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.
and 
Prove that 
be real numbers such that 
. Find the minimum of 
and 
Prove that 
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 
has real roots 
where 
.
has real roots 
where 
.
(s non zero), then 
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 .
, and for a positive integer 
, prove that
.
be positive reals such that 
. Find the maximum value of 
Given 
non-zero reals 
such that their sum is 
. Prove that there exists two numbers 
such that 
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.)
with degree 
is reflexive if there is an integer 
such that 
for every 
, where 
for 
. Let 
be an integer and 
be a polynomial with integer coefficients. Prove that there exist reflexive polynomials 
, 
with integer coefficients such that![\[(1+x+x^2+\dots+x^{\ell-1})p(x)=q(x)+x^\ell r(x)\]](http://latex.artofproblemsolving.com/6/6/7/66729b54a8b87c6ddbfdf7e7e5932dbb8b262439.png)
, for which 
and 
.
be a natural number. We write out the fractions 
, 
, 
, 
such that they are all in their simplest form. Let the sum of the numerators be 
. For what 
is one of and 
odd, but the other is even?
find 
. Please put your solution in [code][/code] tags so you don't spoil the solution for others (I think this is a really nice problem!)
find . Please put your solution in 
satisfy that: 
.
and 
.
[hide][/hide]
and 
don't depend on the parity of 
and 
so really just 
needs to be negative meaning that either or can be the negative, which somehow i didn't realize lol
?
can be or not distances, i.e. they are positive or not
.
, 
. Hence
.
. 
. So 
. Using Heron formula we have 
, because 
. Hence 
, i.e. 
.
? because 
and 
both work and give an answer of 
so maybe there are two possible answers?
., . Hence.. . So . Using Heron formula we have , because . Hence , i.e. .
,
.
meaning that 
.
.
and 
.
and 
.
.
works.
We have 
so 
and thus 
so finally 
but forgot adding the equations would give 
as terms, not 
as terms.
LoC looks best here.
,
,
,
.