[/quote]
,
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 
.
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 
.
.
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. 
,
,
,
,
,
,
,
)
-
(i.e. an ordered 
and 
.
be 
positive integers such that the 
products![\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]](http://latex.artofproblemsolving.com/0/2/c/02cbff15c4d0118156a9d24be09360facb264d5a.png)
form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
be complex numbers such that 
and 
.
be positive real numbers such that 
and 
,
is defined below. 
Show that for all nonnegative integer 
is a positive integer.
Y by
Six boys and six girls are participating at a tango course. They meet every evening for three weeks (a total of 21 times). Each evening, at least one boy-girl pair is selected to dance in front of the others. At the end of the three weeks, every boy-girl pair has been selected at least once. Prove that there exists a person who has been selected on at least 5 distinct evenings.
Note: a person can be selected twice on the same evening.
Note: a person can be selected twice on the same evening.
be number of boys and girls dancing on 
day, respectively. Then at 
dances. If 
or 
,
and there is only one equality case. By rearreagment inequality, we know that sum of 
and 
.
and 
we can see that 
for all 
and 
.
and equality can hold only if 
.
,
.
,
.
times, so we have our problem.
