3D geometry theorem

by KAME06, Apr 21, 2025, 10:18 PM

Let $M$ a point in the space and $G$ the centroid of a tetrahedron $ABCD$ . Prove that:
$\frac{1}{4}(AB^2+AC^2+AD^2+BC^2+BD^2+CD^2)+4MG^2=MA^2+MB^2+MC^2+MD^2$

geometry

3D geometry

domino question

by kjhgyuio, Apr 21, 2025, 10:02 PM

........

Attachments:

domino

algebra

demonic monic polynomial problem

by iStud, Apr 21, 2025, 9:51 PM

(a) Let $P(x)$ be a monic polynomial so that there exists another real coefficients $Q(x)$ that satisfy
$P(x^2-2)=P(x)Q(x)$ Determine all complex roots that are possible from $P(x)$
(b) For arbitrary polynomial $P(x)$ that satisfies (a), determine whether $P(x)$ should have real coefficients or not.

algebra

polynomial

monic

fun set problem

by iStud, Apr 21, 2025, 9:47 PM

Given a set $S$ with exactly 9 elements that is subset of $\{1,2,\dots,72\}$ . Prove that there exist two subsets $A$ and $B$ that satisfy the following:
- $A$ and $B$ are non-empty subsets from $S$ ,
- the sum of all elements in each of $A$ and $B$ are equal, and
- $A\cap B$ is an empty subset.

Sets

combinatorics

basically INAMO 2010/6

by iStud, Apr 21, 2025, 9:31 PM

Call $n$ kawaii if it satisfies $d(n)+\varphi(n)+1=n$ ( $d(n)$ is the number of positive factors of $n$ , while $\varphi(n)$ is the number of integers not more than $n$ that are relatively prime with $n$ ). Find all $n$ that is kawaii.

number theory

relatively prime

trolling geometry problem

by iStud, Apr 21, 2025, 9:28 PM

Given a cyclic quadrilateral $ABCD$ with $BC<AD$ and $CD<AB$ . Lines $BC$ and $AD$ intersect at $X$ , and lines $CD$ and $AB$ intersect at $Y$ . Let $E,F,G,H$ be the midpoints of sides $AB,BC,CD,DA$ , respectively. Let $S$ and $T$ be points on segment $EG$ and $FH$ , respectively, so that $XS$ is the angle bisector of $\angle{DXA}$ and $YT$ is the angle bisector of $\angle{DYA}$ . Prove that $TS$ is parallel to $BD$ if and only if $AC$ divides $ABCD$ into two triangles with equal area.

Funny easy transcendental geo

by qwerty123456asdfgzxcvb, Apr 21, 2025, 7:23 PM

Let $\mathcal{S}$ be a logarithmic spiral centered at the origin (ie curve satisfying for any point $X$ on it, line $OX$ makes a fixed angle with the tangent to $\mathcal{S}$ at $X$ ). Let $\mathcal{H}$ be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.

Prove that for a point $P$ on the spiral, the polar of $P$ wrt. $\mathcal{H}$ is tangent to the spiral.

This post has been edited 3 times. Last edited by qwerty123456asdfgzxcvb, 4 hours ago

My hardest algebra ever created (only one solve in the contest)

by mshtand1, Apr 19, 2025, 9:37 PM

Find all functions $f: (0, +\infty) \to (0, +\infty)$ for which, for all $x, y > 0$ , the following identity holds:
$f(x) f(yf(x)) + y f(xy) = \frac{f\left(\frac{x}{y}\right)}{y} + \frac{f\left(\frac{y}{x}\right)}{x}$
Proposed by Mykhailo Shtandenko

algebra

two tangent circles

by KPBY0507, May 8, 2021, 1:19 PM

The incenter and $A$ -excenter of $\triangle{ABC}$ is $I$ and $O$ . The foot from $A,I$ to $BC$ is $D$ and $E$ . The intersection of $AD$ and $EO$ is $X$ . The circumcenter of $\triangle{BXC}$ is $P$ .
Show that the circumcircle of $\triangle{BPC}$ is tangent to the $A$ -excircle if $X$ is on the incircle of $\triangle{ABC}$ .

p^3 divides (a + b)^p - a^p - b^p

by 62861, Feb 23, 2017, 5:14 PM

Prove that there are infinitely many triples $(a, b, p)$ of positive integers with $p$ prime, $a < p$ , and $b < p$ , such that $(a + b)^p - a^p - b^p$ is a multiple of $p^3$ .

Noam Elkies

This post has been edited 1 time. Last edited by 62861, May 18, 2018, 11:58 PM

number theory

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 KAME06, Apr 21, 2025, 10:18 PM

by kjhgyuio, Apr 21, 2025, 10:02 PM

by iStud, Apr 21, 2025, 9:51 PM

by iStud, Apr 21, 2025, 9:47 PM

by iStud, Apr 21, 2025, 9:31 PM

by iStud, Apr 21, 2025, 9:28 PM

by qwerty123456asdfgzxcvb, Apr 21, 2025, 7:23 PM

by mshtand1, Apr 19, 2025, 9:37 PM

by KPBY0507, May 8, 2021, 1:19 PM

by 62861, Feb 23, 2017, 5:14 PM