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2017-04-24 Intermediate

         

The last digit of 2 is 2.

The last digit of 222^2 is 2+22+2.

The last digit of 2222^{2^2} is 2+2+22+2+2.

The last digit of 22222^{2^{2^2}} is _________.\text{\_\_\_\_\_\_\_\_\_} .

ABCDABCD and ABCDA'BC'D are both rectangles with side lengths AD=AD=2,AB=AB=6.AD = A'D = 2,\quad AB =A'B = 6. Find the area of shaded region (to 2 decimal places).

The following is a conversation between Gabriel and Heather:

Gabriel: "I am thinking of two distinct single-digit numbers. Can you guess the sum of these two numbers?"
Heather: "No. Can you give me a clue?"

Gabriel: "The last digit of the product of the two numbers is your house number."
Heather: "Now I know the sum of the two numbers."

So, what is the sum of the two numbers?


Note: It is possible that the product of the two numbers is a single-digit.

What is the diameter of the semicircle in the diagram below?

Note: The diagram is not drawn to scale.

A bead slides under the pull of gravity, g,g, down a frictionless wire segment in the shape of the curve y=exy = e^{-x}, where xx is the horizontal direction and yy is the vertical direction. The bead starts from rest at (x,y)=(0,1)(x,y) = (0,1).

The time it takes for the particle to travel between x=ax=a and x=bx=b can be expressed as

ta,b=12gab1+PeQx1+ReSxdx,\large{t_{a,b} = \frac{1}{\sqrt{2g}} \int_a^b \sqrt{\frac{1 + P e^{Q x}}{1 + R e^{S x} }}\,dx},

where P,Q,R,P,Q,R, and SS are integers.

Determine P+Q+R+S.P + Q + R + S.

Note: The constant ee is Euler's number.

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