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.
distinct divisors of a positive integer number 
form an arithmetic progression. Prove that the number has at least 
divisors.
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
be a positive integer. A positive integer 
is called a -good number if it satisfies has exactly 
digits in decimal representation (it cannot have leading zeros). digits and the last digits of are considered as two separate -digit.
is a 
-good number because![\[(20 + 25)^2 = 2025.\]](http://latex.artofproblemsolving.com/5/c/5/5c51bca485cea6c3750c48c09091eb5fd72b5192.png)
Find all 
-good numbers.
be the set of real numbers. Let 
be a function such that ![\[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\]](http://latex.artofproblemsolving.com/5/d/3/5d3476fd9195d085d0030d6036a9fde1cf6756a7.png)
for every 
. Assume that the inequality is strict for some 
.
for every 
or 
for every .
such that, for all integers that satisfy 
, the following equality holds:![\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]](http://latex.artofproblemsolving.com/a/9/e/a9edf1a3207d29facc970b6ee288d4fabdd43a4b.png)
denotes the set of integers.)
has how many digits?
has how many digits?
has how many digits?
is called -scared if all of the factors of are not factors of . Find the number of 
-scared numbers less than 
. Note that in the context of this problem, 
does NOT count as a factor.
is called -scared if all of the factors of are not factors of . Find the number of -scared numbers less than .
satisfy that: 
.
and 
Find the value range of 
has 
digits, and 
has 
,
.
digits (excluding the 
has 
digits, you can figure it out from there. So there are 
digits
.
(5 digits) because I just have that memorized. 
for a grand total of 
digits.
when we put this to the 16th power we get 2^32*5^16 we can make 16 0's because we have until 5^16 so we just have to add the 2^16 at the end and when we do this..... we get 
then we already know that it has to be 5 digits so we get 
be the number of candies the twins get combined, let 
and 
be the number of candies the other people get. 
,
,
(13 ways)
(11 ways)
(9 ways)
(7 ways)
,
(5 ways)
,
(3 ways)
,
(1 way)
digits
Taking 
on the inequality,
Thus, 
satisfies 
and 
.
.
.
,
,
,
gives us 
implying 
.
and plugging in 
gives us 
.
shows that 
as the function. Then, 
so the equation of the line must be 
-
.
and 
which gives the ordered pair 
,