[/quote]
,
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 
for all 
such that for all real numbers 
and 
,![\[f(xf(y))+f(y)=f(x+y)+f(xy).\]](http://latex.artofproblemsolving.com/d/2/f/d2ff6ed448cf39e7ec9bce6b944965d5e89b9878.png)
table filled with natural numbers, we say it is a divisor table if:
-
,
-
,
for every 
.
is given. Determine the smallest natural number 
,
to tangent 
disjoint from 
on 
and 
.
is tangent to two fixed circles.
be a triangle and let 
be its incircle. Denote by 
and 
the points where 
and 
,
and 
the points on sides 
and 
,
the point of intersection of segments 
and 
.
is denoted by 
.
.
![\begin{align*}
\frac{a}{c} &= b-\sqrt{b}+c \\
\sqrt{\frac{a}{c}} &= \sqrt{b}+1 \\
\sqrt[4]{\frac{a}{c}} &=\sqrt[3]{b}-1
\end{align*}](http://latex.artofproblemsolving.com/0/c/d/0cd22d36acedbcb10d76d1b884aa35418273ed52.png)
be positive integers. Prove that the expression![\[\frac{\gcd(a+b,ab)}{\gcd(a,b)}\]](http://latex.artofproblemsolving.com/1/b/7/1b78a2027c3b7156fbb566901d2cfd088cc09f00.png)
is always a positive integer, and determine all possible values it can take.
be a convex pentagon such that 
,
,
.
to 
are concurrent.
that satisfies: for all different 
,
?
and internal bisector 
.
again at 
and 
cuts 
.
lie on 
.
at 
respectively. Let circumcircle of 
cuts 
.
.
satisfying
Prove
the following equation :
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
Y by khan.academy, RANDOM__USER
Let 
and 
be positive integers. In an 
grid, two cells are considered neighboring if they share a common edge. Kritesh performs the following actions:
1. He begins by writing
in any cell of the grid.
2. He then fills each remaining cell with a non-negative integer such that the absolute difference between the numbers in any two neighboring cells is exactly
.
Kritesh aims to fill the grid in a way that maximizes the sum of the numbers written in all the cells. Determine the maximum possible sum that Kritesh can achieve in terms of and .
(Kritesh Dhakal, Nepal)
and 
be positive integers. In an 
grid, two cells are considered neighboring if they share a common edge. Kritesh performs the following actions:1. He begins by writing
in any cell of the grid.2. He then fills each remaining cell with a non-negative integer such that the absolute difference between the numbers in any two neighboring cells is exactly
.Kritesh aims to fill the grid in a way that maximizes the sum of the numbers written in all the cells. Determine the maximum possible sum that Kritesh can achieve in terms of
and .(Kritesh Dhakal, Nepal)
.
is 
since there is a path between 
.
for fixed 
.
where 
constant 
maximum occurs when 
.
and this maximum value is equal to 
