and 
are natural numbers such that 
and 
and 
(
we mean a number from the set 
which is congruent with 
modulo 
.
such that ![\[p^q+q^p+p+q-5pq\]](http://latex.artofproblemsolving.com/b/a/b/bab8a0ee3555cf1136b6de56179012b920d0a2ad.png)
is a perfect square.
satisfying![\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]](http://latex.artofproblemsolving.com/a/e/6/ae65e3d53931fe9d60d0de76dc79540ccf44c26e.png)
and let 
denote the largest real number satisfying 
for all positive integers 
?
with circumcircle 
,
on 
,
,
be points on lines 
,
,
,
,
,
concur at a point 
.
be a point on line 
,
be a point on line 
be a point on line 
,
,
concur at a single point.
is a perfect square.
,
a line passing through 
be the points where the reflections of lines 
with respect to 
respectively. Prove that 
and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
,
are all greater than 
.
has its center at the incenter of 
is internally tangent to 
Y by
Given 
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
. What will be displayed after three successive runs: 5,2,5,1,4 | 5,2,5,1,11 | 5,2,5,1,20?
One answer is possible:
a)10;5;0;
b)20;7;0;
c)20;7;1;
d)10;10;0;
e)10;7;0;
f)20;5;0,
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
. What will be displayed after three successive runs: 5,2,5,1,4 | 5,2,5,1,11 | 5,2,5,1,20?One answer is possible:
a)10;5;0;
b)20;7;0;
c)20;7;1;
d)10;10;0;
e)10;7;0;
f)20;5;0,
This post has been edited 3 times. Last edited by MihaiT, Yesterday at 4:29 AM
