AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
of positive integers, with 
prime, such that 
is a perfect square.
consisting of a prime number and two positive integers 
and 
such that 
and 
are both powers of .
be positive integers such that 
and 
. Let 
be the set of values attained by 
as 
runs through the positive integers. Show that is the set of all positive divisors of some positive integer.
be a set of integers (not necessarily positive) such that
with 
; and are elements of (possibly equal), then 
also belongs to . is the set of all integers.
the set of all positive integers. Find all functions 
such that for all positive integers 
and , the integer 
is nonzero and divides 
.
of positive integers such that 
is divisible by 
.
be an odd integer. Determine all functions 
from the set of integers to itself, such that for all integers and the difference 
divides 
for which there exist infinitely many positive integers 
such that ![\[ \frac{a^m+a-1}{a^n+a^2-1} \]](http://latex.artofproblemsolving.com/2/8/a/28a68e409451bbdcaee7c1286341f6aeea9ffdad.png)
is itself an integer.
such that ![\[ a^2b-1 \mid ab^3-1. \]](http://latex.artofproblemsolving.com/7/3/5/735fd61de9a18113846ce2048a00889dcc5ce8a4.png)
, 
be positive integers such that 
is a multiple of 
for all positive integers . Prove that 
. for which ![$\dfrac{n^2+1}{[\sqrt{n}]^2+2}$](http://latex.artofproblemsolving.com/7/e/d/7ed6c67114c0c7d262f67e5b7a733daab7997415.png)
is an integer. Here ![$[r]$](http://latex.artofproblemsolving.com/2/f/c/2fcabe2c2b8ebe2c428830acb9874cb8a8df790f.png)
denotes the greatest integer less than or equal to 
.
of positive integers such that 
is multiple of 
and 
is multiple of 
.
![\[\alpha =1-\cfrac{1}{2a_1-\cfrac{1}{2a_2-\cfrac{1}{2a_3-\cdots}}}\]](http://latex.artofproblemsolving.com/3/0/9/309e0411f481b6d77ccdfe8f18af9852074375aa.png)
where coefficients 
are positive integers, infinitely many of which are greater than 
. Prove that for every positive integer 
at least half of the numbers 
are even.
where coefficients are positive integers, infinitely many of which are greater than . Prove that for every positive integer at least half of the numbers are even.
satisfying
Prove
be a real number and 
.
.
appears either 
times in the sequence, and the latter happens precisely when 
for some 
where 
.
and 
for ![\[\beta = \cfrac{1}{\cfrac{1}{\alpha}-n} = (2a_{n} - 1) - \cfrac{1}{2a_{n+1} - \cfrac{1}{2a_{n+2} - \dots }}\]](http://latex.artofproblemsolving.com/a/c/5/ac55403d033fb4da2d56775563c376944844d6fb.png)
has more even numbers than odd numbers for all readable 
.
be the sequence defining 
is readable.
has more even numbers expressible as 
.
since 
.
has more even numbers than odd numbers. But, this conclusion stays the same no matter how we translate 
by an even number. In particular, if we replace 
,
and 
we obtain that the reader has successfully solved the problem.