$\textbf{Red Cards}$

There are $7$ cards. Two of them have both sides red, two of them have both sides black, the rest three have one side red and one side black. Joe draws one card randomly and finds one side is red, what is the probability that the other side is red too?

$\textbf{My Name}$

I am a son of a chemist and a mathematician. People called me "Iron59" though I have a common English name. What is my name?$\textbf{Heist}$

The chief detective hurries down to the police station after hearing big news: there is a heist at Pi National Bank! The police has brought in seven known gang members seen leaving the crime scene. They belong to the nefarious True/False Gang, so named because each member is required to either always tell the truth or always lie. Although everyone is capable of engaging in wrongdoing, the chief also knows from his past cases that any crime committed by this gang always includes one truth teller. When the chief shows up, he asks the gang members the following questions:- Are you guilty?
- How many of the seven of you are guilty?
- How many of the seven of you tell the truth?

Here are their responses:

- Person $1$: Yes; $1$; $1$
- Person $2$: Yes; $3$; $3$
- Person $3$: No; $2$; $2$
- Person $4$: No; $4$; $1$
- Person $5$: No; $3$; $3$
- Person $6$: No; $3$; $3$
- Person $7$: Yes; $2$; $2$

After looking these answers over, the chief correctly arrests those responsible gang members.
Who out of these seven are arrested?

$\textbf{Class Substitute}$

Kurt, a math professor, needs a substitute for one of his classes today. He sends an email to his three closest co-workers: Julia, Michael, and Mary asking if anyone can help. However, Prof Kurt forgets to give the details of his class. Julia, the department chair, knows which class Kurt teaches, but does not know the time nor the building. Michael plays racquetball with Kurt often, so he knows what time Kurt teaches, but does not know other details. Mary happens to know which building Kurt's class is in, but neither the class itself nor the time.The possible candidates for Prof Kurt's class are list below.

- Calc $1$ at $9$ in North Hall
- Calc $2$ at noon in West Hall
- Calc $1$ at $3$ in West Hall
- Calc $1$ at $10$ in East Hall
- Calc $2$ at $10$ in North Hall
- Calc $1$ at $10$ in South Hall
- Calc $1$ at $10$ in North Hall
- Calc $2$ at $11$ in East Hall
- Calc $3$ at noon in West Hall
- Calc $2$ at noon in South Hall

After looking over the list, Julia says, "Does anyone know which class it is?" Michael and Mary Ellen immediately respond, "Well, you don't." Julia asks, "Do you?" Michael and Mary Ellen both shake their heads. Julia then smiles and says, "I now know." Which class does Kurt need a substitute for?

Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7$. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?

The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying \[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\]is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p+q$?