AoPS Academy
. [/center]
, and 
, 
. This is proven in solution 1a-1. We will refer to this as Trick 1.
+ bc, 
+ ac,
+ ab are powers of 2. (source, Sozopol 2023, Grades 8-9)
such that![\[ p \mid 2^{p^2+3p+2} + 3^{p^2+3p+2}. \]](http://latex.artofproblemsolving.com/d/d/f/ddfb5749fefd0c0b5d5a05e33a03d17ad0acdc0b.png)
, 
, for which 
is divisible by 
. (Note that is a prime number.) is a prime number which divides infinitely many numbers of the form 
where is a positive integer. What is the sum of all possible values of ?
satisfy 
. If the minimum possible value of 
equals 
, find 
.
be positive real numbers such that
and 
Prove that
Let and 
Prove that
Let and 
Prove that
Let and 
Prove that
be a triangle such that 
let 
be a point in 
and 
is the midpoint of and let 
and 
the incenter of 
and 
Prove that 
are concyclic
such that 
,
be a paralellogram with 
and 
. The circle 
passes by 
where is the midpoint of . The point 
is the intersection of and the line 
. The line 
is tangent to . Determine the length of the segment 
.
, the value of 
is always even!
denotes the largest integer that is less than or equal to 
, a contradiction
(after that it gives an error, because 
, where is greater than is quite large and apparently is too large for integer division), and the only contradictions it found were 
and 
. I'm wondering if there are any other contradictions other than these four...
Prove that
n = int(input())
counter = 0
for i in range(1, n+1):
if (((2**i)/i) - int((2**i)/i)) < 0.5 and int((2**i)/i) % 2 == 1:
counter += 1
print(i)
print(counter)
