__The following questions can be answered in the next three days!__

Hello, everyone!

I am Blueeighthnote, and I am responsible for this week's questions. Since I figured that the questions should be coming out in a faster pace, the questions will be in intervals of 4 or 3 days instead!

The following questions can be answered until August 17, 12:00 pm (UTC).

## Questions

1. For the number 19/27, what is the 122th digit after the decimal point? [15 points]

2. What is the **remainder** of $ 40^{255} $ divided by 13? [25 points]

3. In the Math Arts Performance Competition, the host unit made a rule: To avoid subjective bias, all 13 judges should first rate the contestant a score. Next, find the **arithmetic average** first, and then **remove** scores that are 13 points away from the average. At last, the result is recalculated as the final score.

In this round, the contestant received the arithmetic average of 79 points. 3 scores in the extreme range were removed, which were 97, 41, and 59. After that, what is the final score that the contestant received? [30 points]

(For question number 3, feel free to ask the host some questions if there are any wording that causes some ambiguity.)

## Previous Answers (8/7 ~ 13)

**For number 1: The answer was A, C, D.**

A. While pi (π) is not irrational, 3.1416 actually **is rational** (3.1416 = $ \tfrac{31416}{10000} $). Sorry about this trick question!

B. Square root of 3 is irrational (1.73205080757……)

C. log_{10}√5 + log_{10}√2 = log_{10}√10 = 1/2 log_{10}10 = 1/2

D. $ \dfrac{\sin15^\circ}{\cos15^\circ} + \dfrac{\cos15^\circ}{\sin15^\circ} = \dfrac{sin^215^\circ + cos^215^\circ}{cos15^\circ \cdot sin15^\circ} = \dfrac{2}{2sin15^\circ \cdot cos15^\circ} = \dfrac{2}{sin30^\circ} = 4 $

E. With Linear factor test and determination of rational root, the possible real roots of x^{3} - 2^{2} + x - 1 = 0 that are rational can only be positive or negative 1, and they both failed. Therefore, **x** must be irrational.

Dalek James was partially correct! With two options incorrect, 1/5 of the score is earned.

**For number 2:**
Dalek-James got it correct! Originally it was going to be harder and harder to prove when more people are answering.

**For number 3:**
$ (\tfrac{1}{5})^{17} $ is actually 1/762939453125. Each question answered is completely independent to the other, and the chances of answering each question correct was 1/5. There are 17 questions, therefore $ (\tfrac{1}{5})^{17} $ = 1/762939453125 is the answer ($ (\tfrac{1}{5})^{17} $ was also accepted).

## Leaderboards

FORMER HOST. 3primetime3: 36313 points (3 achvmt pkgs)

1. **Blueeighthnote: 13843 points (2 achvmt pkgs) - One more to get the next lucky prime badge!While being the host, the host cannot earn any points during the period.**

2. Minipop56: 12206 points - Four more to get the next lucky prime badge!

3. TimBluesWin: 755 points (1 achvmt pkg)

4. Imamadmad: 745 points (1 achvmt pkg)

5. Dalek-James: **730 points (+350 [Badge]), (+ 45)**

6. Julianthewiki: 525 points (1 achvmt pkg) - RETIRED

7. Wikia Contributor 69.235...: 230 points (1 achvmt pkg)

8. Lefty7788: 300 points - One more to get the next lucky prime badge!

9. Supermario3459: 145 points (1 achvmt pkg)

10. ZombieBird4000: 50 points (1 achvmt pkg)

11. Wildoneshelper: 40 points (1 achvmt pkg)

12. Emmaelise401: 50 points

13. CalzoneManiac: 30 points

14. 21qmcgagin: 20 points

15. Kyodaisuu: 21 points

16. Googleaarex: 10 points

## Achievements

The Badges will still be continued further.

### Skillz Badges!

Usually awarded for having skills to solving hard questions, etc.

1. Quick Thinker! (+10 points) - Awarded for posting an answer before I write "Please leave your answers here." FIRST FOUNDED BY BLUEEIGHTHNOTE

2. RARE: Leaderboards Expert! (+300 points) - Awarded for being 1st place on the leaderboards for 30+ days. FIRST FOUNDED BY BLUEEIGHTHNOTE

3. RARE: More than a Millenium! (+100 points) - Awarded for being first in the leaderboards and scoring more than 1000 points from the runner-up. FIRST FOUNDED BY BLUEEIGHTHNOTE

4. Questions in a question (+20 points)? Awarded for finding three ambiguities in a single question. EARNED BY IMAMADMAD

### Earth Badges

Most rare ones are found here.

1. RARE: Top Ten (+30 points) - Awarded for being the first top 10 people ONLY to participate on the question of the day! EARNED BY THE FIRST 10 PARTICIPATING PEOPLE

2. RARE: Cyclone Centurion! (+250 points) - Awarded for scoring at least 10000 points in the QOTD system! FIRST FOUNDED BY BLUEEIGHTHNOTE.

### Sun Badges

All epic ones are found here.

1. EPIC: Lucky Primer (+100 points): Congratulations! You have guessed the lucky prime correctly. Get the Lucky Prime two more times to earn the next achievement! FIRST FOUNDED BY JULIANTHEWIKI.

2. EPIC: Very Lucky Primer (+200 points): Congratulations! You have guessed the lucky prime correctly two more times. Get the Lucky Prime three more times to earn the next achievement! FIRST FOUNDED BY MINIPOP56 AND BLUEEIGHTHNOTE.

3. REALLLLY LUCKY PRIMER (+300 points): Congratulations! You have guessed the lucky prime correctly three more times. Get the Lucky Prime four more times to earn the next achievement! FIRST FOUNDED BY MINIPOP56.

**4. EPICEPIC: First of 2015! (+350 points): Congratulations for being the first to answer the questions in 2015 QOTD! FIRST FOUNDED BY DALEK-JAMES.**

### Galaxy Badges

Mysterious badges from a far-out universe. Each one found here is worth 10000+ points and/or 5+ achievement packages.

1. MYSTERY: ATLAS (+10000 points) - ??? FIRST FOUNDED BY BLUEEIGHTHNOTE.

2. FOUNDER: TIME KEEPER (+36313 points) -- Being the host of QOTD for more than 1 year! Special thanks to your dedications! FIRST FOUNDED BY 3PRIMETIME3!

### Uniqueness Badges

Unique things one can do in the answers section. ONLY ONE PERSON CAN EARN EACH BADGE.

1. Analyzer! (+10 points) - Awarded for drawing a diagram. FOUNDED BY IMAMADMAD

2. Inspiration! (+5 points) - A question was dedicated to you! FOUNDED BY BLUEEIGHTHNOTE

### Trustworthiness Badges

Awarded by how much I trust someone on this wiki. Please do not ask me about these badges.

1. Reliability (+100 points) - Trusted to write a question of a day. FIRST FOUNDED BY BLUEEIGHTHNOTE, AND IMAMADMAD

2. Scoreboard Tinkerer (+100 points) - Trusted to change the scoreboards. FIRST FOUNDED BY BLUEEIGHTHNOTE AND JULIANTHEWIKI

3. Prime Staff (+100 points) - Become a chat moderator, rollbacker, sysop, or bureaucrat on this wikia! FIRST FOUNDED BY BLUEEIGHTHNOTE, JULIANTHEWIKI, TIMBLUESWIN, IMAMADMAD, SUPERMARIO3459, MINIPOP56, AND DALEK-JAMES

### Duration Badges

Yes! YOU KNOW IT! PARTICIPATION EVERY DAY!

1. 4-Leaf Clover! (+100 points) - Awarded for participating on the Question of the Day everyday during the month of March! FIRST FOUNDED BY MINIPOP56.

2. April Fools! (+100 points) - Awarded for participating on the Question of the Day everyday during the month of April! FIRST FOUNDED BY MINIPOP56.