__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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