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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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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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The product of two p-pods is a p-pod
MellowMelon   10
N 3 minutes ago by Mathandski
Source: USA TST 2011 P3
Let $p$ be a prime. We say that a sequence of integers $\{z_n\}_{n=0}^\infty$ is a $p$-pod if for each $e \geq 0$, there is an $N \geq 0$ such that whenever $m \geq N$, $p^e$ divides the sum
\[\sum_{k=0}^m (-1)^k {m \choose k} z_k.\]
Prove that if both sequences $\{x_n\}_{n=0}^\infty$ and $\{y_n\}_{n=0}^\infty$ are $p$-pods, then the sequence $\{x_ny_n\}_{n=0}^\infty$ is a $p$-pod.
10 replies
MellowMelon
Jul 26, 2011
Mathandski
3 minutes ago
J
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Nordic squares!
mathisreaI   36
N 7 minutes ago by awesomehuman
Source: IMO 2022 Problem 6
Let $n$ be a positive integer. A Nordic square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a valley. An uphill path is a sequence of one or more cells such that:

(i) the first cell in the sequence is a valley,

(ii) each subsequent cell in the sequence is adjacent to the previous cell, and

(iii) the numbers written in the cells in the sequence are in increasing order.

Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square.

Author: Nikola Petrović
36 replies
mathisreaI
Jul 13, 2022
awesomehuman
7 minutes ago
J
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Monochromatic bipartite subgraphs
L567   4
N 14 minutes ago by ihategeo_1969
Source: STEMS Mathematics 2023 Category B P6
For a positive integer $n$, let $f(n)$ denote the largest integer such that for any coloring of a $K_{n,n}$ with two colors, there exists a monochromatic subgraph of $K_{n,n}$ isomorphic to $K_{f(n), f(n)}$. Is it true that for each positive integer $m$ we can find a natural $N$ such that for any integer $n \geqslant N$, $f(n) \geqslant m$?

Proposed by Suchir
4 replies
L567
Jan 8, 2023
ihategeo_1969
14 minutes ago
J
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Tilted Students Thoroughly Splash Tiger part 2
DottedCaculator   18
N 44 minutes ago by MathLuis
Source: ELMO 2024/5
In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$, let $M$ be the midpoint of $\overline{BC}$. Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$. Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$. Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$, and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$. Prove that $A$, $Y$, and $Z$ are collinear.

Tiger Zhang
18 replies
DottedCaculator
Jun 21, 2024
MathLuis
44 minutes ago
J
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Putnam 1958 February A5
sqrtX   4
N 5 hours ago by Safal
Source: Putnam 1958 February
Show that the integral equation
$$f(x,y) = 1 + \int_{0}^{x} \int_{0}^{y} f(u,v) \, du \, dv$$has at most one solution continuous for $0\leq x \leq 1, 0\leq y \leq 1.$
4 replies
sqrtX
Jul 18, 2022
Safal
5 hours ago
J
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Miklós Schweitzer 1956- Problem 1
Coulbert   1
N Today at 1:30 PM by NODIRKHON_UZ
1. Solve without use of determinants the following system of linear equations:

$\sum_{j=0}{k} \binom{k+\alpha}{j} x_{k-j} =b_k$ ($k= 0,1, \dots , n$),

where $\alpha$ is a fixed real number. (A. 7)
1 reply
Coulbert
Oct 9, 2015
NODIRKHON_UZ
Today at 1:30 PM
J
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D1021 : Does this series converge?
Dattier   3
N Today at 1:21 PM by Dattier
Source: les dattes à Dattier
Is this series $\sum \limits_{k\geq 1} \dfrac{\ln(1+\sin(k))} k$ converge?
3 replies
Dattier
Apr 26, 2025
Dattier
Today at 1:21 PM
J
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If a matrix exponential is identity, does it follow the initial matrix is zero?
bakkune   5
N Today at 12:45 PM by loup blanc
This might be a really dumb question, but I have neither a rigorous proof nor a counter example.

For any square matrix $\mathbf{A}$, define
$$ e^{\mathbf{A}} = \mathbf{I} + \sum_{n=1}^{+\infty} \frac{1}{n!}\mathbf{A}^n $$where $\mathbf{I}$ is the identity matrix. If for some matrix $\mathbf{A}$ that $e^{\mathbf{A}}$ is identity, does it follow that $\mathbf{A}$ is zero?
5 replies
bakkune
Mar 4, 2025
loup blanc
Today at 12:45 PM
J
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Range of 2 parameters and Convergency of Improper Integral
Kunihiko_Chikaya   3
N Today at 11:37 AM by Mathzeus1024
Source: 2012 Kyoto University Master Course in Mathematics
Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.
3 replies
Kunihiko_Chikaya
Aug 21, 2012
Mathzeus1024
Today at 11:37 AM
J
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Matrix Row and column relation.
Schro   6
N Today at 6:20 AM by Schro
If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...
6 replies
Schro
Apr 28, 2025
Schro
Today at 6:20 AM
J
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A small problem in group theory
qingshushuxue   2
N Today at 4:42 AM by qingshushuxue
Assume that $G,A,B,C$ are group. If $G=\left( AB \right) \bigcup \left( CA \right)$, prove that $G=AB$ or $G=CA$.

where $$A,B,C\subset G,AB\triangleq \left\{ ab:a\in A,b\in B \right\}.$$
2 replies
qingshushuxue
Today at 2:06 AM
qingshushuxue
Today at 4:42 AM
J
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Putnam 1958 February A4
sqrtX   2
N Today at 2:14 AM by centslordm
Source: Putnam 1958 February
If $a_1 ,a_2 ,\ldots, a_n$ are complex numbers such that
$$ |a_1| =|a_2 | =\cdots = |a_n| =r \ne 0,$$and if $T_s$ denotes the sum of all products of these $n$ numbers taken $s$ at a time, prove that
$$ \left| \frac{T_s }{T_{n-s}}\right| =r^{2s-n}$$whenever the denominator of the left-hand side is different from $0$.
2 replies
sqrtX
Jul 18, 2022
centslordm
Today at 2:14 AM
J
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analysis
Hello_Kitty   2
N Yesterday at 10:37 PM by Hello_Kitty
what is the range of $f=x+2y+3z$ for any positive reals satifying $z+2y+3x<1$ ?
2 replies
Hello_Kitty
Yesterday at 9:59 PM
Hello_Kitty
Yesterday at 10:37 PM
J
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Putnam 1958 February A1
sqrtX   2
N Yesterday at 10:32 PM by centslordm
Source: Putnam 1958 February
If $a_0 , a_1 ,\ldots, a_n$ are real number satisfying
$$ \frac{a_0 }{1} + \frac{a_1 }{2} + \ldots + \frac{a_n }{n+1}=0,$$show that the equation $a_n x^n + \ldots +a_1 x+a_0 =0$ has at least one real root.
2 replies
sqrtX
Jul 18, 2022
centslordm
Yesterday at 10:32 PM
J
a