Double Sum

by P162008, Apr 25, 2025, 3:22 AM

Let $\xi = \lim_{m \to\infty} \sum_{n=4}^{m} \sum_{k=2}^{n-2} \frac{1}{\binom{n}{k}}.$ If the value of $\xi$ can be written as $\frac{m}{n}$ where $m$ and $n$ are co-prime positive integers then compute the value of $m^3 + n^3.$

This post has been edited 1 time. Last edited by P162008, Today at 3:23 AM
Reason: Typo

Polynomial Limit

by P162008, Apr 25, 2025, 1:47 AM

Let $p = \lim_{y\to\infty} \left(\frac{2}{y^2} \left(\lim_{z\to\infty} \frac{1}{z^4} \left(\lim_{x\to\infty} \frac{((y^2 + y + 1)x^k + 1)^{z^2 + z + 1} - ((z^2 + z + 1)x^k + 1)^{y^2 + y + 1}}{x^{2k}}\right)\right)\right)^y$ where $k \in N$ and $q = \lim_{n\to\infty} \left(\frac{\binom{2n}{n}. n!}{n^n}\right)^{1/n}$ where $n \in N$ . Find the value of $p.q.$

This post has been edited 1 time. Last edited by P162008, Today at 1:47 AM
Reason: Typo

Octagon Problem

by Shiyul, Apr 24, 2025, 11:41 PM

The vertices of octagon $ABCDEFGH$ lie on the same circle. If $AB = BC = CD = DE = 11$ and $EF = FG = GH = HA = sqrt2$ , what is the area of octagon $ABCDEFGH$ ?

I approached this problem by noticing that the area of the octagon is the area of the eight isoceles triangles with lengths $r$ , $r$ , and $sqrt2$ or 11. However, I didn't know how to find the radius. Can anyone give me a hint?

geometry unsolved

Sequence problem I never used

by Sedro, Apr 24, 2025, 4:19 PM

Let $\{a_n\}_{n\ge 1}$ be a sequence of reals such that $a_1=1$ and $a_{n+1}a_n = 3a_n+2$ for all positive integers $n$ . As $n$ grows large, the value of $a_{n+2}a_{n+1}a_n$ approaches the real number $M$ . What is the greatest integer less than $M$ ?

algebra

Sequences

linear recurrence

Polynomial Limit

by P162008, Apr 23, 2025, 11:55 AM

If $P_{n}(x) = \prod_{k=1}^{n} \left(x + \frac{1}{2^k}\right) = \sum_{k=0}^{n} a_{k} x^k$ then find the value of $\lim_{n \to \infty} \frac{a_{n - 2}}{a_{n - 4}}.$

This post has been edited 2 times. Last edited by P162008, Today at 1:25 AM
Reason: Typo

Theory of Equations

by P162008, Apr 23, 2025, 11:27 AM

Let $a,b,c,d$ and $e\in [-2,2]$ such that $\sum_{cyc} a = 0, \sum_{cyc} a^3 = 0, \sum_{cyc} a^5 = 10.$ Find the value of $\sum_{cyc} a^2.$

This post has been edited 2 times. Last edited by P162008, Wednesday at 11:28 AM
Reason: Typo

A problem involving modulus from JEE coaching

by AshAuktober, Apr 21, 2025, 8:44 AM

Solve over $\mathbb{R}$ :
$|x-1|+|x+2| = 3x.$
(There are two ways to do this, one being bashing out cases. Try to find the other.)

This post has been edited 2 times. Last edited by AshAuktober, Apr 21, 2025, 2:47 PM
Reason: TYPO CORRECTED< I AM SO SORRY

absolute value

Fun & Simple puzzle

by Kscv, Apr 13, 2025, 1:46 AM

$\angle DCA=45^{\circ},$ $\angle BDC=15^{\circ},$ $\overline{AC}=\overline{CB}$

$\angle ADC=?$

Attachments:

This post has been edited 1 time. Last edited by Kscv, Apr 13, 2025, 1:49 AM

geometry

Inequalities from SXTX

by sqing, Feb 18, 2025, 12:25 PM

T702. Let $a,b,c>0$ and $a+2b+3c=\sqrt{13}.$ Prove that $\sqrt{a^2+1} +2\sqrt{b^2+1} +3\sqrt{c^2+1} \geq 7$ S
T703. Let $a,b$ be real numbers such that $a+b\neq 0.$ . Find the minimum of $a^2+b^2+(\frac{1-ab}{a+b} )^2.$
T704. Let $a,b,c>0$ and $a+b+c=3.$ Prove that $\frac{a^2+7}{(c+a)(a+b)} + \frac{b^2+7}{(a+b)(b+c)} +\frac{c^2+7}{(b+c)(c+a)} \geq 6$ S

Attachments:

This post has been edited 5 times. Last edited by sqing, Feb 22, 2025, 3:48 AM

inequalities

algebra

FB = BK , circumcircle and altitude related (In the World of Mathematics 516)

by parmenides51, Apr 19, 2020, 2:09 AM

Let $BT$ be the altitude and $H$ be the intersection point of the altitudes of triangle $ABC$ . Point $N$ is symmetric to $H$ with respect to $BC$ . The circumcircle of triangle $ATN$ intersects $BC$ at points $F$ and $K$ . Prove that $FB = BK$ .

(V. Starodub, Kyiv)

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