[/quote]
,
.
and 
be the feet of the altitudes from 
to 
to 
and 
as the reflections of 
be the circumcircle of 
.
the second intersection of line 
with 
with 
is the circumcenter of 
can be inscribed within a circle.
,
,
at 
,
be the midpoints of 
and 
,
and 
lie on 
so that 
and 
.
and 
intersect on the circumcircle of 
.
,
.
and 
,
is acute, and let 
be its orthocentre. Points 
and 
,
and 
and 
and 
.
intersect the circumcircle of 
be the midpoint of 
and let the perpendicular bisector of 
intersect 
at 
.
,
.
be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form 
,
and 
are not necessarily distinct. Determine 
.
be a cyclic quadrilateral with 
.
not containing 
and 
.
,
.
and find out the set of numbers 
(unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers 
.
and 
.
?
be the number of positive integers that divide the integer 
.
,
?
is a square of an integer.
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
let 
be the circumcircle of 
and let 
be its radius. Point 
such taht 
and point 
is the foot of the perpendicular from 
mets 
such that 
and 
lie on a line in that order. Finally, let 
and 
.
Y by
Given a right triangle 
, the point 
is the midpoint of the hypotenuse 
. A circle is circumscribed around the triangle 
, which intersects the segment 
at a point 
other than 
. It turned out that the segment 
is twice as large as the side 
. Find the acute angles of triangle .
(Mykola Moroz)
, the point 
is the midpoint of the hypotenuse 
. A circle is circumscribed around the triangle 
, which intersects the segment 
at a point 
other than 
. It turned out that the segment 
is twice as large as the side 
. Find the acute angles of triangle .(Mykola Moroz)
This post has been edited 1 time. Last edited by parmenides51, Feb 24, 2024, 8:12 PM
