AoPS Academy
be two 
complex matrices of the same rank, and let 
be a positive integer. Prove that 
if and only if 
.
to find the third side 
He uses logarithms as follows. He finds 
and doubles it; adds to that the double of 
subtracts the sum of the logarithms of 
and 
divides the result by 
and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?
be a sequence of positive numbers and suppose that 
is divergent. Let further 
. Show that there exists a sequence 
of positive numbers such that 
is convergent and
. (S. 8)
be given real numbers and let 
for all 
. Show that the sequence 
converges and find its limit.
, 
, 
and 
.
in the Cartesian plane such that all side lengths of are positive integers and line segments 
and 
are both sides of .
.
be an real matrix with minimal polynomial 
. Prove that the trace of 
is zero.
of positive integers fast-growing if 
for all 
. Also, for a sequence 
of real numbers and a sequence 
of positive integers, let us denote![\[
S(a, b) = \sum_{n=1}^{\infty} \left| a(b_n) + a(b_n + 1) + \cdots + a(b_{n+1} - 1) \right|.
\]](http://latex.artofproblemsolving.com/2/f/c/2fc080e56fa9d8d6b1cf8a0bb2cb404a65d9ec73.png)
, 
such that for every sequence , if all the series![\[
\sum_{n=1}^{\infty} a(n), \quad S(a, b) \quad \text{and} \quad S(a, c)
\]](http://latex.artofproblemsolving.com/0/8/a/08aa112d16a9c772145ff03a74fb512b4d02340d.png)
are convergent, then the series 
is also convergent?, , 
such that for every sequence , if all the series![\[
S(a, b), \quad S(a, c) \quad \text{and} \quad S(a, d)
\]](http://latex.artofproblemsolving.com/5/8/9/589be33c766fde9e227f56d280d02322585e2ba3.png)
are convergent, then the series is also convergent?
be a ring with 
its centre (
), 
the set of invertible elements and 
.
has the property 
if there exists a subgroup 
of group 
such that 
, 
and 
for any 
. is commutative;
has the property , then 
, for any 
.
and 
are the identity elements for 
and 
does the series 
converge (and absolutely converge)?
greater than 
and the set
and we are supposed to count the number of sets (not pairs!) 
, such that 
, 
and such that the Euclidean distance between 
and 
is an integer. Ok?
that will qualify for this count will be such that and have either the same 
-coordinate or the same 
-coordinate. Let's call these trivial pairs and count them first. For every point 
there will be exactly ten other points in 
with the same -coordinate and ten more with the same -coordinate, and as there is no overlap, there are 20 such points in total for every pair . As there are 
points in , this produces 
pairs of points we need to count as trivial pairs. However, as sets, there are just 
of them. is an element in such that 
share neither - nor -coordinate, then there are exactly two points 
such that 
is a non-degenerate rectangle with each side parallel to one of the coordinate axes, and qualifies for our count if and only if the two sidelengths of that rectangle and the length of its diagonal form a Pythagerean triple, and, due to the geometric dimensions of , the sidelengths, i.e. the first two elements of the Pythagorean triple, must be smaller or equal to 10. But there are just two such Pythagorean triples, namely 
and 
. with sidelengths either 3 and 4 or 6 and 8 such that 
and such that the sides of these rectangles are parallel to the coordinate axes. Every such rectangle contains two pairs of points, and 
that qualify. We do the counting by keeping track of the bottom-left point of the possible rectangles.-axis, the bottom-left corner can vary from 
to 
, which produces 
such rectangles, and if the side with length 3 is parallel to the -axis, we get another 56 rectangles. That's 112 rectangles here.-axis, the bottom-left corner can vary from to 
, which produces 
such rectangles, and in the other orientation we get another 15 rectangles, so 30 rectangles here.
Prove that
