Interesting inequalities

by sqing, Apr 23, 2025, 6:07 AM

Let $a,b,c\geq 0 ,b+c-ca=1$ and $c+a-ab=3.$ Prove that
$a+\frac{19}{10}b-bc\leq 2-\sqrt 2$ $a+\frac{17}{10}b+c-bc\leq 3$ $a^2+\frac{9}{5}b-bc\leq 6-4\sqrt 2$ $a^2+\frac{8}{5}b^2-bc\leq 6-4\sqrt 2$ $a+1.974873b-bc\leq 2-\sqrt 2$ $a+1.775917b+c-bc\leq 3$

inequalities proposed

inequalities

ALGEBRA INEQUALITY

by Tony_stark0094, Apr 23, 2025, 12:17 AM

$a,b,c > 0$ Prove that $\frac{a^2+bc}{b+c} + \frac{b^2+ac}{a+c} + \frac {c^2 + ab}{a+b} \geq a+b+c$

Inequality

inequalities

Combo problem

by soryn, Apr 22, 2025, 6:33 AM

The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.

combinatorics

Inspired by hlminh

by sqing, Apr 22, 2025, 4:43 AM

Let $a,b,c$ be real numbers such that $a^2+b^2+c^2=1.$ Prove that $|a-kb|+|b-kc|+|c-ka|\leq \sqrt{3k^2+2k+3}$ Where $k\geq 0 .$

inequalities proposed

algebra

Apple sharing in Iran

by mojyla222, Apr 20, 2025, 4:17 AM

Ali is hosting a large party. Together with his $n-1$ friends, $n$ people are seated around a circular table in a fixed order. Ali places $n$ apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).

Find all values of $n$ such that after some number of steps, the situation reaches a point where each person has exactly one apple.

combinatorics

Iran 2nd Round

infinite grid

Iran second round 2025-q1

by mohsen, Apr 19, 2025, 10:21 AM

Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.

number theory

Easy Number Theory

by math_comb01, Jan 21, 2024, 12:08 PM

Let $p$ be an odd prime and $a,b,c$ be integers so that the integers $a^{2023}+b^{2023},\quad b^{2024}+c^{2024},\quad a^{2025}+c^{2025}$ are divisible by $p$ .
Prove that $p$ divides each of $a,b,c$ .
$\quad$
Proposed by Navilarekallu Tejaswi

This post has been edited 3 times. Last edited by math_comb01, Jan 22, 2024, 8:22 AM

number theory

modular arithmetic

Divisibility

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

A Familiar Point

by v4913, Apr 16, 2023, 10:00 PM

Let $ABC$ be a triangle with circumcircle $\Omega$ . Let $S_b$ and $S_c$ respectively denote the midpoints of the arcs $AC$ and $AB$ that do not contain the third vertex. Let $N_a$ denote the midpoint of arc $BAC$ (the arc $BC$ including $A$ ). Let $I$ be the incenter of $ABC$ . Let $\omega_b$ be the circle that is tangent to $AB$ and internally tangent to $\Omega$ at $S_b$ , and let $\omega_c$ be the circle that is tangent to $AC$ and internally tangent to $\Omega$ at $S_c$ . Show that the line $IN_a$ , and the lines through the intersections of $\omega_b$ and $\omega_c$ , meet on $\Omega$ .

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

CASIO BASIC programming tips: Text display functions

by sonone, May 9, 2022, 11:26 PM

Text display functions
In programs, you can utilize two different display windows: the graphing window and the text window. The graphing window if where graphing and any type of drawing happens, and the text window is for almost everything else (I say almost, because there are list, table and matrix views, which are different beasts in themselves). I will be focusing on the text window display functions which display text and strings.

The first and easiest is just string output. Simply place a string in its own line and it will print in the console. For example,

"TEXT"

just prints out TEXT to the console.
Akin to this is the output function. It is a little triangle in the pgrm menu which acts the same as the simple string output, except that it requires you to press EXE to continue program execution. If the argument is a string, the display is left flushed, but if it is a number or variable, it is right flushed. For example (/ means the output function):

"PRESS EXE"/
3/
"TEXT"

gives

PRESS EXE
                            -Disp-

PRESS EXE
                                 3
                             -Disp-

PRESS EXE
                                  3
                             -Disp-
TEXT

Note: if you use the output function as the last call of your program, it will re-display the last outputted value once you press EXE and then exit the program if you press it again.

The last display function, found in the I/O of prgm menu, is Locate. This is my favorite display function because I can put anything I want on the text window anywhere. The syntax is

Locate <COL 1-21>, <ROW 1-7>, <arg (str, var, num etc)>

The following code puts 7 A's in a diagonal line:

Locate 1,1,"A"
Locate 2,2,"A"
Locate 3,3,"A"
Locate 4,4,"A"
Locate 5,5,"A"
Locate 6,6,"A"
Locate 7,7,"A"

A more efficient code using a for loop

For 1->I To 7
Locate I,I,"A"
Next

Output:

A
 A
  A
   A
    A
     A
      A

For my coding I generally use the first two display functions in computational programs, while I almost exclusively use Locate for any games. I will cover the basic graphing displays in another post.

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:

Connect Four with CASIO BASIC

by sonone, Jun 16, 2021, 3:56 PM

Hey everyone!
I made a Connect Four (two players) program on my CASIO fx-9750GII using the CASI BASIC programming language.
Here are the features:
Press the left and right arrows to move columns
The EXE button places your chip
When a column is full, the $\uparrow$ will become a $\times$ on the column
You win by getting four of the same chips in a row vertically, horizontally or diagonally
Here is some other useful info:
Uses 1084 bytes of memory
119 lines

BAISC code

ClrText
[[0,0,0,0,0,0,0][0,0,0,0,0,0,0][0,0,0,0,0,0,0][0,0,0,0,0,0,0][0,0,0,0,0,0,0][0,0,0,0,0,0,0]]->Mat A
{0,0,0,0,0,0,0}->List 1
"<DOUBLE VERTICAL LINES>"->Str 1
"="->Str 2
"#"->Str 3
"<UPARROW>"->Str 4
"<XCROSS>"->Str 5
For 1->I To 7
If I≠7
Then Locate 13,I,Str 1
Locate 21,I,Str 1
IfEnd
Locate I+13,7,Str 2
Next
Locate 13,7,Str 3
Locate 21,7,Str 3
Locate 1,6,"press EXE"
Locate 1,7,"place chip"
1->C
1->K
0->Q
Locate 14,7,Str 4
Lbl M
Locate 1,1,"player"
Locate 8,1,K
Locate 1,2,"column:"
If Getkey=38 And C>1
Then Locate 13+C,7,Str 2
C-1->C
If List 1[C]≥5
Then Locate 13+C,7,Str 5
Else Locate 13+C,7,Str 4
IfEnd
IfEnd
If Getkey=27 And C<7
Then Locate 13+C,7,Str 2
C+1->C
If List 1[C]≥5
Then Then Locate 13+C,7,Str 5
Else Locate 13+C,7,Str 4
IfEnd
IfEnd
Locate 8,2,C
List 1[C]->D
If Getkey=31 And D<6
Then Q+1->Q
If K=1
Then 2->K
Else 1->K
IfEnd
K->Mat A[6-D,C]
If K=1
Then Locate 13+C,6-D,"<FILLED CIRCLE>"
Else Locate 13+C,6-D,"<OPEN CIRCLE>"
IfEnd
D+1->D
D->List 1[C]
D=6=>Locate 13+C,7,Str 5
0->W
For 1-I To 6
If Mat A[I,C]=K
Then W+1->W
W=4=>Goto F
Else 0->W
IfEnd
Next
0->W
For 1-I to 7
If Mat A[7-D,I]=K
Then W+1->W
W=4=>GoTo F
Else 0->W
IfEnd
Next
0->W
If C≥D
Then C-D+1->L
1->R
Else 1->L
D-C+1->R
IfEnd
While L≤7 And R≤6
If Mat A[7-R,L]=K
Then W+1->W
Else 0->W
IfEnd
W=4=>Goto F
R+1->R
L+1->L
WhileEnd
0->W
If 8-C≥D
Then 1->R
C+D-1->L
Else 7->L
D+C-8->R
IfEnd
While L≥1 And R≤6
If Mat A[7-R,L]=K
Then W+1->W
Else 0->W
IfEnd
W=4=>Goto F
L-1->L
R+1->R
WhileEnd
Q=42=>Goto T
IfEnd
While Getkey=31
WhileEnd
Goto M
ClrList 1
ClrMat A
Lbl T
Locate 1,4,"tie :("
Stop
Lbl F
Locate 1,4,"you win!"

P.S. I learned CASIO BASIC from the software guide (too large to attach), if anyone is interested. Page 212 (8-1)

code

Connect4

casio BASIC

Iran Team Selection Test 2016

by MRF2017, Jul 15, 2016, 7:46 PM

Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$ .Points $X,Y$ lie on $AB,AC$ ,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$ .Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$ .

geometry

circumcircle

geometric transformation

reflection

Two permutations

by Nima Ahmadi Pour, Apr 24, 2006, 11:11 AM

Suppose that $a_1$ , $a_2$ , $\ldots$ , $a_n$ are integers such that $n\mid a_1 + a_2 + \ldots + a_n$ .
Prove that there exist two permutations $\left(b_1,b_2,\ldots,b_n\right)$ and $\left(c_1,c_2,\ldots,c_n\right)$ of $\left(1,2,\ldots,n\right)$ such that for each integer $i$ with $1\leq i\leq n$ , we have
$n\mid a_i - b_i - c_i$

Proposed by Ricky Liu & Zuming Feng, USA

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