[/quote]
,
with real coefficients such that for all reals 
such that 
we have the following relations![\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]](http://latex.artofproblemsolving.com/4/4/9/4491a53f9c6f96c18e42eb984b8e83566140bfdb.png)
be a triangle with 
and 
,
be the midpoint of 
.
be a variable point of the circumcircle of 
be the point on the segment 
such that 
.
at a unique point 
.
Compute -![\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg]
\end{align*}](http://latex.artofproblemsolving.com/8/6/f/86fb48db976012106bd30f74cb6ada8c6e6d7800.png)
For 
define 
Determine the open interval(s) (if any) where 
is decreasing and all the open interval(s) (if any) where 
Determine all the local minima of 
be a parallelogram 
be a point in its interior such that 
denote the set of all positive integers less than 
that are relatively prime to 
.
reducible over the rational numbers, given that it has integer coefficients?
such that ![\[f(x+f(x)+2f(y))+f(2f(x)-y)=4x+f(y)\]](http://latex.artofproblemsolving.com/4/4/2/4423651bb829b92debd65a23f6cee3d4d4b8530f.png)
holds for all reals 
and 
.
Prove that
.
children, each of whom selects
tables, with 
and 
.
Let 
.
.
Where 
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
.
be the circle with diameter 
.
be the intersection of 
and 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
respectively. Let 
be the intersection of 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
![\[ p = \frac{2ab}{a + b} \]](http://latex.artofproblemsolving.com/0/f/4/0f4297ec59fe97c92fda75af4cc835be6b9a2507.png)
, ![\[ p, a, b \in \mathbb{C} \]](http://latex.artofproblemsolving.com/b/4/4/b4499970ac49d770a810d7345fedef7773cd03f5.png)
Prove that
Y by Amir Hossein, Davi-8191, rustam, Kayak, Tenee, Purple_Planet, centslordm, jhu08, megarnie, Adventure10, Mango247, lian_the_noob12, Sedro
A rectangle 
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even.
Proposed by Jeck Lim, Singapore
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even.Proposed by Jeck Lim, Singapore
This post has been edited 2 times. Last edited by MarkBcc168, Jul 15, 2018, 12:57 PM
.
board. Color 
and 
with 
if 
,
if 
,
if 
.
.
.
,
.
is even and so 
is odd. Repeat in the other direction (up and down instead of left and right).
,
tile it with dominoes and for 
mark the leftmost corner and tile the remaining rectangle with dominoes. Observe that each tiling 
maps to a tiling 
consisting only of dominoes and boxes such that 
white if 
,
squares have at most 2 corners.
,
be the side of one other small rectangle as well?
are both even or odd; we also know for sure that the distance from the horizontal sides of this rectangle to 
![$[0,a]\times [0,b]$](http://latex.artofproblemsolving.com/6/0/1/601076a1c0a357d6bd2c71085669c8c3074b0129.png)
and we are considering ![$[x,x+c]\times [y,y+d]$](http://latex.artofproblemsolving.com/3/6/b/36b9f1541a35e6b750fd01d3ff4dbb3765c805c2.png)
,
,
are odd and 
is possible. Let us just focus on the first half. We can note that by weighting each cell by 
,
must be odd, so there must be some rectangle with 
odd.
.
.
,
black corners. Iff the rectangle's distances from the four sides are all odd or all even, then it has 
white and 
,
in the dissection has the following property: the distance from 
,
to be the number of black squares contained in it, and 
to be the number of white squares in it. Then the condition on our required rectangle 
.
for every rectangle. Then 
where 
refers to the rectangle covering the entire of 
,
Such a small rectangle would have odd area and thus odd side lengths, meaning the distances from each pair of parallel sides have the same parity (i.e. the distance from the top and bottom would both be odd or both be even)
,
,
and 
rectangles. Suppose such a rectangle doesn't exists. Then, for any rectangle 
which is contained in some division of 
If a smaller rectangle 
and 
or in a 
and 
will never generate rectangles with that.
By a 
rotation, it suffices to consider the first case. Note that the piece 
squares. Color the cells chessboard-style with the corners all white and number the white cells 
with even area
with even area
with even area
with odd area
with odd area
with odd area
![\[
f(\mathcal{D}) = \int\int_\mathcal{D} e^{\pi i(x+y)}
\]](http://latex.artofproblemsolving.com/7/2/e/72e842bd369b5836cf555d4b33a018fc33ed45fd.png)
is nonzero iff it has all odd side lengths. Since 
is nonzero, the 
