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.
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
, 
, 
be nonzero real numbers such that 
and 
. Find the value of
.
kawaii if it satisfies 
(
is the number of positive factors of , while 
is the number of integers not more than that are relatively prime with ). Find all that is kawaii.
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
be a function satisfying
for all integers 
and 
. Show that there exist positive integers 
and such that 
for all integers .
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x + y^2) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
of positive integers with 
prime, 
, and 
, such that 
is a multiple of 
.
a point in the space and 
the centroid of a tetrahedron 
. Prove that:
be a logarithmic spiral centered at the origin (ie curve satisfying for any point 
on it, line 
makes a fixed angle with the tangent to at ). Let 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.
on the spiral, the polar of wrt. is tangent to the spiral.
be a monic polynomial so that there exists another real coefficients 
that satisfy![\[P(x^2-2)=P(x)Q(x)\]](http://latex.artofproblemsolving.com/6/9/7/697faa929e4fda7a6e9b1cd97849bd42ffc14306.png)
Determine all complex roots that are possible from that satisfies (a), determine whether should have real coefficients or not.
with exactly 9 elements that is subset of 
. Prove that there exist two subsets 
and 
that satisfy the following: and are non-empty subsets from , and are equal, and
is an empty subset.
-excenter of 
is 
and 
. The foot from 
to 
is 
and 
. The intersection of 
and 
is . The circumcenter of 
is .
is tangent to the -excircle if is on the incircle of .
that 
where 
is the canonical form of with 
.
where is the canonical form of is true for all positive integer . that 
.
, we get that 
for all 
.
where 
. Modulo 
gives us 
, impossible since all 
s are distinct.
. This gives 
for some prime number . that not smaller than 
is 
.
that where is the canonical form of with . where is the canonical form of is true for all positive integer . that ., we get that for all . where . Modulo gives us , impossible since all s are distinct.. This gives for some prime number . that not smaller than is .
and 
, and them being the only zeroes of this function. One can also say 
doesn't take any negative numbers.
, where 
, and assume this number satisfies the given condition
We get that 
. Rewriting 
gives 
It should be obvious that and 
. Since 
(Or else 
), we have . All that is left to find the smallest prime that satisfies 
, which is 
such that :
such that 
)))))