be reals such that 
. Prove that
Let 
. Prove that
be a positive integer and 
a prime number. Consider the sequence 
defined by![\[
a_n = (p - 2)^n + p^n - m, \quad \text{for } n \geq 1.
\]](http://latex.artofproblemsolving.com/5/9/e/59e2074875b2d41f1e2ac272d06e65f3ebab85ff.png)
Let 
be the smallest prime such that some term of the sequence is divisible by 
for which 
.
be a sequence defined by:![\[
a_1 = 850, \quad \text{and} \quad a_{n+1} = \frac{a_n^2}{a_n - 1} \quad \text{for all } n \geq 1.
\]](http://latex.artofproblemsolving.com/2/0/c/20cb7d9045b7d7ce8a2bf7e8ec93ea458095367e.png)
Determine all values of 
for which 
.
and 
.
and 
.
Write only the answers to the questions given.
to be more exact:
in years 2002-08 time was 90' for part A and 120' for part B
red balls and 
white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...
unit. The area of the hexagon is ...
and 
.
is ...
so that 
and 
is ...
,
,
,
has an average of 
and a median of 
.
are integers greater than 
which are four consecutive quarters of an arithmetic row with 
.
and 
are squares of two consecutive natural numbers, then the smallest value of 
is ...
,
,
and 
.
and 
lies on the line segment 
.
and 
.
is ...
meet 
for each natural number 
is ....
provided that the difference of any two numbers at least 
is ...
which satisfies 
are as many as ...
and 
.
.
is a point on the side extension 
so that 
is perpendicular to 
and 
.
is ...
consists of 
with 
positive reals is ....
with 
is ....
square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of 
coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of 
.
such that 
are the lengths of a right triangle.
Would anyone tell me whether the number of ways for proving Nesbitt inequality more than one hundred ?
with 
such that
.
Y by
An odd integer is written in each cell of a 2009 2009 table. For 1 i 2009 let Ri be
the sum of the numbers in the ith row, and for 1 j 2009 let Cj be the sum of the
numbers in the jth column. Finally, let A be the product of the Ri, and B the product of
the Cj . Prove that A + B is different from zero.
I really need help on this. Can people give me hints? I don’t know where to start.
the sum of the numbers in the ith row, and for 1 j 2009 let Cj be the sum of the
numbers in the jth column. Finally, let A be the product of the Ri, and B the product of
the Cj . Prove that A + B is different from zero.
I really need help on this. Can people give me hints? I don’t know where to start.
