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Interesting problem from a friend
v4913   10
N 37 minutes ago by OronSH
Source: I'm not sure...
Let the incircle $(I)$ of $\triangle{ABC}$ touch $BC$ at $D$, $ID \cap (I) = K$, let $\ell$ denote the line tangent to $(I)$ through $K$. Define $E, F \in \ell$ such that $\angle{EIF} = 90^{\circ}, EI, FI \cap (AEF) = E', F'$. Prove that the circumcenter $O$ of $\triangle{ABC}$ lies on $E'F'$.
10 replies
v4913
Nov 25, 2023
OronSH
37 minutes ago
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Number of roots of boundary preserving unit disk maps
Assassino9931   3
N Yesterday at 2:12 AM by bsf714
Source: Vojtech Jarnik IMC 2025, Category II, P4
Let $D = \{z\in \mathbb{C}: |z| < 1\}$ be the open unit disk in the complex plane and let $f : D \to D$ be a holomorphic function such that $\lim_{|z|\to 1}|f(z)| = 1$. Let the Taylor series of $f$ be $f(z) = \sum_{n=0}^{\infty} a_nz^n$. Prove that the number of zeroes of $f$ (counted with multiplicities) equals $\sum_{n=0}^{\infty} n|a_n|^2$.
3 replies
Assassino9931
May 2, 2025
bsf714
Yesterday at 2:12 AM
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|A/pA|<=p, finite index=> isomorphism - OIMU 2008 Problem 7
Jorge Miranda   2
N Thursday at 8:00 PM by pi_quadrat_sechstel
Let $A$ be an abelian additive group such that all nonzero elements have infinite order and for each prime number $p$ we have the inequality $|A/pA|\leq p$, where $pA = \{pa |a \in A\}$, $pa = a+a+\cdots+a$ (where the sum has $p$ summands) and $|A/pA|$ is the order of the quotient group $A/pA$ (the index of the subgroup $pA$).

Prove that each subgroup of $A$ of finite index is isomorphic to $A$.
2 replies
Jorge Miranda
Aug 28, 2010
pi_quadrat_sechstel
Thursday at 8:00 PM
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Prove the statement
Butterfly   8
N Thursday at 7:32 PM by oty
Given an infinite sequence $\{x_n\} \subseteq [0,1]$, there exists some constant $C$, for any $r>0$, among the sequence $x_n$ and $x_m$ could be chosen to satisfy $|n-m|\ge r $ and $|x_n-x_m|<\frac{C}{|n-m|}$.
8 replies
Butterfly
May 7, 2025
oty
Thursday at 7:32 PM
J
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Functional equation from limit
IsicleFlow   1
N Thursday at 4:22 PM by jasperE3
Is there a solution to the functional equation $f(x)=\frac{1}{1-x}f(\frac{2 \sqrt{x} }{1-x}), f(0)=1$ Such That $ f(x) $ is even?
Click to reveal hidden text
1 reply
IsicleFlow
Jun 9, 2024
jasperE3
Thursday at 4:22 PM
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f(m+n)≤f(m)f(n) implies existence of limit
Etkan   2
N Thursday at 3:19 PM by Etkan
Let $f:\mathbb{Z}_{\geq 0}\to \mathbb{Z}_{\geq 0}$ satisfy $f(m+n)\leq f(m)f(n)$ for all $m,n\in \mathbb{Z}_{\geq 0}$. Prove that$$\lim \limits _{n\to \infty}f(n)^{1/n}=\inf \limits _{n\in \mathbb{Z}_{>0}}f(n)^{1/n}.$$
2 replies
Etkan
May 15, 2025
Etkan
Thursday at 3:19 PM
J
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Collinearity in a Harmonic Configuration from a Cyclic Quadrilateral
kieusuong   0
Thursday at 2:26 PM
Let \((O)\) be a fixed circle, and let \(P\) be a point outside \((O)\) such that \(PO > 2r\). A variable line through \(P\) intersects the circle \((O)\) at two points \(M\) and \(N\), such that the quadrilateral \(ANMB\) is cyclic, where \(A, B\) are fixed points on the circle.

Define the following:
- \(G = AM \cap BN\),
- \(T = AN \cap BM\),
- \(PJ\) is the tangent from \(P\) to the circle \((O)\), and \(J\) is the point of tangency.

**Problem:**
Prove that for all such configurations:
1. The points \(T\), \(G\), and \(J\) are collinear.
2. The line \(TG\) is perpendicular to chord \(AB\).
3. As the line through \(P\) varies, the point \(G\) traces a fixed straight line, which is parallel to the isogonal conjugate axis (the so-called *isotropic line*) of the centers \(O\) and \(P\).

---

### Outline of a Synthetic Proof:

**1. Harmonic Configuration:**
- Since \(A, N, M, B\) lie on a circle, their cross-ratio is harmonic:
\[ (ANMB) = -1. \]- The intersection points \(G = AM \cap BN\), and \(T = AN \cap BM\) form a well-known harmonic setup along the diagonals of the quadrilateral.

**2. Collinearity of \(T\), \(G\), \(J\):**
- The line \(PJ\) is tangent to \((O)\), and due to harmonicity and projective duality, the polar of \(G\) passes through \(J\).
- Thus, \(T\), \(G\), and \(J\) must lie on a common line.

**3. Perpendicularity:**
- Since \(PJ\) is tangent at \(J\) and \(AB\) is a chord, the angle between \(PJ\) and chord \(AB\) is right.
- Therefore, line \(TG\) is perpendicular to \(AB\).

**4. Quasi-directrix of \(G\):**
- As the line through \(P\) varies, the point \(G = AM \cap BN\) moves.
- However, all such points \(G\) lie on a fixed line, which is perpendicular to \(PO\), and is parallel to the isogonal (or isotropic) line determined by the centers \(O\) and \(P\).

---

**Further Questions for Discussion:**
- Can this configuration be extended to other conics, such as ellipses?
- Is there a pure projective geometry interpretation (perhaps using polar reciprocity)?
- What is the locus of point \(T\), or of line \(TG\), as \(P\) varies?

*This configuration arose from a geometric investigation involving cyclic quadrilaterals and harmonic bundles. Any insights, counterexamples, or improvements are warmly welcomed.*
0 replies
kieusuong
Thursday at 2:26 PM
0 replies
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Find solution of IVP
neerajbhauryal   2
N Thursday at 1:50 PM by Mathzeus1024
Show that the initial value problem \[y''+by'+cy=g(t)\] with $y(t_o)=0=y'(t_o)$, where $b,c$ are constants has the form \[y(t)=\int^{t}_{t_0}K(t-s)g(s)ds\,\]

What I did
2 replies
neerajbhauryal
Sep 23, 2014
Mathzeus1024
Thursday at 1:50 PM
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fourier series?
keroro902   2
N Thursday at 12:54 PM by Mathzeus1024
f(x)=$\sum _{n=0}^{\infty } \text{cos}(nx)/2^{n}$
f(x) = ?
thanks
2 replies
keroro902
May 14, 2010
Mathzeus1024
Thursday at 12:54 PM
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Sets on which a continuous function exists
Creativename27   1
N Thursday at 10:49 AM by alexheinis
Source: My head
Find all $X\subseteq R$ that exist function $f:R\to R$ such $f$ continuous on $X$ and discontinuous on $R/X$
1 reply
Creativename27
Thursday at 9:50 AM
alexheinis
Thursday at 10:49 AM
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Japanese Olympiad
parkjungmin   6
N May 15, 2025 by mathNcheese_aops
It's about the Japanese Olympiad

I can't solve it no matter how much I think about it.

If there are people who are good at math

Please help me.
6 replies
parkjungmin
May 10, 2025
mathNcheese_aops
May 15, 2025
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Another two parallels
jayme   2
N Apr 28, 2025 by jayme
Dear Mathlinkers,

1. ABCD a square
2. (A) the circle with center at A passing through B
3. P the points of intersection of the segment AC and (A)
4. I the midpoint of AB
5. Q the point of intersection of the segment IC and (A)
6. M the foot of the perpendicular to (AB) through P.
7. Y the point of intersection of the segment MC and (A)
8. X the point of intersection de AY and BC.

Prove : QX is parallel to AB.

Jean-Louis
2 replies
jayme
Apr 28, 2025
jayme
Apr 28, 2025
J
Another two parallels
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jayme
9795 posts
jayme
#1 h  Apr 28, 2025, 9:21 AM
Y by
Dear Mathlinkers,

1. ABCD a square
2. (A) the circle with center at A passing through B
3. P the points of intersection of the segment AC and (A)
4. I the midpoint of AB
5. Q the point of intersection of the segment IC and (A)
6. M the foot of the perpendicular to (AB) through P.
7. Y the point of intersection of the segment MC and (A)
8. X the point of intersection de AY and BC.

Prove : QX is parallel to AB.

Jean-Louis
This post has been edited 2 times. Last edited by jayme, Apr 28, 2025, 1:29 PM
Reason: typo
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jayme
9795 posts
jayme
#2 h  Apr 28, 2025, 1:34 PM
Y by
Sorry, I have corrected some typos...

Sincerely
Jean-Louis
Z K Y
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jayme
9795 posts
jayme
#3 h  Apr 28, 2025, 5:36 PM
Y by
Any ideas?

Sincerely
Jean-Louis
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