Good divisors and special numbers.

by Nuran2010, Apr 29, 2025, 4:52 PM

$N$ is a positive integer. Call all positive divisors of $N$ which are different from $1$ and $N$ beautiful divisors.We call $N$ a special number when it has at least $2$ beautiful divisors and difference of any $2$ beautiful divisors divides $N$ as well. Find all special numbers.

Vasc = 1?

by Li4, Apr 26, 2025, 1:33 PM

Find all integer tuples $(a, b, c)$ such that
\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]
Proposed by Li4, Untro368, usjl and YaWNeeT.

hard problem

by Cobedangiu, Apr 21, 2025, 1:51 PM

Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$

Counting graph theory

by MathSaiyan, Mar 17, 2025, 2:00 PM

Let $m$ and $n$ be positive integers. For a connected simple graph $G$ on $n$ vertices and $m$ edges, we consider the number $N(G)$ of orientations of (all of) its edges so that, in the resulting directed graph, every vertex has even outdegree.
Show that $N(G)$ only depends on $m$ and $n$, and determine its value.

2 variable functional equation in integers

by Supercali, Dec 20, 2022, 12:08 PM

Find all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ satisfying
$$f(x+f(xy))=f(x)+xf(y)$$for all integers $x,y$.

Another perpendicular to the Euler line

by darij grinberg, Mar 11, 2022, 1:01 PM

Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be a point in the plane such that $AP \perp BC$. Let $Q$ and $R$ be the reflections of $P$ in the lines $CA$ and $AB$, respectively. Let $Y$ be the orthogonal projection of $R$ onto $CA$. Let $Z$ be the orthogonal projection of $Q$ onto $AB$. Assume that $H \neq O$ and $Y \neq Z$. Prove that $YZ \perp HO$.

[asy]
import olympiad;
unitsize(30);
pair A,B,C,H,O,P,Q,R,Y,Z,Q2,R2,P2;
A = (-14.8, -6.6);
B = (-10.9, 0.3);
C = (-3.1, -7.1);
O = circumcenter(A,B,C);
H = orthocenter(A,B,C);
P = 1.2 * H - 0.2 * A;
Q = reflect(A, C) * P;
R = reflect(A, B) * P;
Y = foot(R, C, A);
Z = foot(Q, A, B);
P2 = foot(A, B, C);
Q2 = foot(P, C, A);
R2 = foot(P, A, B);
draw(B--(1.6*A-0.6*B));
draw(B--C--A);
draw(P--R, blue);
draw(R--Y, red);
draw(P--Q, blue);
draw(Q--Z, red);
draw(A--P2, blue);
draw(O--H, darkgreen+linewidth(1.2));
draw((1.4*Z-0.4*Y)--(4.6*Y-3.6*Z), red+linewidth(1.2));
draw(rightanglemark(R,Y,A,10), red);
draw(rightanglemark(Q,Z,B,10), red);
draw(rightanglemark(C,Q2,P,10), blue);
draw(rightanglemark(A,R2,P,10), blue);
draw(rightanglemark(B,P2,H,10), blue);
label("$\textcolor{blue}{H}$",H,NW);
label("$\textcolor{blue}{P}$",P,N);
label("$A$",A,W);
label("$B$",B,N);
label("$C$",C,S);
label("$O$",O,S);
label("$\textcolor{blue}{Q}$",Q,E);
label("$\textcolor{blue}{R}$",R,W);
label("$\textcolor{red}{Y}$",Y,S);
label("$\textcolor{red}{Z}$",Z,NW);
dot(A, filltype=FillDraw(black));
dot(B, filltype=FillDraw(black));
dot(C, filltype=FillDraw(black));
dot(H, filltype=FillDraw(blue));
dot(P, filltype=FillDraw(blue));
dot(Q, filltype=FillDraw(blue));
dot(R, filltype=FillDraw(blue));
dot(Y, filltype=FillDraw(red));
dot(Z, filltype=FillDraw(red));
dot(O, filltype=FillDraw(black));
[/asy]

Integer Functional Equation

by mathlogician, Sep 11, 2020, 11:52 PM

Let $f\colon\mathbb{N} \to \mathbb{N}$ be a function that satisfies$$\frac{ab}{f(a)} + \frac{ab}{f(b)} = f(a+b)$$for all positive integer pairs $(a,b).$ Find all possible functions $f.$

(Here, we define $\mathbb{N}$ as the set of all positive integers.)

H not needed

by dchenmathcounts, May 23, 2020, 11:00 PM

Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar.

Robin Son
This post has been edited 2 times. Last edited by v_Enhance, Oct 25, 2020, 6:01 AM
Reason: backdate

\sqrt{(1^2+2^2+...+n^2)/n}$ is an integer.

by parmenides51, Mar 26, 2020, 5:08 PM

Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.
This post has been edited 1 time. Last edited by parmenides51, Mar 28, 2020, 3:42 AM

Intersection of circumcircles of MNP and BOC

by Djile, Apr 8, 2013, 3:13 PM

Let $M$, $N$ and $P$ be midpoints of sides $BC, AC$ and $AB$, respectively, and let $O$ be circumcenter of acute-angled triangle $ABC$. Circumcircles of triangles $BOC$ and $MNP$ intersect at two different points $X$ and $Y$ inside of triangle $ABC$. Prove that \[\angle BAX=\angle CAY.\]

Old material is mostly Asymptote, new material is calculator programming

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  • 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

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