be real numbers . Prove that
:
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 real numbers . Prove that
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 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
A non-degenerate triangle 
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
, let 
be the reflection of 
in sides 
respectively. Define a function 
such that 
is the circumcircle of 
. Is 
injective?
Note: The function is called injective if for any 
, 
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
, let 
be the reflection of 
in sides 
respectively. Define a function 
such that 
is the circumcircle of 
. Is 
injective?Note: The function
is called injective if for any 
, 
This post has been edited 1 time. Last edited by Project_Donkey_into_M4, Apr 20, 2025, 6:24 PM
