[/quote]
,
is lovely if there exist a positive integer 
and (not necessarily distinct) positive integers 
,
,
such that 
and 
for 
.
,
be an odd prime. Prove that:![\[\displaystyle\sum_{k=1}^{p-1}k^{2p-1} \equiv \frac{p(p+1)}{2} \pmod{p^2}\]](http://latex.artofproblemsolving.com/a/5/5/a55e75d4d9769ceec3a3ce359aac676c7c7875fd.png)
,
,
,
'![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
,
,
,
.
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
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.)
be a function such that
for all 
.
.
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.
with integer coefficients such that 
and 
is a square of an integer for all nonnegative integers 
.
is defined as follows: 
if 
and 
if 
.
,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
.
is a benefactor of 
because 
.
benefactors.
denotes the least natural such that
Find all naturals 
such that 
.
square is divided into unit squares. Is it possible to fill each unit square with a number 
in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?
be an odd prime number. How many 
of 
are there, the sum of whose elements is divisible by 
be a natural number. The numbers 
are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the 
GCDs obtained?
Y by Adventure10, Mango247, cubres
A sequence is generated as follows. The first term is 
. There after each term is derived from the previous term by substituting 
for and for 
Thus the first few terms are , , 
,
It is easy to see that each term is an initial segment for all later terms.
Which position in S is held by the hundredth?
Show that S is not periodic.
(Tournament of the towns 2nd half of 1988 Senior A level)
. There after each term is derived from the previous term by substituting 
for and for 
Thus the first few terms are , , 
,
It is easy to see that each term is an initial segment for all later terms.
Which position in S is held by the hundredth
?Show that S is not periodic.
(Tournament of the towns 2nd half of 1988 Senior A level)
This post has been edited 1 time. Last edited by SMOJ, May 13, 2014, 12:14 PM
.
defined as 
.
,
.
,
is the concatenation of two words 
.
.
is 
,
.
.
as 
.
.
.
,
.
