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.
prime and 
a positive integer. Determine all pairs 
satisfying the equation: 
be a positive integer and let 
and 
be sets of integers satisfying the following conditions:
, 
and is a subset of 
, 
iff 
.
consecutive even numbers, such that the square of their product divides the sum of their squares.
be a function that satisfies the following equation for all real numbers 
:![\[
f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x + 17.
\]](http://latex.artofproblemsolving.com/f/6/2/f625b922656061221157fdb5a3b958cfa7d57108.png)
Find the value of 
. be a prime such that: 
. Simplify:
be a given integer. The Mint issues coins of 
different values 
, where each 
is a positive integer (the number of coins of each value is unlimited). A set of values 
is called lucky, if the sum 
can be collected in a unique way (namely, by taking one coin of each value).
with 
(b) Prove that every lucky set of values satisfies 
is a given prime number.
is a set which 
.
non-empty subsets of such that 
can be written in the form
where 
are non negative integers, such that
for every 
.
Prove that
Where 
Prove that
be real numbers such that 
. Find the smallest possible value of 
.
with the property that for every real numbers 
and 
it holds that
along the perimeter of a circle, then every word of the length 
consisting of letters and (i.e. 
, 
, 
, 
, 
, 
, 
, 
) occurs exactly once on the perimeter. Decide whether it is possible to write a sequence of letters from a 
-element alphabet along the perimeter of a circle in such a way that every word of the length 
(i.e. an ordered -tuple of letters) occurs exactly once on the perimeter.
consecutive integers then they must have all prime factors of 
. Primes below are 
so checking their factors in we get the numbers as 
and 
.
with the following property: for 
consecutive positive integers, each one is a divisor of 