AoPS Academy
squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.
satisfies
for any non-negative integers 
and 
. Determine the number of possible sets of integers 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
, centroid 
and orthocenter 
. Prove that 
is constant.
+1 w
be the positive real number. Prove that
converge?
be distinct irrational numbers. Show that there exists 
such that ![$\forall S\subset [n]\neq\varnothing,$](http://latex.artofproblemsolving.com/5/8/2/582af4a2239de777263d5c915e85f66cbbc31b55.png)
be differentiable on 
. Prove that there exists ![$x \in [0, 1]$](http://latex.artofproblemsolving.com/a/f/e/afe25008185bb22130658a17e0903591635b60ed.png)
such that
derivable on 
with 
and be the origin, 
In space given the vessel formed by the revolution of the segment 
about 
-axis. Denote the hight from to the surface of water by 
When the vessel is filled with water, drain the water out such that the displacement per time is 
at the time of 
that is to say, if the total volume of the water drained out till the time 
is 
then 
holds. Find the time required for draining out water.
converge?
![$$\int_{-\sqrt[3]{2}}^{1}(x^3+x^6)(x^2+2x^5)dx$$](http://latex.artofproblemsolving.com/4/a/6/4a605ab36af1ef8fdb64fbbd5b4009f422112e9e.png)
![$$\int \frac{d}{dx}\left [ \frac{e^x}{\ln(x)} \right] \cdot \frac{\ln(x)}{e^x} dx$$](http://latex.artofproblemsolving.com/0/f/8/0f85946a83d7b8a210c470cf8876a2a701a0051d.png)
such that 
![\[f(x)+f(\lfloor x\rfloor)f(\{x\})=x,\]](http://latex.artofproblemsolving.com/6/4/1/6411452c9f27ac7a8a25141927ed2f34e810e6dd.png)
and draw its graph. Find all 
for which the equation 
has solutions for any 
.
where 
and 
where 
.
and 
for all 
,
and so we must have 
and 
for some constants 
.
,
.