[/quote]
,
. Prove that
:
with 
.![\[
\frac{a^5}{b^2 + 2c^3} + \frac{2b^5}{3c + a^6} + \frac{c^7}{a + b^4} \geq 2.
\]](http://latex.artofproblemsolving.com/f/f/5/ff5043705025592aeefc55a764a8b7170e13ff4b.png)
and 
are coprime positive integers and 
an integer, prove that the greatest common divisor of 
and 
divides 
.
,
?
.
.
.
and 
Then the number of elements in the set 
and 
If each person remains there for 
minutes, what's the probability that all three people meet each other?
and solve for the area of a hexagon in that 
square.
satisfy 
.
and the incircle 
of 
at 
,
meets 
at 
and intersects 
at 
,
meets 
at 
.
(different from 
)
and 
.
.
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
,
lie on a semicircle 
in this order with diameter 
,
and 
lie on segments 
respectively such that 
and 
.
passes through 
.
,
are concurrent.
,
of 
points in a plane 
-
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 
