be the orthocenter of acute triangle 
,
be the foot of the altitude from 
to 
,
be the reflection of 
.
intersects line 
and 
.
.
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?
.
