be a general quadrangle which does not form an orthocentric system. Let 
,
,
be the Euler-Poncelet point 
)
)
)
)
be the orthocenter of 
,
,
,
cyclically.
)
)
,
,
,
,
,
,
,
,
,
,
,
lie on the same conic, here denoted by 
.
be the nine-point conic of 
,
,
be the intersections, other than 
passes through 
be a metric space and 
be a function such that 
.
Prove that for all 
,
exists, where 
is 
.
Prove that the amount of the limit does not depend on choosing 
.
![\[\iint_{R} \sin(xy) \,dx\,dy, \quad R = \left[0, \frac{\pi}{2}\right] \times \left[0, \frac{\pi}{2}\right]\]](http://latex.artofproblemsolving.com/a/2/0/a205868dd99f4794de53ad1db50f34b17c70a923.png)
![\[y''+by'+cy=g(t)\]](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?
, 
,
be a positive sequence and let 
be a constant such that ![\[ \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\]](http://latex.artofproblemsolving.com/5/3/c/53ccb27390a368e1a74afae1da4f7566ad86af8a.png)
Prove that there exists a constant 
such that ![\[ \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\]](http://latex.artofproblemsolving.com/e/7/4/e744b1b1179f879dbea84a53622afc5ca8e45fe2.png)
be a real function such that![\[ \lim_{x \rightarrow +\infty} \frac{f(x)}{e^x}=1\]](http://latex.artofproblemsolving.com/7/f/e/7fe200614381665ef11aed463fbbee51f8461bb3.png)
for all sufficiently large 
.![\[ \lim_{x \rightarrow +\infty} \frac{f'(x)}{e^x}=1.\]](http://latex.artofproblemsolving.com/0/9/6/096b5b71111efdd7565d60424240e20ee3251382.png)
be real numbers. Find the ranges of 
converges.
is defined on the interval 
.
is said to be a fixed point of 
.
and 
.
,
.
be the set of continuous functions ![$f: [0,1] \to (0, \infty )$](http://latex.artofproblemsolving.com/4/5/0/45018aa878524f722851ddc968a4f6aa10310bd5.png)
such that 
.
,
Determine 
.
Y by
Let 
be triangle and point 
be 
altitude of 
.
Let
be a point on 
and 
such that 
and 
. Point 
is the intersection of 
and 
. Point 
be intersection of 
and 
. Line 
intersect circumcircle of again at 
.
Prove that
(a)
.
(b)
bisect .
be triangle and point 
be 
altitude of 
.Let
be a point on 
and 
such that 
and 
. Point 
is the intersection of 
and 
. Point 
be intersection of 
and 
. Line 
intersect circumcircle of again at 
.Prove that
(a)
.(b)
bisect .
