[/quote]
,
which satisfy ![\[f(x)\cdot f(y)\mid (1+2x)\cdot f(y)+(1+2y)\cdot f(x)\]](http://latex.artofproblemsolving.com/6/4/f/64f2720634f57b4d88db823b9931379386ac1c27.png)
for all non-negative integers 
and 
.
be natural numbers such that: 
and 
.
?
,
and 
are taken on the legs 
and 
,
and 
.
and 
.
be the real polonomial such that all roots are real and distinct. Prove that there is a rational number 
that all roots of 
are real numbers
from the reals to the reals such that ![\[ \left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz) \]](http://latex.artofproblemsolving.com/6/1/0/610e13a6f27341b5eed393923bb98e182537e254.png)
for all real 
specifies the greatest common divisor of 
and 
.
, such that 
Prove that :
be nonnegative real numbers such that 
.
Where 
be the Fibonacci sequence: 
if 
.
and that 
such that 
.
Y by
Hello guys, I want to know what was wrong with my PIE approach.
First here's the problem:
For a positive integer
, define the positive integer 
to be -safe if differs in absolute value by more than 
from all multiples of . For example, the set of 
-safe numbers is 
. Find the number of positive integers less than or equal to 
which are simultaneously 
-safe, 
-safe, and 
-safe.
My approach was finding the number of 7-unsafe numbers, 11-unsafe, and 13-unsafe, then finding 77-unsafe, 91-unsafe, 143-unsafe, and finally 1001-unsafe numbers and using PIE for a complementary counting approach. But somehow I got a number over than 10,000 for the total number of unsafe numbers. Is my approach valid and have I made arithmetic errors or does the PIE approach just not work?
First here's the problem:
For a positive integer
, define the positive integer 
to be -safe if differs in absolute value by more than 
from all multiples of . For example, the set of 
-safe numbers is 
. Find the number of positive integers less than or equal to 
which are simultaneously 
-safe, 
-safe, and 
-safe.My approach was finding the number of 7-unsafe numbers, 11-unsafe, and 13-unsafe, then finding 77-unsafe, 91-unsafe, 143-unsafe, and finally 1001-unsafe numbers and using PIE for a complementary counting approach. But somehow I got a number over than 10,000 for the total number of unsafe numbers. Is my approach valid and have I made arithmetic errors or does the PIE approach just not work?
.
