and 
respectively be the incentre, circumcentre and circumcircle of triangle 
.
are chosen on 
,
is the foot of the altitude from 
to 
.
are similarly defined, prove that lines 
and 
concurr on 
.
and let 
and 
be tangency points of the incircle of 
,
and 
respectively. Let the circle with centre 
passing through 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
that is not isosceles at 
,
be its incircle, which is tangent to 
at 
,
and 
intersect the line passing through 
at 
and 
,
and perpendicular to 
intersect 
and 
at 
and 
,
are collinear and that 
are concurrent.
positive integer numbers such that the least common multiple (LCM) of all these numbers is not a perfect square. Mykhailo consecutively hides one of these numbers and writes down the LCM of the remaining 
numbers that are not hidden. What is the maximum number of the 
have the property that 
is a perfect square for infinitely many positive integers 
Y by
Let 
be a positive integer. For every positive integer 
the sequence 
is defined, where 
are integers. Among these sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by ?
be a positive integer. For every positive integer 
the sequence 
is defined, where 
are integers. Among these sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by ?
