be an integer. Prove that there exists a set 
of 
positive integers satisfying the following property: For every 
the set 
.
,
,
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
,
is an integer. Prove that there is a positive integer 
such that 
for all 
.
be a positive integer. On the table, we have 
ornaments in 
different colours.
grid of real numbers. Determine the greatest possible number of cells 
in the grid such that the entry in 
is called a 
if it satisfies the following conditions:
contains at least two elements.
For all 
with 
,
,
and 
,
For all 
,
is irrational.
being a natural number, let 
be a divisor of 
satisfying 
.
as the sum of the digits of a natural number 
.![\[
\{x\} \text{ as the fractional part of } x.
\]](http://latex.artofproblemsolving.com/b/f/7/bf7b4cc20cbdcee755c73f967c6d231a0896b4b4.png)
The smallest natural number ![\[
\left\{\frac{n}{s(n)}\right\} = \frac{1}{6}
\]](http://latex.artofproblemsolving.com/e/9/3/e9324df5f1fe12512254c2a8e273629b6b824126.png)
is 
.
,
,
positive numbers that are not all necessarily different but all 
that contain at least 3 elements and do not have any two elements whose difference is exactly 7 is ...
with exactly 2 distinct prime factors. Prove that 
![\[
19^{17^{15^{13^{11^{9^{7^{5^3}}}}}}} \mod 100
\]](http://latex.artofproblemsolving.com/d/3/d/d3d8657e2ab69cdc0886e2498ab03f7fd959056f.png)
Y by
Given 
and centroid 
. 
is the reflection of 
over , similarily define 
and 
.
1) Prove that if we reflect over , it is the same as reflecting over 
Now, do same for reflecting over and over , 
over and over .
2) Is the big triangle similar to?
and centroid 
. 
is the reflection of 
over , similarily define 
and 
.1) Prove that if we reflect
over , it is the same as reflecting over 
Now, do same for reflecting
over and over , 
over and over .2) Is the big triangle similar to
?
bump
is the midpoint of 
and consider the homothety with centre 
then 
and 
,
,
(and it is not close to being similar). Maybe you can provide a diagram to see what you mean?
