,
.
.
are fixed positive integers.
be the set of all functions 
with the following properties:
the following equality holds: 
such that
,
.
such that:
,
bisects neither the angle 
nor the angle 
.
lies inside ![\[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\]](http://latex.artofproblemsolving.com/c/2/0/c20761f3eadd054958f40259f3d1c05f26279783.png)
Prove that 
.
![\[y''+by'+cy=g(t)\]](http://latex.artofproblemsolving.com/7/8/8/7882dc954ab6dba2445b68d370d630ce1c66a6db.png)
with 
,
are constants has the form ![\[y(t)=\int^{t}_{t_0}K(t-s)g(s)ds\,\]](http://latex.artofproblemsolving.com/6/3/c/63c34df1c0013bf659bc76506a93873a5f879070.png)
and use variation of parameters in each case to find the solution. Is there any short way to calculate solution?
![$f:[1,8]\to R$](http://latex.artofproblemsolving.com/c/e/c/cec97ea0372e37977e8a77e8e77518a2344c1c43.png)
that:
that exist function 
such 
and discontinuous on 
![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
with 
and 
.
to be the projection of point 
such that 
is a square.
is a finite group of 
matrices with respect to multiplication. Prove that:
that 
be an infinite set of translated copies (i.e., obtained by parallel translation) of a given ellipse 
in the plane, and let 
be a fixed straight line. It is known that every straight line parallel to 
fibonacci numbers 
)
and 
that 
'strong' and 
'strong'
Y by Adventure10, Mango247
Given 
points 
on the 
-plane.
Let
. Denote by 
the distance between 
and the line 
. Let 
.
Answer the following questions:
(1) Express
in terms of 
.
(2) When
moves in the range of 
, express the maximum and minimum value of in terms of 
.
points 
on the 
-plane.Let
. Denote by 
the distance between 
and the line 
. Let 
.Answer the following questions:
(1) Express
in terms of 
.(2) When
moves in the range of 
, express the maximum and minimum value of in terms of 
.
and define 
as the perpendicular distance from point 
to 
.
on 
and 
;
;![$d_{k}^{2} = \frac{[1+\cot^{2}(\theta)]x_{k}^{2} + 2[\tan(\theta)+\cot(\theta)]x_{k}y_{k} + [1+\tan^{2}(\theta)]y_{k}^{2}}{[\tan(\theta)+\cot(\theta)]^2}$](http://latex.artofproblemsolving.com/7/c/2/7c263d679079d4bdd6b703dce356cc8a0e0d9d7c.png)
;![$d_{k}^{2} = \frac{\csc^{2}(\theta)x_{k}^{2} + 2[\tan(\theta)+\cot(\theta)]x_{k}y_{k} + \sec^{2}(\theta)y_{k}^{2}}{[\tan(\theta)+\cot(\theta)]^2} = \cos^{2}(\theta)x_{k}^{2} + 2\sin(\theta)\cos(\theta)x_{k}y_{k} +\sin^{2}(\theta)y_{k}^{2}$](http://latex.artofproblemsolving.com/1/0/4/104f062d859dc750fa96b4633e88e862789be897.png)
;
.
yields the critical values:
or 
or 
;
;
for 
(hence, a maximum) and 
for 
(hence, a minimum). Thus, we ultimately obtain:
;
.
