[/quote]
,
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 
)
(
well). i want to study so if you guys have some easy or normal problems please send me
has a right angle at 
.
be the point on line 
such that 
and 
lies between 
.
is chosen so that 
and 
is the bisector of 
.
is chosen so that 
and 
is the bisector of 
.
be the midpoint of 
be the point such that 
is a parallelogram. Prove that 
and 
are concurrent.
satisfying the following conditions:
for all 
Define the set ![\[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\]](http://latex.artofproblemsolving.com/d/e/b/deb50830b9fe10c86ea169c044c1b26f515b2514.png)
then 
，
satisfying: 
is such that 
for any positive integers 
.
such that 
for all positive integers 
.
.
,
,
,
,
in some order.
be an acute triangle with orthocenter 
.
be the point such that the quadrilateral 
is a parallelogram. Let 
be the point on the line 
such that 
.
at 
.
.
and 
be points on circle 
divided into three equal parts. Construct three equal circles 
,
,
tangent to 
be any point on arc 
,
,
,
to circles 
.
,
,
,
be points on circle 
equal parts. Construct 
tangent to 
,
,
to circles 
of 
(where 
)
and 
with sum 
.![\[\sum_{i=1}^n \min\{{x_i^2, x_{i+1}}\}\leq \frac{1}{2}.\]](http://latex.artofproblemsolving.com/7/f/0/7f01a8d065c8e03233fe2254946522a5762f80e2.png)
where 
.
of the form 
with 
,
Determine all ordered pairs of polynomials 
with integer coefficients such that 
.
where 
denotes the Fibonacci sequence.**
is defined by the conditions
and 
for a set of indices 
.
satisfying the equation![\[
(I_p + A)^n = B^n + C^n.
\]](http://latex.artofproblemsolving.com/2/d/7/2d75d47203353160ccac356251659bd16a4cb5c8.png)
Y by Adventure10
Let 
be a probability distribution on the set of nonnegative integers. Select a number according to this distribution and repeat the selection independently until either a zero or an already selected number is obtained. Write the selected numbers in a row in order of selection without the last one. Below this line, write the numbers again in increasing order. Let 
denote the event that the number 
has been selected and that it is in the same place in both lines. Prove that the events 
are mutually independent, and 
.
T. F. Mori
be a probability distribution on the set of nonnegative integers. Select a number according to this distribution and repeat the selection independently until either a zero or an already selected number is obtained. Write the selected numbers in a row in order of selection without the last one. Below this line, write the numbers again in increasing order. Let 
denote the event that the number 
has been selected and that it is in the same place in both lines. Prove that the events 
are mutually independent, and 
.T. F. Mori
,
(Side note: the probability of obtaining an infinite sequence is 
.
.
be an 
in the altered process. Notice that 
Summing over all outcome sequences in the altered process gives
