[/quote]
,
be nonnegative real numbers such that 
.
Prove that
denote the sum of the sum of the factors of all positive 
,
is the sum of the factors of 
added to the sum of the factors of 
and so on until the sum of the factors of 
,
.
denote 
.
or find a counterexample
such that
are natural numbers satisfy that: 
and 
are prime numbers. Prove that 
is a perfect squared number
be nonnegative real numbers such that 
Prove that
Let 
Prove that
and 
with conditions,
is a prime number.
is a perfect square.
is also perfect square.
such that the numbers![\[n, f(n),f(f(n)),\dots,f^{m-1}(n)\]](http://latex.artofproblemsolving.com/b/3/0/b303b6dd25b8e172f41737ffa6c136dcc4db2266.png)
are distinct modulo 
for all integers 
with 
.
is defined by 
and 
for 
.
such that for every prime number 
and natural number 
,
is a complete residue system modulo 
for every natural number 
and 
.
,![\[abc\sqrt{1-d}+bcd\sqrt{1-a}+cda\sqrt{1-b}+dab\sqrt{1-c}\leq 1.\]](http://latex.artofproblemsolving.com/6/9/8/698270a4b6b055b71f9191de370eb674f1ff72f5.png)
be the circumcenter of triangle 
and let 
be the height from 
)
and 
be the midpoints of 
and 
respectively. Let 
and 
be the circumcircles of triangles 
and 
.
.
for 
Y by Adventure10 and 1 other user
After the schoolday is over, Juku must attend an extra math class. The teacher
writes a quadratic equation
with integer coefficients on the blackboard and Juku has to find its solutions. If they are not both integers, Jukumay go home. If the solutions are integers, then the teacher writes a new equation 
where 
and 
are the solutions of the previous equation taken in some order, and everything starts all over. Find all possible values for 
and 
such that the teacher can hold Juku at school forever.
writes a quadratic equation
with integer coefficients on the blackboard and Juku has to find its solutions. If they are not both integers, Jukumay go home. If the solutions are integers, then the teacher writes a new equation 
where 
and 
are the solutions of the previous equation taken in some order, and everything starts all over. Find all possible values for 
and 
such that the teacher can hold Juku at school forever.
