Coaxial circles related to Gergon point

by Headhunter, Apr 3, 2025, 2:48 AM

Hi, everyone.

In $\triangle$$ABC$, $Ge$ is the Gergon point and the incircle $(I)$ touch $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively.
Let the circumcircles of $\triangle IDGe$, $\triangle IEGe$, $\triangle IFGe$ be $O_{1}$ , $O_{2}$ , $O_{3}$ respectively.

Reflect $O_{1}$ in $ID$ and then we get the circle $O'_{1}$
Reflect $O_{2}$ in $IE$ and then the circle $O'_{2}$
Reflect $O_{3}$ in $IF$ and then the circle $O'_{3}$

Prove that $O'_{1}$ , $O'_{2}$ , $O'_{3}$ are coaxial.
geometry
circumcircle
geometric transformation
reflection
L

pretty well known

by dotscom26, Apr 3, 2025, 2:03 AM

Let $\triangle ABC$ be a scalene triangle such that $\Omega$ is its incircle. $AB$ is tangent to $\Omega$ at $D$. A point $E$ ($E \notin \Omega$) is located on $BC$.

Let $\omega_1$, $\omega_2$, and $\omega_3$ be the incircles of the triangles $BED$, $ADE$, and $AEC$, respectively.

Show that the common tangent to $\omega_1$ and $\omega_3$ is also tangent to $\omega_2$.
geometry
Inversion
L

Is this FE solvable?

by Mathdreams, Apr 1, 2025, 6:58 PM

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ such that \[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]for all reals $x$ and $y$.
algebra
functional equation
L

Thanks u!

by Ruji2018252, Mar 26, 2025, 8:45 AM

Find all $f:\mathbb{R}\to\mathbb{R}$ and
\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]
This post has been edited 1 time. Last edited by Ruji2018252, Mar 26, 2025, 9:30 AM
Reason: Sori
functional equation
IMO
L

f((x XOR f(y)) + y) = (f(x) XOR y) + y

by the_universe6626, Feb 21, 2025, 1:23 PM

Find all functions $f:\mathbb{Z}_{\ge0}\rightarrow\mathbb{Z}_{\ge0}$ such that
\[f((x\oplus f(y))+y)=(f(x)\oplus y)+y\]Note: $\oplus$ denotes the bitwise XOR operation. For example, $1001_2 \oplus 101_2 = 1100_2$.

(Proposed by ja.)
algebra
number theory
L

Modular NT

by oVlad, Jul 31, 2024, 12:29 PM

Find all the positive integers $a{}$ and $b{}$ such that $(7^a-5^b)/8$ is a prime number.

Cosmin Manea and Dragoș Petrică
number theory
modular arithmetic
prime numbers
L

no numbers of the form 80...01 are squares

by Marius_Avion_De_Vanatoare, Jun 10, 2024, 2:37 PM

Prove that a number of the form $80\dots01$ (there is at least 1 zero) can't be a perfect square.
number theory
L

2024 8's

by Marius_Avion_De_Vanatoare, Jun 10, 2024, 2:28 PM

Prove that the number $ \underbrace{88\dots8}_\text{2024\; \textrm{times}}$ is divisible by 2024.
number theory
L

Equation with powers

by a_507_bc, May 25, 2024, 5:27 PM

Find all non-negative integers $x, y$ and primes $p$ such that $$3^x+p^2=7 \cdot 2^y.$$
number theory
L

An nxn Checkboard

by MithsApprentice, Oct 3, 2005, 10:41 PM

Some checkers placed on an $n \times n$ checkerboard satisfy the following conditions:

(a) every square that does not contain a checker shares a side with one that does;

(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.

Prove that at least $(n^{2}-2)/3$ checkers have been placed on the board.
induction
combinatorics proposed
combinatorics
L

The oldest, shortest words — "yes" and "no" — are those which require the most thought.

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