Problem 1
Using proper combinatorial argument prove
C(n+r+1, r) = summation from j=0 to j=r {C(n+j, j)}
Where C(n,r) = n!/ r!(n-r)!
Problem 2
Find a closed formula for
S(subscript n) = summation k=1 to k=n {{C(n,k)} (k square)}
Use combinatorics.
Problem 3
Prove the inequality
Given: (x>y>0)
square root (xy) < {(x-y) / (lnx - lny)} <{(x+y)/2}
Problem 4
How many spheres are needed to shield a point source of light?
Problem 5
a) Prove that the following equation has no solution in positive integer
x square+y square +z square +u square = 2xyzu
b) In a triangle with sides a, b, c it is known that ab + bc + ca = 12. Between which bounds does the perimeter lie?
Problem 6
A student has sleeping habit as follows:
If he sleeps on a particular night sleep to next night has chance to sleep is 0.6. If he does not sleep the night the probability
that to sleep the next night is 0.9. In the long run how many % of night the student is likely to sleep?
