Austrian Regional MO 2025 P4
by BR1F1SZ, Apr 18, 2025, 4:35 PM
Let 
be a positive integer that is not divisible by 
. Furthermore, let 
be a positive integer. Prove that none of the numbers of the form 
is a square number.
(Walther Janous)
be a positive integer that is not divisible by 
. Furthermore, let 
be a positive integer. Prove that none of the numbers of the form 
is a square number.(Walther Janous)
This post has been edited 1 time. Last edited by BR1F1SZ, 17 minutes ago
different bus lines in a city, each stopping at exactly 
stations and running in both directions. Nevertheless, for every two different stations there is always a bus line connecting these two stations. Determine the maximum number of stations in this city.
be an isosceles triangle with 
and circumcircle 
.
perpendicular to 
is denoted by 
.
be any point on 
with center 
intersects 
once more at point 
and the circumcircle 
.
and 
lie on a straight line.![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x^2 + y) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/c/7/c/c7cd7c9a4d4372afc23e4503bb1c05fc7d07e788.png)
be positive real numbers such that![\[
x + y + z = 3.
\]](http://latex.artofproblemsolving.com/1/c/6/1c6c52fbb7220d19c20a6f543c87b8eaa6bdf9aa.png)
Find the minimum value of![\[
x^2 + y^2 + z^3
\]](http://latex.artofproblemsolving.com/b/1/6/b1618ed295b6c950115cbec15f882cb4f9f0fcfa.png)
and express it in the form 
,
.
.
such that 
and has exactly 
such that:
with all 
is real number
of positive integers is called central if for every positive integer 
terms of the sequence is equal to 
of positive integers such that for every central sequence 
there are infinitely many positive integers 
.
be an acute triangle. Points 
,
.
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
,
and 
and 
and 
be real numbers such that 
Find the maximum value of the expression 
May 2020
January 2020