Polynomials

by P162008, Apr 25, 2025, 2:05 AM

Define a family of polynomials by $P_{0}(x) = x - 2$ and $P_{k}(x) = \left(P_{k - 1} (x)\right)^2 - 2$ if $k \geq 1$ then find the coefficient of $x^2$ in $P_{k}(x)$ in terms of $k.$

This post has been edited 2 times. Last edited by P162008, Today at 2:06 AM
Reason: Typo

algebra

polynomial

Irrational equation

by giangtruong13, Apr 24, 2025, 1:44 PM

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, Yesterday at 1:44 PM

equation

Inspired by SXTX (4)2025 Q712

by sqing, Apr 24, 2025, 11:59 AM

Let $a ,b,c>0$ and $(a+b)^2+2(b+c)^2+(c+a)^2=12.$ Prove that $abc(a+b+c) \leq \frac{9}{5}$ Let $a ,b,c>0$ and $2(a+b)^2+ (b+c)^2+2(c+a)^2=12.$ Prove that $abc(a+b+c) \leq \frac{9}{8}$

inequalities proposed

algebra

inequalities

2^x+3^x = yx^2

by truongphatt2668, Apr 22, 2025, 3:38 PM

Prove that the following equation has infinite integer solutions:
$2^x+3^x = yx^2$

number theory

\frac{1}{9}+\frac{1}{\sqrt{3}}\geq a^2+\sqrt{a+ b^2} \geq \frac{1}{4}

by sqing, Oct 3, 2023, 8:11 AM

Let $a,b\geq 0$ and $3a+4b =1 .$ Prove that
$\frac{2}{3}\geq a +\sqrt{a^2+ 4b^2}\geq \frac{6}{13}$ $\frac{1}{9}+\frac{1}{\sqrt{3}}\geq a^2+\sqrt{a+ b^2} \geq \frac{1}{4}$ $2\geq a+\sqrt{a^2+16b} \geq \frac{2}{3}\geq a+\sqrt{a^2+16b^3} \geq \frac{2(725-8\sqrt{259})}{729}$

This post has been edited 2 times. Last edited by sqing, Oct 3, 2023, 8:55 AM

inequalities proposed

algebra

inequalities

Inequality

Operations on Pebbles

by MarkBcc168, Jul 9, 2023, 4:39 AM

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$ ) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

combinatorics

IMO Shortlist

AZE IMO TST

Functional equation over the integers

by Jutaro, Oct 28, 2020, 8:44 PM

Find all the functions $f: \mathbb{Z}\to \mathbb{Z}$ satisfying the following property: if $a$ , $b$ and $c$ are integers such that $a+b+c=0$ , then

$f(a)+f(b)+f(c)=a^2+b^2+c^2.$

algebra

functional equation

function

Find all possible values of BT/BM

by va2010, Jul 7, 2016, 8:59 PM

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$ . A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$ . Determine all possible values of $\frac{BT}{BM}$ .

This post has been edited 1 time. Last edited by v_Enhance, Jul 7, 2016, 9:23 PM
Reason: improve title

geometry

f(x+y) = max(f(x), y) + min(f(y), x)

by Zhero, Jul 5, 2012, 1:18 AM

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$ .

George Xing.

Functions

by Potla, Feb 21, 2009, 1:05 PM

Find all functions $f: \mathbb{R}\longrightarrow \mathbb{R}$ such that
$f(x+y)+f(y+z)+f(z+x)\ge 3f(x+2y+3z)$
for all $x, y, z \in \mathbb R$ .

function

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

98 shouts

sonone's place

by P162008, Apr 25, 2025, 2:05 AM

by giangtruong13, Apr 24, 2025, 1:44 PM

by sqing, Apr 24, 2025, 11:59 AM

by truongphatt2668, Apr 22, 2025, 3:38 PM

by sqing, Oct 3, 2023, 8:11 AM

by MarkBcc168, Jul 9, 2023, 4:39 AM

by Jutaro, Oct 28, 2020, 8:44 PM

by va2010, Jul 7, 2016, 8:59 PM

by Zhero, Jul 5, 2012, 1:18 AM

by Potla, Feb 21, 2009, 1:05 PM