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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.
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 .
with 
and 
. Lines and intersect at , and lines 
and 
intersect at 
. Let 
be the midpoints of sides 
, respectively. Let and 
be points on segment 
and 
, respectively, so that 
is the angle bisector of 
and 
is the angle bisector of 
. Prove that 
is parallel to 
if and only if 
divides into two triangles with equal area.
for which, for all 
, the following identity holds:![\[
f(x) f(yf(x)) + y f(xy) = \frac{f\left(\frac{x}{y}\right)}{y} + \frac{f\left(\frac{y}{x}\right)}{x}
\]](http://latex.artofproblemsolving.com/9/0/c/90c180110402e1a32b70edb2b0a03a28727457d1.png)
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
.
and 
again at 
respectively. Prove that 
are concurrent.
gives that 
and 
are cyclic, so radax
,
are concyclic, respectively. This is proved by![\[
\measuredangle AWZ
=\measuredangle CWZ
=\measuredangle CDZ
=\measuredangle CDB
=\measuredangle ABD
\]](http://latex.artofproblemsolving.com/8/c/f/8cf6ee67dfd80862f1a5ff05b28afedf37f9ef2a.png)
and same for