The last digit of 2 is 2.
The last digit of \(2^2\) is \(2+2\).
The last digit of \(2^{2^2} \) is \(2+2+2\).
The last digit of \(2^{2^{2^2}} \) is \(\text{_________} . \)
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\(ABCD\) and \(A'BC'D\) are both rectangles with side lengths \[AD = A'D = 2,\quad AB =A'B = 6.\] Find the area of shaded region (to 2 decimal places).
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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.
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What is the diameter of the semicircle in the diagram below?
Note: The diagram is not drawn to scale.
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A bead slides under the pull of gravity, \(g,\) down a frictionless wire segment in the shape of the curve \(y = e^{-x}\), where \(x\) is the horizontal direction and \(y\) is the vertical direction. The bead starts from rest at \((x,y) = (0,1)\).
The time it takes for the particle to travel between \(x=a\) and \(x=b\) can be expressed as
\[\large{t_{a,b} = \frac{1}{\sqrt{2g}} \int_a^b \sqrt{\frac{1 + P e^{Q x}}{1 + R e^{S x} }}\,dx},\]
where \(P,Q,R,\) and \(S\) are integers.
Determine \(P + Q + R + S.\)
Note: The constant \(e\) is Euler's number.
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