Find the value

by sqing, Apr 20, 2025, 9:57 AM

Let $A=\lim_{n\to\infty}(\frac{\pi}{4}+\frac{1}{n})^n .$ Find the value of $[100A] .$

This post has been edited 1 time. Last edited by sqing, 23 minutes ago

algebra

value

Algebra polynomial problem

by Pi-rate_91, Apr 20, 2025, 9:46 AM

If $p(x)$ is polynomial with minimum degree such that $p(x)=\frac{x}{x^2+3x+2}$ for $x=0,1,2,...,10$ , find $p(-1)$

polynomial

algebra

Number theory

by XAN4, Apr 20, 2025, 9:44 AM

Prove that there exists infinitely many positive integers $x,y,z$ such that $x,y,z\ne1$ and $x^x\cdot y^y=z^z$ .

number theory

number theory unsolved

Diophantine equation

From Recursion to Inequality

by mojyla222, Apr 20, 2025, 5:01 AM

$\{a_n\}_{n\geq 1}$ is a sequence of real numbers with $a_1=1,\;a_2 =2$ such that for all $n\geq 1$
$a_{n+2}=\dfrac{a_{n+1}^{2}}{1+a_{n}}+a_{n+1}.$ Prove that

$\dfrac{1}{1+a_{1}+a_{2}}+\dfrac{1}{1+a_{2}+a_{3}}+\cdots + \dfrac{1}{1+a_{1403}+a_{1404}}>\dfrac{2^{1403}-1}{2^{1404}}.$
Proposed by Mojtaba Zare

inequalities

density over modulo M

by SomeGuy3335, Apr 20, 2025, 4:19 AM

Let $M$ be a positive integer and let $\alpha$ be an irrational number in $(0,1)$ . Show that for every integer $0\leq a < M$ , there exists a positive integer $n$ such that $M \mid \lfloor{n \alpha}\rfloor-a$ .

This post has been edited 2 times. Last edited by SomeGuy3335, Today at 4:20 AM

abstract algebra

number theory

irrational number

Test from Côte d'Ivoire Diophantine equation

by MENELAUSS, Apr 19, 2025, 3:05 PM

determine all triplets $(x;y;z)$ of natural numbers such that
$y \quad \text{is prime }$
$y \quad \text{and} \quad 3 \quad \text{does not divide} \quad z$
$x^3-y^3=z^2$

number theory

Diophantine equation

Petya and vasya are playing with ones

by egxa, Apr 18, 2025, 5:13 PM

$100$ ones are written in a circle. Petya and Vasya take turns making $10^{10}$ moves each. In each move, Petya chooses 9 consecutive numbers and decreases each by $2$ . Vasya chooses $10$ consecutive numbers and increases each by $1$ . They alternate turns, starting with Petya. Prove that Vasya can act in such a way that after each of his moves, there are always at least five positive numbers, regardless of how Petya plays.

This post has been edited 2 times. Last edited by egxa, Friday at 5:20 PM

combinatorics

game

Guessing Point is Hard

by MarkBcc168, Jul 17, 2024, 12:01 PM

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$ . Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$ . Let $CO$ meet $AB$ at $X$ , and $BO$ meet $AC$ at $Y$ .

Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$ .

Ivan Chan Kai Chin, Malaysia

This post has been edited 1 time. Last edited by MarkBcc168, Jul 19, 2024, 2:11 PM

geometry

IMO Shortlist

Hard geo from UKR TST

by mshtand1, May 5, 2023, 1:28 PM

Let $ABC$ be a triangle and let its incircle, centred at $I$ , touches the side $BC$ at $D$ . A line through $A$ intersects the lines $BC$ , $BI$ and $CI$ at $X$ , $Y$ and $Z$ , respectively. The circle $(ABC)$ intersects the circles $(AIY)$ and $(AIZ)$ again at $U$ and $V$ , respectively. Prove that the points $D$ , $U$ , $V$ and $X$ are concyclic.
Proposed by Fedir Yudin and Mykhailo Shtandenko

geometry

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

SUBSHOT: program in CASIO BASIC

by sonone, May 3, 2022, 8:49 PM

SUBSHOT: Submarine Sinking Game
This is one of my earlier games inspired when I was learning projectile motion in physics. The object of the game is simple: sink as many subs as you can in ten shots. Depending on where you shoot determines how much damage the sub takes. The game methodology is as follows:

Draw board
Position gun
Fire gun and calculate hit
Update stats

The shooting mechanism is vector based. There is a pixel you can move around the screen that acts as the head of the initial velocity vector. The longer the vector the father it shoots etc. Using this information the program calculates the final location using 2D kinematics equations.

I have attached two pictures of what the game looks like on the calculator, one is the initial screen and the other is after the first shot.

The code is rather lengthy and not appealing to type out right now. If the code is asked for I will endeavor to type it up ASAP.

Attachments:

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...

IMO Shortlist 2014 G4

by hajimbrak, Jul 11, 2015, 8:52 AM

Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$ . For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$ , let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$ . Prove that as $P$ varies, the point $Q$ lies on a fixed circle.

Proposed by Jack Edward Smith, UK

This post has been edited 3 times. Last edited by v_Enhance, Jan 12, 2016, 3:46 PM
Reason: Fix typo

IMO Shortlist

geometry

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