AoPS Academy
such that satisfies two following statements?
which contains 
.
if 
is an odd natural number, 
if is an even natural number ( is different from 0).
. Prove that for all polynomial 
on 
, we have
Can I change 1 by a constant 
?
be holomorphic on 
, continuous on 
and such that the limit 
exists. Prove that 
attains its maximum on the circle 
.
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.
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)?
be a function defined in any real numbers 
with 
Prove that on the 
plane, the line 
passes through the fixed point which isn't on the 
axis in regardless of the value of if only if is a linear function in .
be a function defined in any real numbers with Prove that on the plane, the line passes through the fixed point which isn't on the axis in regardless of the value of if only if is a linear function in .
is slope-intercept form, then 
Knowing that passes through a fixed point 
with 
, we obtain:
;
is linear.
