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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.

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[*]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
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Element sum of k others
akasht   19
N 2 minutes ago by ezpotd
Source: ISL 2022 A2
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.
19 replies
akasht
Jul 9, 2023
ezpotd
2 minutes ago
J
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Least swaps to get any labeling of a regular 99-gon
Photaesthesia   9
N 14 minutes ago by Blast_S1
Source: 2024 China MO, Day 2, Problem 6
Let $P$ be a regular $99$-gon. Assign integers between $1$ and $99$ to the vertices of $P$ such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An operation is a swap of the integers assigned to a pair of adjacent vertices of $P$. Find the smallest integer $n$ such that one can achieve every other assignment from a given one with no more than $n$ operations.

Proposed by Zhenhua Qu
9 replies
Photaesthesia
Nov 29, 2023
Blast_S1
14 minutes ago
J
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Angles in a triangle with integer cotangents
Stear14   0
23 minutes ago
In a triangle $ABC$, the point $M$ is the midpoint of $BC$ and $N$ is a point on the side $BC$ such that $BN:NC=2:1$. The cotangents of the angles $\angle BAM$, $\angle MAN$, and $\angle NAC$ are positive integers $k,m,n$.
(a) Show that the cotangent of the angle $\angle BAC$ is also an integer and equals $m-k-n$.
(b) Show that there are infinitely many possible triples $(k,m,n)$, some of which consisting of Fibonacci numbers.
0 replies
Stear14
23 minutes ago
0 replies
J
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R+ FE f(f(xy)+y)=(x+1)f(y)
jasperE3   1
N an hour ago by maromex
Source: p24734470
Find all functions $f:\mathbb R^+\to\mathbb R^+$ such that for all positive real numbers $x$ and $y$:
$$f(f(xy)+y)=(x+1)f(y).$$
1 reply
jasperE3
3 hours ago
maromex
an hour ago
J
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Reducing the exponents for good
RobertRogo   0
Yesterday at 6:38 PM
Source: The national Algebra contest (Romania), 2025, Problem 3/Abstract Algebra (a bit generalized)
Let $A$ be a ring with unity such that for every $x \in A$ there exist $t_x, n_x \in \mathbb{N}^*$ such that $x^{t_x+n_x}=x^{n_x}$. Prove that
a) If $t_x \cdot 1 \in U(A), \forall x \in A$ then $x^{t_x+1}=x, \forall x \in A$
b) If there is an $x \in A$ such that $t_x \cdot 1 \notin U(A)$ then the result from a) may no longer hold.

Authors: Laurențiu Panaitopol, Dorel Miheț, Mihai Opincariu, me, Filip Munteanu
0 replies
RobertRogo
Yesterday at 6:38 PM
0 replies
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Differential equations , Matrix theory
c00lb0y   3
N Yesterday at 12:26 PM by loup blanc
Source: RUDN MATH OLYMP 2024 problem 4
Any idea?? Diff equational system combined with Matrix theory.
Consider the equation dX/dt=X^2, where X(t) is an n×n matrix satisfying the condition detX=0. It is known that there are no solutions of this equation defined on a bounded interval, but there exist non-continuable solutions defined on unbounded intervals of the form (t ,+∞) and (−∞,t). Find n.
3 replies
c00lb0y
Apr 17, 2025
loup blanc
Yesterday at 12:26 PM
J
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The matrix in some degree is a scalar
FFA21   4
N Yesterday at 12:06 PM by FFA21
Source: MSU algebra olympiad 2025 P2
$A\in M_{3\times 3}$ invertible, for an infinite number of $k$:
$tr(A^k)=0$
Is it true that $\exists n$ such that $A^n$ is a scalar
4 replies
FFA21
Yesterday at 12:11 AM
FFA21
Yesterday at 12:06 PM
J
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Prove the statement
Butterfly   10
N Yesterday at 10:16 AM by oty
Given an infinite sequence $\{x_n\} \subseteq [0,1]$, there exists some constant $C$, for any $r>0$, among the sequence $x_n$ and $x_m$ could be chosen to satisfy $|n-m|\ge r $ and $|x_n-x_m|<\frac{C}{|n-m|}$.
10 replies
Butterfly
May 7, 2025
oty
Yesterday at 10:16 AM
J
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Weird integral
Martin.s   0
Yesterday at 9:33 AM
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1 - e^{-2} \cos\left(2\left(u + \tan u\right)\right)} {1 - 2e^{-2} \cos\left(2\left(u + \tan u\right)\right) + e^{-4}} \, \mathrm{d}u \]
0 replies
Martin.s
Yesterday at 9:33 AM
0 replies
J
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hard number theory problem
danilorj   4
N Yesterday at 9:01 AM by c00lb0y
Let \( a \) and \( b \) be positive integers. Prove that
\[ a^2 + \left\lceil \frac{4a^2}{b} \right\rceil \]is not a perfect square.
4 replies
danilorj
May 18, 2025
c00lb0y
Yesterday at 9:01 AM
J
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Unsolving differential equation
Madunglecha   2
N Yesterday at 8:44 AM by vanstraelen
For parameter t
I made a differential equation :
y"=y*(x')^2
for here, '&" is derivate and second order derivate for t
could anyone tell me what is equation between y&x?
2 replies
Madunglecha
May 18, 2025
vanstraelen
Yesterday at 8:44 AM
J
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maximum dimention of non-singular subspace
FFA21   1
N Yesterday at 8:27 AM by alexheinis
Source: MSU algebra olympiad 2025 P1
We call a linear subspace in the space of square matrices non-singular if all matrices contained in it, except for the zero one, are non-singular. Find the maximum dimension of a non-singular subspace in the space of
a) complex $n\times n$ matrices
b) real $4\times 4$ matrices
c) rational $n\times n$ matrices
1 reply
FFA21
Yesterday at 12:02 AM
alexheinis
Yesterday at 8:27 AM
J
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functional equation
pratyush   4
N Yesterday at 8:00 AM by Mathzeus1024
For the functional equation $f(x-y)=\frac{f(x)}{f(y)}$, if f ' (0)=p and f ' (5)=q, then prove f ' (-5) = q
4 replies
pratyush
Apr 4, 2014
Mathzeus1024
Yesterday at 8:00 AM
J
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a product that is never a square
FFA21   1
N Yesterday at 7:21 AM by ohiorizzler1434
Source: MSU algebra olympiad 2025 P3
Show that the product $7*77*777*7777*77777...$ is never a square of an integer.
1 reply
FFA21
Yesterday at 12:18 AM
ohiorizzler1434
Yesterday at 7:21 AM
J
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J
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interesting function equation (fe) in IR
skellyrah   2
N Apr 23, 2025 by jasperE3
Source: mine
find all function F: IR->IR such that $$ xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy) $$
2 replies
skellyrah
Apr 23, 2025
jasperE3
Apr 23, 2025
J
interesting function equation (fe) in IR
G H J
Reveal topic tags
functional equation
function
real numbers
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Source: mine
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skellyrah
25 posts
skellyrah
#1 h  Apr 23, 2025, 9:51 AM
Y by
find all function F: IR->IR such that $$ xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy) $$
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CrazyInMath
459 posts
CrazyInMath
#2 h  Apr 23, 2025, 11:37 AM
Y by
solution
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jasperE3
11364 posts
jasperE3
#4 h  Apr 23, 2025, 8:33 PM
Y by
skellyrah wrote:
find all function F: IR->IR such that $$ xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy) $$

If IR means irrational numbers, and the problem is to find all $f$ such that $xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy)$ for all $x,y\in\mathbb{IR}$ then setting $x=f(y)^{-1}$ gives a contradiction
This post has been edited 1 time. Last edited by jasperE3, Apr 23, 2025, 9:43 PM
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