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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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0 replies
jlacosta
May 1, 2025
0 replies
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functional equation with exponentials
produit   5
N a few seconds ago by GreekIdiot
Find all solutions of the real valued functional equation:
f(\sqrt{x^2+y^2})=f(x)f(y).
Here we do not assume f is continuous
5 replies
produit
3 hours ago
GreekIdiot
a few seconds ago
J
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FE over R
IAmTheHazard   20
N a minute ago by shanelin-sigma
Source: ELMO Shortlist 2024/A3
Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$,
$$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$$
Andrew Carratu
20 replies
IAmTheHazard
Jun 22, 2024
shanelin-sigma
a minute ago
J
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AP = PC wanted, circumcircle and 2 midpoints
parmenides51   1
N 6 minutes ago by venhancefan777
Source: Mathematics Regional Olympiad of Mexico Center Zone 2018 P2
Let $\vartriangle ABC$be a triangle and let $\Gamma$ its circumscribed circle. Let $M$ be the midpoint of the side $BC$ and let $D$ be the point of intersection of the line $AM$ with $\Gamma$. By $D$ a straight line is drawn parallel to $BC$, which intersects $\Gamma$ at a point $E$. Let $N$ be the midpoint of the segment $AE$ and let $P$ be the point of intersection of $CN$ with $AM$. Show that $AP = PC$.
1 reply
+1 w
parmenides51
Nov 13, 2021
venhancefan777
6 minutes ago
J
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Interesting inequalities
sqing   2
N 7 minutes ago by SunnyEvan
Source: Own
Let $ a,b,c,d\geq 0 , a+b+c+d \leq 4.$ Prove that
$$a(kbc+bd+cd) \leq \frac{64k}{27}$$$$a (b+c) (kb c+ b d+ c d) \leq \frac{27k}{4}$$Where $ k\geq 2. $
2 replies
1 viewing
sqing
3 hours ago
SunnyEvan
7 minutes ago
J
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Classifying Math with Symbols Based on Behavior
Midevilgmer   2
N Today at 4:06 AM by Midevilgmer
I have been working on a new math idea that prioritizes the behavior of numbers, equations, and expressions rather than their exact values. Numbers are described using symbols like P (Positive), N (Negative), Z (Zero), I (Imaginary), and D (Decimal). The goal is to create a system that uses symbols that allows you to perform operations like P*D and P+N and determine the behavioral outcomes based on the properties involved. For example, instead of identifying a number as 3, you would describe it as a positive, odd, whole, prime number, allowing you work with those traits individually or together. I would like to mention that I already have created an Addition, Subtraction, Multiplication, Division, Square Root, Exponent, and Factorial table that shows how these different behaviors work in basic operations. Finally, I want to mention that my current background includes a knowledge of geometry, algebra, and a very little amount of calculus. Any thoughts or ideas would be appreciated.
2 replies
Midevilgmer
Today at 12:55 AM
Midevilgmer
Today at 4:06 AM
J
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36x⁴ + 12x² - 36x + 13 > 0
fxandi   3
N Today at 1:48 AM by fxandi
Prove that for any real $x \geq 0$ holds inequality $36x^4 + 12x^2 - 36x + 13 > 0.$
3 replies
fxandi
May 5, 2025
fxandi
Today at 1:48 AM
J
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Weird integral
Martin.s   2
N Today at 12:43 AM by ADus
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1 - e^{-2} \cos\left(2\left(u + \tan u\right)\right)} {1 - 2e^{-2} \cos\left(2\left(u + \tan u\right)\right) + e^{-4}} \, \mathrm{d}u \]
2 replies
Martin.s
May 20, 2025
ADus
Today at 12:43 AM
J
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IMC 2018 P4
ThE-dArK-lOrD   18
N Yesterday at 8:50 PM by jonh_malkovich
Source: IMC 2018 P4
Find all differentiable functions $f:(0,\infty) \to \mathbb{R}$ such that
$$f(b)-f(a)=(b-a)f’(\sqrt{ab}) \qquad \text{for all}\qquad a,b>0.$$
Proposed by Orif Ibrogimov, National University of Uzbekistan
18 replies
ThE-dArK-lOrD
Jul 24, 2018
jonh_malkovich
Yesterday at 8:50 PM
J
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convergence
Soupboy0   2
N Yesterday at 6:33 PM by fruitmonster97
If the function $\zeta(n) = \frac{1}{1^n}+\frac{1}{2^n}+\frac{1}{3^n}+....$ diverges for $n=1$ (harmonic sequence) but converges for $n=2$ because $\frac{\pi^2}{6}$, is there a value between $n=1$ and $n=2$ such that $\zeta(n)$ converges

(i dont know the answer could someone please help me)
2 replies
Soupboy0
Yesterday at 6:13 PM
fruitmonster97
Yesterday at 6:33 PM
J
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a^2=3a+2imatrix 2*2
zolfmark   3
N Yesterday at 2:00 PM by Mathzeus1024
A
matrix 2*2

A^2=3A+2i
A^3=mA+Li


i means identity matrix,

find constant m ، L
3 replies
zolfmark
Feb 23, 2019
Mathzeus1024
Yesterday at 2:00 PM
J
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polynomial having a simple root
FFA21   1
N Yesterday at 1:59 PM by Doru2718
Source: MSU algebra olympiad 2025 P4
$f(x)\in R[x]$ show that $f(x)+i$ has at least one root of multiplicity one
1 reply
FFA21
May 20, 2025
Doru2718
Yesterday at 1:59 PM
J
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non-solvable group has subgroup that is not isomorphic to any normal subgroup
FFA21   1
N Yesterday at 1:45 PM by Doru2718
Source: MSU algebra olympiad 2025 P7
Show that in every finite non-solvable group there is a subgroup that is not isomorphic to any normal subgroup
1 reply
FFA21
May 20, 2025
Doru2718
Yesterday at 1:45 PM
J
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Reduction coefficient
zolfmark   1
N Yesterday at 1:26 PM by Mathzeus1024

find Reduction coefficient of x^10

in(1+x-x^2)^9
1 reply
zolfmark
Jul 17, 2016
Mathzeus1024
Yesterday at 1:26 PM
J
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Metric space
wiseman   3
N Yesterday at 10:33 AM by alinazarboland
Source: IMS 2014 - Day1 - Problem4
Let $(X,d)$ be a metric space and $f:X \to X$ be a function such that $\forall x,y\in X : d(f(x),f(y))=d(x,y)$.
$\text{a})$ Prove that for all $x \in X$, $\lim_{n \rightarrow +\infty} \frac{d(x,f^n(x))}{n}$ exists, where $f^n(x)$ is $\underbrace{f(f(\cdots f(x)}_{n \text{times}} \cdots ))$.
$\text{b})$ Prove that the amount of the limit does not depend on choosing $x$.
3 replies
wiseman
Oct 2, 2014
alinazarboland
Yesterday at 10:33 AM
J
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J
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Irrational equation
giangtruong13   5
N Apr 25, 2025 by pooh123
Solve the equation : $$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$
5 replies
giangtruong13
Apr 24, 2025
pooh123
Apr 25, 2025
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Irrational equation
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giangtruong13
148 posts
giangtruong13
#1 h  Apr 24, 2025, 1:44 PM
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Solve the equation : $$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$
This post has been edited 1 time. Last edited by giangtruong13, Apr 24, 2025, 1:44 PM
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Tuvshuu
11 posts
Tuvshuu
#2 h  Apr 24, 2025, 2:20 PM
Y by
$x = 4$.
I think it's only solution
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Tuvshuu
11 posts
Tuvshuu
#3 h  Apr 24, 2025, 2:37 PM
Y by
Let $f(x) = (\sqrt{x} + 1)[2 - (x - 6) \sqrt{x - 3}]$.
We can easily check $f''(x) \leq 0$ then $f$ is concave function.
$x + 8$ is the $f$'s tangent on the $x = 4$ then only solution is $x = 4$ ($f'(4) = 1$ and $f(4) = 12$).
Attachments:
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navier3072
121 posts
navier3072
#4 h  Apr 24, 2025, 2:58 PM
Y by
How did you show $f''(x)\leq 0$? It seems like a very complicated function
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giangtruong13
148 posts
giangtruong13
#5 h  Apr 25, 2025, 8:19 AM
Y by
boom chalaka
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pooh123
67 posts
pooh123
#6 h  Apr 25, 2025, 1:31 PM
Y by
giangtruong13 wrote:
Solve the equation : $$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$

The domain of the equation is \( x \geq 3 \). We have:

\[ (\sqrt{x} + 1)\left[2 - (x-6)\sqrt{x-3}\right] = x + 8 \]
\[ \Leftrightarrow x + 8 + (\sqrt{x} + 1)\left[(x-6)\sqrt{x-3} - 2\right] = 0 \]
\[ \Leftrightarrow x + 8 - 4\sqrt{x} - 4 + (\sqrt{x} + 1)\left[(x-6)\sqrt{x-3} + 2\right] = 0 \]
\[ \Leftrightarrow (\sqrt{x} - 2)^2 + (\sqrt{x} + 1)\left[(x-3)\sqrt{x-3} - 3\sqrt{x-3} + 2\right] = 0 \]
\[ \Leftrightarrow (\sqrt{x} - 2)^2 + (\sqrt{x} + 1)(\sqrt{x-3}+2)(\sqrt{x-3}-1)^2 = 0 \]
Since each term on the left-hand side is non-negative, the left-hand side must be non-negative.
Therefore, \(\sqrt{x} - 2 = 0\) and \(\sqrt{x-3} - 1 = 0\), so \(x = 4\), which is in the domain.

Hence the only solution to the equation is \(x = 4\).
This post has been edited 2 times. Last edited by pooh123, Apr 25, 2025, 1:42 PM
Reason: typo
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