is a prime number. We have a number of cards, each of which has a number written on it such that each of the numbers 
appears at most once and 
exactly once. To design a game, for each pair of cards 
and 
,
If card 
,
If card 
,
must not win over card 
.
of natural numbers such that for every pair of natural numbers 
and 
,![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
such that
the following equation :
such that 
and 
.
be real square matrices of the same order, and suppose 
is invertible. Prove that![\[ (A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B \]](http://latex.artofproblemsolving.com/0/6/0/06007271017c2c48faa708366f1adc67db91f245.png)
is differentiable and 
Show that for some 
![$f:[0,1]\to\mathbb R$](http://latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)
is differentiable with 
If ![$|f'(x)|\le f(x)\;\forall\;x\in[0,1],$](http://latex.artofproblemsolving.com/b/9/d/b9d6088657cd85b37190b98abac1d94a18bb5916.png)
then show that 
be the unit circle in the complex plane. Let 
be the map given by 
We define 
and 
for 
The smallest positive integer 
such that 
is called period of 
Determine the total number of points 
of period 
denote the set of natural numbers, and let 
be nine distinct tuples in 
Show that there are 
distinct elements in the set 
whose product is a perfect cube.
and let 
be positive integers such that 
Prove that 
and determine when equality holds.
and 
for some integers 
Find the value of 
Y by
The altitudes of the acute-angled triangle 
intersect at the point 
. On the segments 
and 
, the points 
and 
are marked, respectively, so that 
. It turned out that the center of the circle 
circumscribed around the triangle 
lies on the line 
. Prove that the circle 
, which is circumscribed around the triangle , is tangent to the circle .
intersect at the point 
. On the segments 
and 
, the points 
and 
are marked, respectively, so that 
. It turned out that the center of the circle 
circumscribed around the triangle 
lies on the line 
. Prove that the circle 
, which is circumscribed around the triangle , is tangent to the circle .
and 
are omothetic, if their centers are 
and 
,
,
goes to 
and also 
by simmetry. Therefore, since 
lies on both 
