such that 
which contains 
.
if 
is an odd natural number, 
if 
.
on 
,
Can I change 1 by a constant 
?
,
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 
.
of positive numbers such that 
is convergent and
.
be given real numbers and let 
for all 
.
converges and find its limit.
of positive integers fast-growing if 
for all 
.
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 ![\[
\sum_{n=1}^{\infty} a(n), \quad S(a, b) \quad \text{and} \quad S(a, c)
\]](http://latex.artofproblemsolving.com/0/8/a/08aa112d16a9c772145ff03a74fb512b4d02340d.png)
are convergent, then the series 
is also convergent?
such that for every sequence ![\[
S(a, b), \quad S(a, c) \quad \text{and} \quad S(a, d)
\]](http://latex.artofproblemsolving.com/5/8/9/589be33c766fde9e227f56d280d02322585e2ba3.png)
are convergent, then the series 
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 
.
has the property 
,
.
and 
are the identity elements for 
and 
does the series 
converge (and absolutely converge)?
Y by EntangledElectron99
1)Two players 
and 
play a series of of 
games. Each game either results in a win or loss for . Total number of ways in which can win the series is:
A)
B)
C)
D)
2) A person predicts the outcome of 20 cricket matches of his home team. Each match can either result in a win , loss or tie for the home team. The total number of ways in which he can make the predictions so that 10 predictions is correct is equal to:
A)
B)
C)
D)
Please mention the solutions properly.
and 
play a series of of 
games. Each game either results in a win or loss for . Total number of ways in which can win the series is:A)
B)
C)
D)
2) A person predicts the outcome of 20 cricket matches of his home team. Each match can either result in a win , loss or tie for the home team. The total number of ways in which he can make the predictions so that 10 predictions is correct is equal to:
A)
B)
C)
D)
Please mention the solutions properly.
This post has been edited 1 time. Last edited by AzSolver257, Apr 19, 2025, 4:54 AM
Reason: Correct typing error
Reason: Correct typing error
many possible results. Out of this 
end up in a draw. Thus 
games end up in a win or a loss. Thus 
games are won by 
