by 
and 
such that 
.
is large enough, in every 
square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by 
.
with 
and 
exist such that 
are of the same color.
points such that these 
where 
is a positive real number instead of 
be the solid square region with vertices at![\[
(0, 0), \left(\frac{1}{2}, \frac{1}{2}\right), (1, 0), \left(\frac{1}{2}, -\frac{1}{2}\right).
\]](http://latex.artofproblemsolving.com/a/9/8/a9898f9237f07eaef5d17a9ac689d80ca631863f.png)
consists of two copies of 
:
counterclockwise around the origin and scaled by a factor of 
,
.
below.
be the limiting region of the sequence 
.
for relatively prime positive integers 
and 
.
.
shows that after each species divides and then each amoeba cell preys on exactly one bacterial cell, it turns out kill bacteria. Determine the ratio of the number of amoeba to the number of bacteria on the first day.
.
cm. The first nine balls are placed on the table so that
whose sides are 
cm, 
cm, and 
cm. Find the maximum possible area of the rectangle can be made in the triangle 
people waiting in line to buy tickets to a show with the price of one ticket is 
Rp.. Known 
Rp. in banknotes and the rest is only has a banknote of 
with
regions. Every pair of lines in the same group are parallel. Let 
and 
respectively be the number of lines in groups 1, 2, and 3. If no lines in group 3 go through the intersection of any two lines (in groups 1 and 2, of course), then the least number of lines required in order to have more than 2018 regions is ....
where 
and 
are squares with both planes being parallel. The length of the sides of 
and 
,
.
is perpendicular to the plane 
and 
and calculate the volume of the 3D figure which is the intersection of pyramids 
is on the 3rd row, 4th column, and on the 6th diagonal. Meanwhile the position of the number 
is on the 3rd row, 5th column, and 7th diagonal.
based on its row, column, and diagonal.
.
dan 
.
is a fraction in its simplest form.
.
be given in the following graphs.
or 
,
,
,
,
,
.
.
with 
for all 
where 
is the domain of 
so that 
.
cm and 
cm. All four of its vertices lie on a circle. Calculate the area of quadrilateral 
,
and 
.
,
is a 3-digit number whose number in its tens place is 5. Determine the number/quantity of all possible values of 
(the original problem does not have an overline, which I fixed) is arranged from the set 
.
.
is 
.
,
.
.
that satisfies![\[
a^3+b^3=1957
\]](http://latex.artofproblemsolving.com/5/d/8/5d8d2a40734ec6b0a9149f9a4ae3a5828d5055f3.png)
![\[
(a+b)(a+1)(b+1)=2014
\]](http://latex.artofproblemsolving.com/3/0/0/300f2d16e669a1d8d9c54b357a6eaf7a8eb5d826.png)
Hence, find the value of 
Y by Adventure10, Mango247
Let 
be the centre of the incircle of 
. Points 
are the intersection points of the circles circumscribed about triangles 
respectively with the bisectors of the angles at 
respectively 
. Also 
is the midpoint of segment 
, 
is the reflection of with respect to and 
is the reflection of with respect to line . Prove that the points 
and lie on the same circle.
be the centre of the incircle of 
. Points 
are the intersection points of the circles circumscribed about triangles 
respectively with the bisectors of the angles at 
respectively 
. Also 
is the midpoint of segment 
, 
is the reflection of with respect to and 
is the reflection of with respect to line . Prove that the points 
and lie on the same circle.This post has been edited 1 time. Last edited by StefanS, Apr 13, 2012, 5:21 PM
is on the segment 
is a deltoid
)
.
,
is a parallelogram (their diagonals divide in half), thus 
.
