Question on Balkan SL

by Fmimch, Apr 30, 2025, 12:13 AM

Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!

algebra

combinatorics

geometry

number theory

IMO Shortlist

hard inequalities

by pennypc123456789, Apr 30, 2025, 12:12 AM

Given $x,y,z$ be the positive real number. Prove that

$\frac{2xy}{\sqrt{2xy(x^2+y^2)}} + \frac{2yz}{\sqrt{2yz(y^2+z^2)}} + \frac{2xz}{\sqrt{2xz(x^2+z^2)}} \le \frac{2(x^2+y^2+z^2) + xy+yz+xz}{x^2+y^2+z^2}$

inequalities

Good divisors and special numbers.

by Nuran2010, Apr 29, 2025, 4:52 PM

$N$ is a positive integer. Call all positive divisors of $N$ which are different from $1$ and $N$ beautiful divisors.We call $N$ a special number when it has at least $2$ beautiful divisors and difference of any $2$ beautiful divisors divides $N$ as well. Find all special numbers.

number theory

inequality problem

by pennypc123456789, Apr 29, 2025, 2:42 PM

Given $a,b,c$ be positive real numbers . Prove that
$\frac{ab}{(a+b)^2} +\frac{bc}{(b+c)^2}+\frac{ac}{(a+c)^2} \ge \frac{6abc }{(a+b)(b+c)(a+c)}$

inequalities

Inspired by old results

by sqing, Apr 29, 2025, 12:29 PM

Let $a,b>0 , a^2+b^2+ab+a+b=5 .$ Prove that
$\frac{ 1 }{a+b+ab+1}+\frac{6}{a^2+b^2+ab+1}\geq \frac{7}{4}$ $\frac{ 1 }{a+b+ab+1}+\frac{1}{a^2+b^2+ab+1}\geq \frac{1}{2}$ $\frac{41}{a+b+2}+\frac{ab}{a^3+b^3+2} \geq \frac{21}{2}$

This post has been edited 1 time. Last edited by sqing, Yesterday at 12:49 PM

inequalities proposed

algebra

inequalities

Vasc = 1?

by Li4, Apr 26, 2025, 1:33 PM

Find all integer tuples $(a, b, c)$ such that
$(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1.$
Proposed by Li4, Untro368, usjl and YaWNeeT.

number theory

inequalities

hard problem

by Cobedangiu, Apr 21, 2025, 1:51 PM

Let $a,b,c>0$ and $a+b+c=3$ . Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$

inequalities

Counting graph theory

by MathSaiyan, Mar 17, 2025, 2:00 PM

Let $m$ and $n$ be positive integers. For a connected simple graph $G$ on $n$ vertices and $m$ edges, we consider the number $N(G)$ of orientations of (all of) its edges so that, in the resulting directed graph, every vertex has even outdegree.
Show that $N(G)$ only depends on $m$ and $n$ , and determine its value.

graph theory

combinatorics

Beizer curves

by sonone, Apr 18, 2023, 2:20 PM

Beizer curves

Using complex numbers to represent coordinates, I made a simple program to draw a beizer curve:

Cls
ViewWindow -6.3,6.3,1,-3.1,3.1,1
{0,2-2i,3+3i}->List 1  'curve coordinates
0.05->A                'step size
Plot ReP List 1[1],ImP List 1[1]
For A->I To 1 Step A 
List 1->List 2
Dim List 1->N 
While N>1
Dsz N
For 1->J To N
List 2[J]+I*(List 2[J+1]-List 2[J])->List 2[J]
Next
WhileEnd
Plot ReP List 2[1],ImP List 2[1]
Line
Next

I coded this on my CASIO fx-9860G Slim, but it should work on all other CASIO fx with CASIO-BASIC.

casio BASIC

casio

graphing calculator

Direction field generator for TI 84 plus CE python

by sonone, Jan 30, 2023, 3:12 PM

Have you every got the feeling, when you are taking ordinary differential equations, and they ask you to use technology to graph the direction field for some differential equation $y^\prime=f(x,y)$ ? Did you ever wish that your graphing calculator could be that "technology"? That is how I felt, and after numerous versions I have made a beautiful and simplistic direction field program in python, designed for a TI 84 plus CE with the python app.

Here are the pros and cons of the python program:
Pros:

Thin lines make slopes easy to estimate
Features colored lines (negative slopes= blue, positive slopes = red)
Will trace the approximate solution from a point with a custom step interval

Cons:

You need to update the function in the code itself

To use the program you need to first specify the function in the commented section of the code. When you run it it will ask you for the dimensions of the screen you want to graph and for the spacing of the lines. Then the program will prompt you too trace approximate solution. If you enter 0 (no), it will draw the direction field. If you enter 1 (yes) it will draw the direction field AND the traced line.

installation instructions

Attachments:

DIRFIELD.py (1kb)

This post has been edited 1 time. Last edited by sonone, Jan 31, 2023, 8:22 PM
Reason: Added file

graphing calculator

TI 84 plus CE

Python

AoPS avatar drawer

by sonone, Aug 5, 2022, 9:14 PM

AoPS avatar drawer
I made a program that draws an AoPS avatar, but it can do more! If desired, one can create custom avatars of most any size (I doubt some of the larger sizes render well at all on a 126x62 resolution display). It only supports plane partitions (i.e. a normal avatars), otherwise it will look like scribble scrabble.

The height matrix give information on high each "pillar" is. The AoPS logo has a height matrix of

[[3,3,3][3,2,1][3,1,1]]

The first row is the back right, the columns counting up to the right.

If you so desire to change the size, say to a 5x5x5, go to the beginning of the code and change S to 5 (default is 3). Then edit the assignment for MAT A to a 5x5 height matrix. Then run the program and it should render. If it looks strange, check the height matrix to make sure that the values are correct.

The program is called AOPSAVTR and takes up 492 bytes of memory. The .g1m file is attached below. I had to attach it as a .zip file since AoPS doesn't recognize.g1m files.

Attachments:

AOPSAVTR.g1m.zip (0kb)

This post has been edited 1 time. Last edited by sonone, Aug 5, 2022, 9:26 PM

casio BASIC

graphing calculator

programming

REVIVAL: programming graphing calculators

by sonone, Feb 18, 2022, 10:47 PM

REVIVAL
The reason I haven't been posting lately is mainly I had a lot of school, but also that when I wasn't doing school I was programming my graphing calculator(s). I have become pretty comfortable with CASIO BASIC have made programs ranging from the quadratic formula to a sub-sinking game.
I will be posting about some of the programs I have created, and some little programming tips for CASIO calculators.

I have a CASIO fx-9750GII and an fx-9860G Slim that I code with BASIC, and a TI Nspire CX II CAS, which supports TI BASIC, Python, and lua scripting (I haven't learned lua, yet). The majority of my posts from here on out will be on CASIO programming, but I might occasionally post a little thing about python or TI BASIC.

Lastly, before I finish, I will give some teasers on what programs I have already completed:

Drawing a direction field for a two-variable function
Using Euler's method to approximate the value of a first order DE
Converting a matrix to its reduced row echelon form
My own version of Subway-Surfers
Day-of-the-week finder for date in Gregorian calendar
Radical simplifier
Quadratic formula
Submarine sinking game (ballistic missile game)
And more...

\sqrt{(1^2+2^2+...+n^2)/n}$ is an integer.

by parmenides51, Mar 26, 2020, 5:08 PM

Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.

This post has been edited 1 time. Last edited by parmenides51, Mar 28, 2020, 3:42 AM

Intersection of circumcircles of MNP and BOC

by Djile, Apr 8, 2013, 3:13 PM

Let $M$ , $N$ and $P$ be midpoints of sides $BC, AC$ and $AB$ , respectively, and let $O$ be circumcenter of acute-angled triangle $ABC$ . Circumcircles of triangles $BOC$ and $MNP$ intersect at two different points $X$ and $Y$ inside of triangle $ABC$ . Prove that $\angle BAX=\angle CAY.$

geometric transformation

reflection

Submit

I still exist as well.
by G.G.Otto, Aug 11, 2023, 2:44 AM
hello I'm still here lol
by player01, Aug 6, 2022, 6:24 PM
[REVIVAL] I will start posting more calculator relating posts very soon. Even though school has been busy, I have been programming my calculators a decent amount, so I have a lot to share...
by sonone, Feb 18, 2022, 10:29 PM
wow its been like 2.5 years since geo class
by pieMax2713, Feb 4, 2022, 8:38 PM
@violin21, I've been very busy with school lately and haven't been able to add another lesson. I will when i get a free moment
by sonone, Aug 19, 2021, 12:45 AM
ORZ CODER
by samrocksnature, Aug 9, 2021, 9:57 PM
Could you make more Asymptote lessons on your "How to do Asymptote" blog?
by violin21, Aug 9, 2021, 7:26 PM
You can take it, just C&P the CSS into your CSS area
by sonone, Apr 17, 2021, 10:08 PM
how can we take the CSS if we have permission to not take it?
by GoogleNebula, Apr 17, 2021, 5:22 PM
That is awesome!
by sonone, Apr 15, 2021, 10:09 PM
I modified your dodecahedron and got:
[asy]
import three;
import solids;
size(300);
currentprojection=orthographic(0,1.3,1.2);
light(0,5,10);

real phi=(sqrt(6)+1)/3;
real g=(phi-1)/2;
real s=1/2;
real a=sqrt(1-phi*phi/4-g*g)+phi/2;

triple[] d;
d[0]=(phi
by Andrew2019, Mar 26, 2021, 12:15 AM
Not too many, just changing the color here and there. I really like your CSS!
by sonone, Feb 2, 2021, 10:35 AM
Nice!

I see you're making changes to the CSS.
by G.G.Otto, Feb 1, 2021, 9:26 PM
I'm learning Java now!
by sonone, Feb 1, 2021, 5:56 PM
And I took part of it from CaptainFlint and then added a ton of modifications.
by G.G.Otto, Dec 1, 2020, 8:56 AM
fixed $\text{[math]}$
by sonone, Apr 25, 2020, 8:43 PM
Change in progress...
by sonone, Apr 25, 2020, 8:42 PM
When it says "At least 11 users clicked that button" it's incorrect since one person could have just posted 11 times. It should say "At most 11 users click that button".
by G.G.Otto, Apr 25, 2020, 8:26 PM
see this PM
by sonone, Apr 25, 2020, 7:03 PM
I can't get the cursor to work.
by G.G.Otto, Apr 25, 2020, 6:38 PM
If the cursor is confusing, where you are actually pointing in a front of the ship
by sonone, Apr 25, 2020, 11:04 AM
It's ok $\text{[math]}$
by sonone, Apr 23, 2020, 5:30 PM
Sorry I haven't been posting. I've got a Research Paper (for Roman History) due in a week.
by G.G.Otto, Apr 22, 2020, 11:15 PM
You should download it, as it's really useful when learning python. You can find out more here
by G.G.Otto, Apr 19, 2020, 10:04 PM
No. I don't have IDLE.
by sonone, Apr 18, 2020, 7:49 PM
Do you have IDLE installed on your computer? If so, I can give you a couple really cool codes that made that don't work in the AoPS widgets.
by G.G.Otto, Apr 14, 2020, 6:47 AM
I don't know how to change it....
by G.G.Otto, Apr 5, 2020, 5:58 PM
Do you know how the change the into thing above my avatar a different color? It's too hard to see.
by sonone, Apr 5, 2020, 2:08 PM
WELCOME TO MY BLOG!
by sonone, Mar 31, 2020, 4:56 PM