[/quote]
,
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 
.
and an integer 
,
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 
.
.
be a tetrahedron such that 
A sphere of radius 
Given that 
,
,
,![$$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{9\sqrt[3]{4}}{4}$$](http://latex.artofproblemsolving.com/6/c/6/6c665b4611f5fc8ebeba03ac649bfee4e2634950.png)
with equality at ![$a=b=c=\sqrt[3]{2}.$](http://latex.artofproblemsolving.com/b/b/9/bb9c9671b23cb45e5f85ae01bf25d566c2e43e91.png)
and 
be points on a plane such that 
,
be the set of all points 
,
is a real number. The path that 
.
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?
.
be reals such that 
. Prove that
Let 
. Prove that
and then use that for find 
but I must have gone wrong somewhere.
terms are indeed in an A.P. but I can't really understand what does the 
in the end indicate, as in what am I supposed to do for the later terms in the sequence?
Y by PikaPika999
These are a few questions I wasn't able to solve, any help will be appreciated!
If 
for 
and 
, then find the value of 
.
If the 
th term of an A.P. is 
and the th term is , then find its 
th term.
If 
is a positive real number different from 
, then prove that the numbers 
are in A.P. Also find their common difference.
If 
for 
and 
, then find the value of 
.
If the 
th term of an A.P. is 
and the th term is , then find its 
th term.
If 
is a positive real number different from 
, then prove that the numbers 
are in A.P. Also find their common difference.
therefore 
you should be able to find 
then cd is 
so the 
thus, the 
If 
are in A.P., then prove that 
are also in A.P.
,
,
:
and 
therefore 
and 
If 
are distinct integers in an A.P. such that 
,
.
,
.
,
.
,
.
and 
are you refering to when you mention 
occur? I see no mention of it.
and then between 
and 
.
?
