from the non-zero reals to the non-zero reals, such that ![\[f(x+y) \left(f(x) + f(y)\right) = f(xy)\]](http://latex.artofproblemsolving.com/2/f/e/2fe8a09e628f60faedaa07ceb154891375763a83.png)
for all non-zero reals 
such that 
.
.
Let 
.
Let 
.
.
be a real numbers such that 
P
Where 
numebr on his list? Write its last two digits as your answer.
grid of square tiles. Robert chooses four of these squares uniformly at random. The probability that the centers of these four squares form a square themselves is 
where 
are coprime naturals. Write the largest 
digit odd factor of 
.
.
which touches one face of the cube and passes through all 
vertices of the face opposite to it. Find the integer closest to 
are distinct numbers in AP such that 
are also in AP. Find the smallest possible value of 
(in degrees).
that satisfy 
so that both 
have odd numbers of divisors.
which is defined as all the natural numbers listed in order, side by side, behind a decimal point, without commas? If so, what is it?
and 
are primes, then can 
be a perfect square?
be a trapezoid such that 
lie on a circle with center 
,
is parallel to side 
.
and 
intersect at point 
,
.
.
,
and 
are positive integers and 
.
, where 
always occurs at 
,
is a regular tetrahedron.
be a trapezium in which 
and 
.
be then midpoint of the diagonal 
.![$[ABCD] = n \times [CDE]$](http://latex.artofproblemsolving.com/e/9/3/e9347ba2c8e9396adfe716159e711408a3b9dc69.png)
,![$[t]$](http://latex.artofproblemsolving.com/8/a/8/8a86702dc60b9db916798b1115a6b6822bf828b9.png)
denotes the area of the geometrical figur
.
Y by
Let 
be a positive integer, and let its positive divisors be
![\[
d_1 < d_2 < \cdots < d_k.
\]](//latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.
Find
.
Also, find a general formula for when
![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](//latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers.
be a positive integer, and let its positive divisors be![\[
d_1 < d_2 < \cdots < d_k.
\]](http://latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.Find
.Also, find a general formula for
when![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](http://latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers.
and all 
.
pairs 
with 
.
to 
where 
.
.
.
hence 
.
with the 
distinct. Then we count disjoint subsets 
of 
.
w
hence we find 
,
is multiplicative and 
.
