordan form and canonical base of a matrix
And1viper   2
N Today at 12:49 AM by rchokler
Find the Jordan form and a canonical basis of the following matrix $A$ over the field $Z_5$:
$$A = \begin{bmatrix} 2 & 1 & 2 & 0 & 0 \\ 0 & 4 & 0 & 3 & 4 \\ 0 & 0 & 2 & 1 & 2 \\ 0 & 0 & 0 & 4 & 1 \\ 0 & 0 & 0 & 0 & 2 \end{bmatrix} $$
2 replies
And1viper
Feb 26, 2023
rchokler
Today at 12:49 AM
J
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Putnam 1960 B1
sqrtX   4
N Yesterday at 11:26 PM by KAME06
Source: Putnam 1960
Find all integer solutions $(m,n)$ to $m^{n}=n^{m}.$
4 replies
sqrtX
Jun 18, 2022
KAME06
Yesterday at 11:26 PM
J
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Putnam 1958 November B1
sqrtX   11
N Yesterday at 11:09 PM by Hello_Kitty
Source: Putnam 1958 November
Given
$$b_n = \sum_{k=0}^{n} \binom{n}{k}^{-1}, \;\; n\geq 1,$$prove that
$$b_n = \frac{n+1}{2n} b_{n-1} +1, \;\; n \geq 2.$$Hence, as a corollary, show
$$ \lim_{n \to \infty} b_n =2.$$
11 replies
sqrtX
Jul 19, 2022
Hello_Kitty
Yesterday at 11:09 PM
J
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2025 OMOUS Problem 6
enter16180   1
N Yesterday at 10:43 PM by Doru2718
Source: Open Mathematical Olympiad for University Students (OMOUS-2025)
Let $A=\left(a_{i j}\right)_{i, j=1}^{n} \in M_{n}(\mathbb{R})$ be a positive semi-definite matrix. Prove that the matrix $B=\left(b_{i j}\right)_{i, j=1}^{n} \text {, where }$ $b_{i j}=\arcsin \left(x^{i+j}\right) \cdot a_{i j}$, is also positive semi-definite for all $x \in(0,1)$.
1 reply
enter16180
Yesterday at 11:52 AM
Doru2718
Yesterday at 10:43 PM
J
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2025 OMOUS Problem 1
enter16180   1
N Yesterday at 7:02 PM by KAME06
Source: Open Mathematical Olympiad for University Students (OMOUS-2025)
Aman and Berdi, two biologists, they invented a new type of bacteria such that they can control the division of bacteria into several parts. They are also participants of $OMOUS-2025$ with the aim to train for the first problem of $OMOUS-2025$. They play the following game.
Initially, they take $1$ bacteria and choose a natural number $n$. On each move, the player chooses any $k$ number from $1$ to $n$. Then the player divides each bacterium into $k$ pants. Once chosen, the number $k$ cannot be chosen twice. If after any player's move the number of bacteria population is divisible by $n$ then that player loses. Determine who has the winning strategy depending on the given number $n$ if it's known that Amman starts first.
1 reply
enter16180
Yesterday at 11:44 AM
KAME06
Yesterday at 7:02 PM
J
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Interesting Limit
Riptide1901   0
Yesterday at 6:18 PM
Find $\displaystyle\lim_{x\to\infty}\left|f(x)-\Gamma^{-1}(x)\right|$ where $\Gamma^{-1}(x)$ is the inverse gamma function, and $f^{-1}$ is the inverse of $f(x)=x^x.$
0 replies
Riptide1901
Yesterday at 6:18 PM
0 replies
J
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Sequence of functions
Squeeze   1
N Yesterday at 5:08 PM by Squeeze
Q) let $f_n:[-1,1)\to\mathbb{R}$ and $f_n(x)=x^{n}$ then is this uniformly convergence on $(0,1)$ comment on uniformly convergence on $[0,1]$ where in general it is should be uniformly convergence.

My I am trying with some contradicton method like chose $\epsilon=1$ and trying to solve$|f_n(a)-f(a)|<\epsilon=1$
Next take a in (0,1) and chose a= 2^1/N but not solution
How to solve like this way help.
1 reply
Squeeze
Yesterday at 3:56 AM
Squeeze
Yesterday at 5:08 PM
J
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Integrate lnx/sqrt{1-x^2}
EthanWYX2009   1
N Yesterday at 3:43 PM by GreenKeeper
Determine the value of
\[I=\int\limits_{0}^{1}\frac{\ln x}{\sqrt{1-x^2}}\mathrm dx.\]
1 reply
EthanWYX2009
Yesterday at 2:38 PM
GreenKeeper
Yesterday at 3:43 PM
J
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J
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Perhaps a classic with parameter
mihaig   1
N Apr 16, 2025 by LLriyue
Find the largest positive constant $r$ such that
$$a^2+b^2+c^2+d^2+2\left(abcd\right)^r\geq6$$for all reals $a\geq1\geq b\geq c\geq d\geq0$ satisfying $a+b+c+d=4.$
1 reply
mihaig
Jan 7, 2025
LLriyue
Apr 16, 2025
J
Perhaps a classic with parameter
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Reveal topic tags
inequalities
parameterization
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mihaig
7339 posts
mihaig
#1 h  Jan 7, 2025, 6:01 PM
Y by
Find the largest positive constant $r$ such that
$$a^2+b^2+c^2+d^2+2\left(abcd\right)^r\geq6$$for all reals $a\geq1\geq b\geq c\geq d\geq0$ satisfying $a+b+c+d=4.$
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LLriyue
2 posts
LLriyue
#2 h  Apr 16, 2025, 4:12 AM
Y by
good Question
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