[/quote]
,
be the circumcenter of an acute-angled triangle 
with 
.
meets the side 
at 
.
and 
are 
and 
,
and 
beyond 
,
and 
such that 
and 
.
is a rectangle if and only if 
.
satisfying
Prove
is a positive integer. Call all positive divisors of 
that verify the following equation for all 
:
and circumcircle 
.
passing through 
and 
passing through 
are tangent at 
of 
and 
,
of 
and 
.
and 
meet at 
.
be a point such that 
is tangent to 
is tangent to 
are collinear.
.
When does equality hold?
denote the set of all positive integers less than 
that are relatively prime to 
.
reducible over the rational numbers, given that it has integer coefficients?
blocks, each weighing at least 
.
with 
you can choose a subset of the blocks whose total weight is at least 
.
.
.
.
Y by Mango247
A convex 35-gon is drawn on the board, in which all the diagonals are drawn (it's sides are also drawn). Rostislav and Kostya take turns finding an open polyline in which 
or 
links emerge from each vertex, and erase its edges (the polyline can self-intersect). The one who cannot make a move loses. Rostislav starts. Who wins when both play correctly?
(Arseniy Nikolaev)
or 
links emerge from each vertex, and erase its edges (the polyline can self-intersect). The one who cannot make a move loses. Rostislav starts. Who wins when both play correctly?(Arseniy Nikolaev)
This post has been edited 2 times. Last edited by parmenides51, Jun 6, 2021, 9:22 AM
