7/25/2019 untitled22314

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1. Let n be a nonnegative integer. Prove that

1

2n1

nk=0

k

2n

k

n

22n +

2n

n

.

2. Solve the equation 8

{x} = 9

x + 10

x , wherex and{x} denote the greatest integer xand the fractional part ofx, respectively.

3. Prove thatn

k=2

k21i=1

i= n(n 1)(n + 1)2

6 .

4. Let a, b R with 0 a < b and let f : [a, b] R be twice-differentiable with fcontinuous. If

b

a

f(x)dx=a2

2f(a)

b2

2f(b) + bf(b)

af(a),

show that there exists a c(a, b) such that f(c) = 0.5. Let ABCD be a square. Points Eand F lie on sidesBCand CD, respectively, such that

CE= CF. Let Y be a point such that E C FY is a rectangle. BD meets EY at L andAE meets F Y at M. Let Nbe a point on side AB such that LN AB. Prove that ifEF =B E+ DF, then M N is parallel to BD.

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