[/quote]
,
,
,
be an acute triangle with orthocenter 
,
be a point on the side 
,
and 
.
and 
are the feet of the altitudes from 
is the circumcircle of 
,
is a diameter of 
the circumcircle of triangle 
,
be the point such that 
is a diameter of 
and 
be real numbers.Prove that
and for all 
,
Determine the smallest 
.
![\[T_1 z = \frac{z + 2}{z + 3}, \qquad T_2 z = \frac z{z + 1},\]](http://latex.artofproblemsolving.com/b/7/2/b72b5e302cecdf2649c43a078aea3aa9a88ef2cc.png)
find 
and 
be a polynomial of degree 
we have 
.
be an 
square matrix with eigenvalues 
(each 
having algebraic multiplicity 
,
)
be a polynomial. It is known that if 
is an eigenvalue of 
is an eigenvalue of 
.
(counting multiplicities)?
(with multiplicity 3),
(with multiplicity 2),
(with multiplicity 1),
(with multiplicity 3), 
(with multiplicity 2), and 
(with multiplicity 1)?
vs. equation 
)
Y by greenturtle3141
a) Consider a square ![$[0, n]^2$](//latex.artofproblemsolving.com/b/3/8/b38b78b68b1c5651fa5825bf88eb10f801f4d821.png)
in the plane, for natural 
. Erase all the points which have both coordinates non-integer. We are left with one-dimensional cell complex which we will call 
. Find the maximal 
such that
for any continuous map of to 
there is a point with at least 
preimages.
b) The same for maps to
of the
two-dimensional complex obtained from![$[0, n]^3\subset R^3$](//latex.artofproblemsolving.com/8/1/c/81c72fc51fd486cfe352438f452965f887fbf8f9.png)
by erasing all the points with all coordinates non-integer.
![$[0, n]^2$](http://latex.artofproblemsolving.com/b/3/8/b38b78b68b1c5651fa5825bf88eb10f801f4d821.png)
in the plane, for natural 
. Erase all the points which have both coordinates non-integer. We are left with one-dimensional cell complex which we will call 
. Find the maximal 
such thatfor any continuous map of
to 
there is a point with at least 
preimages.b) The same for maps to
of thetwo-dimensional complex obtained from
![$[0, n]^3\subset R^3$](http://latex.artofproblemsolving.com/8/1/c/81c72fc51fd486cfe352438f452965f887fbf8f9.png)
by erasing all the points with all coordinates non-integer.This post has been edited 2 times. Last edited by FFA21, Mar 24, 2025, 2:23 AM
