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The following questions can be answered in the next four days!

Hi, everyone!

I am Blueeighthnote, and the following questions can be answered until August 21, 12:00 pm (UTC).

Questions

1. Which of the following numbers is the smallest? [20 points]

A)2^{\tfrac{1}{3}}

B)(\dfrac{1}{8})^{-2}

C)2^{-\tfrac{1}{4}}

D)(\dfrac{1}{2})^{\tfrac{1}{2}}

E)8^{-\tfrac{1}{3}}

2. The mean of ALPHA and BETA is 28, and the mean of BETA and DELTA is 39. Therefore:

ALPHA is [Greater than / Equal to / Less than] DELTA. [12 points]

3. For any number x, what is the lowest value of 27^{\left ( x^2+\tfrac{2}{3} \right )}? [25 points]

4. In a regular 3-digit number xyz, how many numbers in total fit the following criteria:

The difference between ones digit (z) and hundreds digit (x) is 2 in absolute value. [20 points]

Previous Answers (8/14 ~ 16)

For number 1: Simply divide 19 by 27. It is equal to 0.703703703...... The 122th digit is equal to the 2nd. Therefore, the answer was 0.

For number 2: Using the binomial theorem,

\begin{align}
40^{255} &= (39 + 1)^{255} \\
&= {255 \choose 0} 39^{255} + {255 \choose 1} 39^{254} + \ldots + {255 \choose 254} 39 +{255 \choose 255} \\
&= 39k + 1(k \in N)\end{align}

Therefore the remainder of 40^{255} is 1.

For number 3:

\frac{79 * 13 - 97 - 41 - 59}{10} = 83

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