Practice (137)
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$\textbf{Who Finishes the Second}$
Adam, Bob, and Charlie are the only three athletes who are competing in a series of track and field events. The first, second and third places in each event are awarded $X$, $Y$ and $Z$ points respectively, where $X > Y > Z$ and all are integers. It is known that
- Adam finishes first with $22$ points overall
- Bob wins the javelin event and finishes with $9$ points overall.
- Charlie also finishes $9$ points overall.
Who finishes second in the $100$-meter dash and why?
$\textbf{Heist}$
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?
$\textbf{The Pet Inc}$
The Pet Inc is owned by three gentlemen pets: a dog, a cat, and a pig. One day, while they are chatting to each other. Mr. Pig says: "Isn't it a bit odd that our surnames match our species, but none of our surnames matches our own species?" The dog replies: "Yes, but does it matter?" Can you relate their surnames and species?
The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?
In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?
