interesting geo config (2/3)
by Royal_mhyasd, May 31, 2025, 11:36 PM
Let
be an acute triangle and
its orthocenter. Let
be a point on the parallel through
to
such that
. Define
and
as points on the parallels through
to
and through
to
similarly. If
are positioned around the sides of
as in the given configuration, prove that
are collinear.















interesting geo config (1\3)
by Royal_mhyasd, May 31, 2025, 11:18 PM
Let
be an acute triangle with
,
its orthocenter and
it's circumcenter. Let
be a point on the parallel through
to
such that
and
and
are on different sides of
. Denote by
the intersection of the circumcircle of
and
, where
is the reflection of
over
,
the midpoint of
,
the intersection of
and the parallel through
to
,
the intersection of
and the perpendicular through
to
and
a point on
such that
, where
is the midpoint of
. Prove that
lie on a line.
fiy it's 2am and i'm bored so i decided to look further into this interesting config that i had already made some observations on, maybe this problem is trivial from some theorem so if that's the case then i'm sorry lol
i'll probably post 2 more problems related to it soon, i'd say they're easier than this though

































fiy it's 2am and i'm bored so i decided to look further into this interesting config that i had already made some observations on, maybe this problem is trivial from some theorem so if that's the case then i'm sorry lol

This post has been edited 1 time. Last edited by Royal_mhyasd, Yesterday at 11:28 PM
pairs (m, n) such that a fractional expression is an integer
by cielblue, May 24, 2025, 8:38 PM
Find all pairs
of positive integers such that
is an integer.


Pentagon with given diameter, ratio desired
by bin_sherlo, May 11, 2025, 7:21 PM















This post has been edited 1 time. Last edited by bin_sherlo, May 11, 2025, 8:04 PM
weird conditions in geo
by Davdav1232, May 8, 2025, 8:24 PM
Let
be an isosceles triangle with
. Let
be a point on
. Let
be a point inside the triangle such that
and
Prove that the circumcenter of triangle
lies on line
.






![\[
CL \cdot BD = BL \cdot CD.
\]](http://latex.artofproblemsolving.com/1/6/7/16771838c95f86e92f79b8d049e46ab473e6287d.png)


Gcd of N and its coprime pair sum
by EeEeRUT, Apr 16, 2025, 1:33 AM
For a positive integer
, let
be all positive integers smaller than
that are coprime to
. Find all
such that
for all 
Here
is the largest positive integer that divides both
and
. Integers
and
are coprime if
.
Proposed by Paulius Aleknavičius, Lithuania







Here






Proposed by Paulius Aleknavičius, Lithuania
This post has been edited 3 times. Last edited by EeEeRUT, May 11, 2025, 11:49 AM
Number Theory
by fasttrust_12-mn, Aug 15, 2024, 11:19 PM
Find all positive intgers
and
such that
and
is a prime number




This post has been edited 1 time. Last edited by fasttrust_12-mn, Aug 15, 2024, 11:19 PM
Rootiful sets
by InternetPerson10, Sep 22, 2020, 11:49 PM
We say that a set
of integers is rootiful if, for any positive integer
and any
, all integer roots of the polynomial
are also in
. Find all rootiful sets of integers that contain all numbers of the form
for positive integers
and
.








Find all sequences satisfying two conditions
by orl, Jul 13, 2008, 1:21 PM
Let
be an integer. Find all sequences
satisfying the following conditions:
![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](//latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](//latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
Author: Dusan Dukic, Serbia


![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
Author: Dusan Dukic, Serbia
This post has been edited 2 times. Last edited by orl, Jan 4, 2009, 8:47 PM
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