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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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Quadratic + cubic residue => 6th power residue?
Miquel-point   0
a minute ago
Source: KoMaL B. 5445
Decide whether the following statement is true: if an infinite arithmetic sequence of positive integers includes both a perfect square and a perfect cube, then it also includes a perfect $6$th power.

Proposed by Sándor Róka, Nyíregyháza
0 replies
1 viewing
Miquel-point
a minute ago
0 replies
J
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Cute property of Pascal hexagon config
Miquel-point   0
5 minutes ago
Source: KoMaL B. 5444
In cyclic hexagon $ABCDEF$ let $P$ denote the intersection of diagonals $AD$ and $CF$, and let $Q$ denote the intersection of diagonals $AE$ and $BF$. Prove that if $BC=CP$ and $DP=DE$, then $PQ$ bisects angle $BQE$.

Proposed by Géza Kós, Budapest
0 replies
Miquel-point
5 minutes ago
0 replies
J
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II_a - r_a = R - r implies A = 60
Miquel-point   0
9 minutes ago
Source: KoMaL B. 5421
The incenter and the inradius of the acute triangle $ABC$ are $I$ and $r$, respectively. The excenter and exradius relative to vertex $A$ is $I_a$ and $r_a$, respectively. Let $R$ denote the circumradius. Prove that if $II_a=r_a+R-r$, then $\angle BAC=60^\circ$.

Proposed by Class 2024C of Fazekas M. Gyak. Ált. Isk. és Gimn., Budapest
0 replies
Miquel-point
9 minutes ago
0 replies
J
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Cheating effectively in game of luck
Miquel-point   0
11 minutes ago
Source: KoMaL B. 5420
Ádám, the famous conman signed up for the following game of luck. There is a rotating table with a shape of a regular $13$-gon, and at each vertex there is a black or a white cap. (Caps of the same colour are indistinguishable from each other.) Under one of the caps $1000$ dollars are hidden, and there is nothing under the other caps. The host rotates the table, and then Ádám chooses a cap, and take what is underneath. Ádám's accomplice, Béla is working at the company behind this game. Béla is responsible for the placement of the $1000$ dollars under the caps, however, the colors of the caps are chosen by a different collegaue. After placing the money under a cap, Béla
[list=a]
[*] has to change the color of the cap,
[*] is allowed to change the color of the cap, but he is not allowed to touch any other cap.
[/list]
Can Ádám and Béla find a strategy in part a. and in part b., respectively, so that Ádám can surely find the money? (After entering the casino, Béla cannot communicate with Ádám, and he also cannot influence his colleague choosing the colors of the caps on the table.)

Proposed by Gábor Damásdi, Budapest
0 replies
+1 w
Miquel-point
11 minutes ago
0 replies
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Number of roots of boundary preserving unit disk maps
Assassino9931   3
N Today 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
Today at 2:12 AM
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|A/pA|<=p, finite index=> isomorphism - OIMU 2008 Problem 7
Jorge Miranda   2
N Yesterday 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
Yesterday at 8:00 PM
J
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Prove the statement
Butterfly   8
N Yesterday 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
Yesterday at 7:32 PM
J
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Functional equation from limit
IsicleFlow   1
N Yesterday 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
Yesterday at 4:22 PM
J
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f(m+n)≤f(m)f(n) implies existence of limit
Etkan   2
N Yesterday 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
Yesterday at 2:22 AM
Etkan
Yesterday at 3:19 PM
J
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Collinearity in a Harmonic Configuration from a Cyclic Quadrilateral
kieusuong   0
Yesterday 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
Yesterday at 2:26 PM
0 replies
J
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Find solution of IVP
neerajbhauryal   2
N Yesterday 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
Yesterday at 1:50 PM
J
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fourier series?
keroro902   2
N Yesterday 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
Yesterday at 12:54 PM
J
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Sets on which a continuous function exists
Creativename27   1
N Yesterday 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
Yesterday at 9:50 AM
alexheinis
Yesterday at 10:49 AM
J
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Japanese Olympiad
parkjungmin   6
N Yesterday at 5:01 AM 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
Yesterday at 5:01 AM
J
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J
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where a, b, c are positive real numbers
eyesofgod1930   2
N Apr 10, 2025 by sqing
where $a, b, c$ are positive real numbers.Prove that
$\frac{4}{\sqrt{a^{2}+b^{2}+c^{2}+4}}-\frac{9}{\sqrt{(a+b)\sqrt{(a+2c)(b+2c)}}}\leq \frac{5}{8}$
2 replies
eyesofgod1930
Jun 8, 2020
sqing
Apr 10, 2025
J
where a, b, c are positive real numbers
G H J
Reveal topic tags
inequalities
inequalities proposed
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eyesofgod1930
58 posts
eyesofgod1930
#1 h  Jun 8, 2020, 2:06 PM
Y by
where $a, b, c$ are positive real numbers.Prove that
$\frac{4}{\sqrt{a^{2}+b^{2}+c^{2}+4}}-\frac{9}{\sqrt{(a+b)\sqrt{(a+2c)(b+2c)}}}\leq \frac{5}{8}$
Z K Y
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duonghy93
40 posts
duonghy93
#2 h  Jun 8, 2020, 2:37 PM • 1 Y
Y by eyesofgod1930
I set t=a+b+c.Use the derivative f (t) => P max=5/8
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sqing
42194 posts
sqing
#3 h  Apr 10, 2025, 2:51 AM
Y by
Let $a,b,c>0$. Prove that$$\frac{8}{\sqrt{a^2+b^2+c^2+1}} - \frac{9}{(a+b)\sqrt{(a+2c)(b+2c)}} \leq \frac{5}{2}$$Let $a,b,c>0$. Prove that$$\frac{2}{(a+1)(b+1)(c+1)}-\frac{1}{\sqrt{a^2+b^2+c^2+1}}\leq 1$$
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