AoPS Academy
: 
such :
be a general quadrangle which does not form an orthocentric system. Let 
, 
, 
, 
be the Euler-Poncelet point (
), isogonal center (
), midray homothetic center (
), inscribed square axes crosspoint (
) of , respectively.
be the orthocenter of 
, and define 
, 
, 
cyclically.
(
).
(
).
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
lie on the same conic, here denoted by 
.
be the nine-point conic of . and have 4 distinct intersection points, and let 
, 
, 
be the intersections, other than , of and . with respect to 
passes through and , and show that the center of and the orthocenter of coincide with .
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 
, where 
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)
, 
, 
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 
. Prove that ![\[ \lim_{x \rightarrow +\infty} \frac{f'(x)}{e^x}=1.\]](http://latex.artofproblemsolving.com/0/9/6/096b5b71111efdd7565d60424240e20ee3251382.png)
be real numbers. Find the ranges of such that the improper integral 
converges.
is defined on the interval 
. A point 
is said to be a fixed point of if 
. Suppose that is a differentiable function such that 
and 
. Show that if 
, then has a fixed point in the interval 
.
be the set of continuous functions ![$f: [0,1] \to (0, \infty )$](http://latex.artofproblemsolving.com/4/5/0/45018aa878524f722851ddc968a4f6aa10310bd5.png)
such that 
. For 
, let 
Determine 
.
with the following property: for each integer 
it is possible to partition a cube into 
smaller cubes.
smaller cubes or 
smaller cubes easily.(
)
smaller cubes we can also split it into 
and 
smaller cubes. Since we can "split" it into 
cube we can split it into 
for any 
. 
. Let 
be such that 
. Let 
, then 
and we can split it into smaller cubes whenever 
. QED
works; 
is the largest positive integer that can't be written as a sum of sevens and 26s. Obviously, we can do better.
could be 
; rather than replacing a 
cube with 
cubes, we replace it with one 
and cubes. Since adding is possible by doing the above and then splitting the large cube into eight pieces, we don't need to include it separately. From sevens and nineteens, we get a new estimate of 
. into a 
and 
cubes; this allows us to add . This is very useful, since 
and 
. The multiples 
form a complete system of residues mod 
, and 
is the last number we can't get as a sum of copies of 
. The new estimate is 
.
, 
, or 
smaller cubes, which effectively replace my dissections into 
and 
cubes respectively. You can look them up at mszew's link, or try to find them yourself if you're feeling clever. That gets down to 
.
be real numbers . Prove that
and use variation of parameters in each case to find the solution. Is there any short way to calculate solution?
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