such that![\[
f(xy) = f(x)f(y) \;-\; f(x + y) \;+\; 1,
\quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/7/a/5/7a51a7be2306d18f4e80db1f5aac6e9bbf25c25c.png)
be an acute-angled triangle with 
.
and 
intersect at 
.
be the midpoint of 
.
is chosen on the extension of 
beyond 
is chosen on the segment 
such that 
.
and 
be the projection of 
onto 
.
bisects 
.
,
,
.
such that 
lies on side 
.
at 
.
?
of the numbers 
satisfy 
and 
?
be an odd positive integer satisfying 
with exactly 2 distinct prime factors. Prove that 
that satisfy the following condition: For every integer k such that 
there always exists a positive integer 
that is divisible by n and 
.
:
is drawn externally on side 
.
.
if 
.
for all 
.
Y by Nuterrow, laikhanhhoang_3011, Hedra, lazizbek42, geometry6, Mango247
These days I was trying to find the official booklet of Balkan MO Shortlist. But apparently, there's no big list of all Balkan shortlists for previous years. Through some sources, I have been able to find the official booklet for the following years. So if people have it for other years too, can they please put it on this thread, so that everything is in one place.
This post has been edited 1 time. Last edited by guptaamitu1, Jun 20, 2022, 11:17 PM
Reason: changing "BMO" to "Balkan MO" in title
Reason: changing "BMO" to "Balkan MO" in title
https://imomath.com/srb/zadaci/2018_bmo_shortlist.pdf
