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

Hi, everyone! Here are some brand new questions to answer!

## Questions

1. Which of the following are true? There are multiple answers. [30 points]

Try not to use the calculator, though. They can all be done with simple calculations without even knowing the exact values of them.

(A) sin 50° < cos 50°
(B) tan 50° < cot 50°
(C) tan 50° < sec 50°
(D) sin 230° < cos 230°
(E) tan 230° < cot 230°

2. On a two-dimensional plane, where do $y = 4^x$ and $y = 2^{(3x+2)}$ intersect? [45 points]No lucky prime hint today. Though, you can still guess it!

For number 1: The answer is easily found by simplifying $\frac{1}{n} + \frac{2}{n} + \cdots + \frac{10}{n}$ into $\frac{1 + 2 + 3 + \cdots + 10}{n} = \frac{55}{n}$, and therefore n must be 1, 5, 11, or 55.

For number 2: Correct! The answer is 4. This can be found by simply multiplying 2 over and over again and find that it loops at each 20 numbers. The same happens for $3^n$, actually, looping every 20 numebrs.

For number 3: I should have used this one as a checkpoint question since there is some difficulty with this one.

$\begin{cases} x + 2y + az = 1 & \text{(1)}\\ 3x + 4y + bz = -1 & \text{(2)}\\ 2x + 10y + 7z = c & \text{(3)} \end{cases}$

A) (If there is a solution) Exactly one set of answers is available.

B) (If there is a solution) 11a - 3b ≠ 7

C) (If there is a solution) c = 14

D) (If there are no solutions) 11a - 3b = 7

E) (If there are no solutions) c ≠ 14

First, a breakdown to the equations.

$\begin{cases} (3) - (1) \times 2: 6y + (7-2a)z = c - 2 \\ (1) \times 3 - (2): 2y + (3a-b)z = 4 \end{cases}$

With the simplifications, it is now easier to go through the options:

A) Now we can see that there is a possibility for multiple sets of answers, or no sets of answers at all. For example, a = 1/2, b = -(1/2). Then if c = 6, then there are multiple solutions; otherwise, no solutions possible.

B) If there is one solution, $\frac{6}{2} \ne \frac{7 - 2a}{3a - b} \Rightarrow 11a - 3b \ne 7$

Still, keep in mind that if the result of a, b, and c causes there to be multiple solutions, $\frac{6}{2} = \frac{7 - 2a}{3a - b} = \frac{c-2}{4} \Rightarrow 11a - 3b = 7 \text{and that } c = 14$

This one is counted as correct because I worded the original option as "a solution", which, originally the question was worded "If solutions are possible". In this QOTD's case, the answer is correct.

C) As described "a solution", c does not always equates to 14 if there is exactly one solution. Only when there are multiple solutions can this be defined.

D) E) If there are no solutions, $11a - 3b = 7 \text{and that } c \ne 14$

User:3primetime3 earned 16 points!

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. Dalek-James: 760 points

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

7. 3primetime3: 398 points (Lucky Primer!, again!)

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

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

10. Supermario3459: 145 points (1 achvmt pkg)

11. ZombieBird4000: 50 points (1 achvmt pkg)

12. Wildoneshelper: 40 points (1 achvmt pkg)

13. Emmaelise401: 50 points

14. CalzoneManiac: 30 points

15. 21qmcgagin: 20 points

16. Kyodaisuu: 21 points

## Achievements

The Badges will still be continued further.

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

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.

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.

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 (+0 points) -- Being the host of QOTD for more than 1 year! Special thanks to your dedications! FIRST FOUNDED BY 3PRIMETIME3!

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

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