Answer: B

Solution(s):

Note that $1$ cm represents $72 \div 12 = 6$ kilometers. This means that $17$ cm represents $6 \cdot 17 = 102$ kilometers.

Thus, B is the correct answer.

Answer: B

Solution(s):

Note that there are $2$ non-zero digits that could be the thousands digit.

After choosing that, we need to arrange the other $3$ digits. There are $3$ spots for the other non-zero digit.

This gives us $2 \cdot 3 = 6$ possible numbers.

Thus, B is the correct answer.

Answer: A

Solution(s):

Note that $12$ meals feed $18$ people. This means that one meal feeds $\dfrac{18}{12} = \dfrac{3}{2}$ people.

This means that they need $12 \div \dfrac{3}{2} = 8$ meals for $12$ people.

Thus, A is the correct answer.

Answer: B

Solution(s):

If there are $3$ starters, then one person must not be starting. Choosing the person who doesn't start determines the starters.

There are $4$ choices for the person who doesn't start.

Thus, B is the correct answer.

Answer: D

Solution(s):

Note that after every game, one team gets eliminated. For there to be one team remaining, $15$ teams must have been eliminated.

This means that $15$ games had to have been played.

Thus, D is the correct answer.

Answer: C

Solution(s):

Sally made $20 \times .55 = 11$ of her first $20$ shots. Then we get that $$\dfrac{11 + x}{25} = .56,$$ which tells us that $$11 + x = 14$$ and $x = 3.$

Thus, C is the correct answer.

Answer: B

Solution(s):

The maximum heart rate for this athlete would be $220 - 26 = 194.$ Then the target heart rate would be $194 \times .8 \approx 155.$

Thus, B is the correct answer.

Answer: B

Solution(s):

Note that the tens digit can range from $1$ to $7,$ and this digit determines the units digit.

Therefore, there are $7$ numbers.

Thus, B is the correct answer.

Answer: D

Solution(s):

The sum of all $5$ numbers is $54 \cdot 5 = 270.$ The sum of the first $2$ numbers is $48 \cdot 2 = 96.$

The sum of the last $3$ numbers is $270 - 96 = 174.$ The average is therefore $174 \div 3 = 58.$

Thus, D is the correct answer.

Answer: E

Solution(s):

Aaron worked $\dfrac{50}{60} = \dfrac{5}{6}$ hours on Tuesday. He worked $2$ hours and $25$ minutes on Wednesday, which equals $2 + \dfrac{25}{60} = \dfrac{29}{12}$ hours.

This means that he would have earned $$3\left(\dfrac{5}{4} + \dfrac{5}{6} + \dfrac{29}{12} + \dfrac{1}{2}\right)$$$$= 3 \cdot 5$$= $ 15.

Thus, E is the correct answer.

Answer: C

Solution(s):

Note that the largest, smallest, and median numbers cannot be the first or last number.

This means that the first and last numbers are $4$ and $9$ in some order. The average is $$(4 + 9) \div 2$$$$= 13 \div 2$$$$= 6.5.$$

Thus, C is the correct answer.

Answer: B

Solution(s):

When not in use, her cell phone uses up $\frac{1}{24}$ of its battery per hour. When it is in use, it uses up $\frac{1}{3}$ of its battery per hour.

Niki's phone has been on for $9$ hours, with $8$ of those hours being idle and $1$ hour being used to talk on the phone.

This means that the phone has used up $\frac{2}{3}$ of its battery. In order to drain the remaining $\frac{1}{3}$ of the battery, the phone can last for $8$ more hours without being used.

Thus, B is the correct answer.

Answer: E

Solution(s):

If Bill is the oldest, we get that statement I and II are true, which is not allowed. This tell us that statement I is false.

If Amy is not the oldest, then Celine is the oldest since Bill isn't. This makes statement III true, which is also not allowed.

Thus, statement III must the true one and I and II are false. Using this, we get the order to be Amy, Celine, Bill.

Thus, E is the correct answer.

Answer: C

Solution(s):

We can use Pick's Theorem which tells us that the area of a polygon whose vertices are lattice points is given by $$A = I + \dfrac{1}{2}B - 1,$$ where $I$ is the number of interior lattice points and $B$ is the number of boundary lattice points.

Alternatively, one could also decompose the quadrilateral into triangles, but it is also possible to prove Pick's Theorem by recursively decomposing arbitrary polygons into triangles with lattice point vertices, though that is left an exercise for the reader.

Counting, we get that $B = 5$ and $I = 21.$ Plugging these in, we get $$A = \dfrac{5}{2} + 21 - 1 = 22 \dfrac{1}{2}.$$

Thus, C is the correct answer.

Answer: C

Solution(s):

Note that the first ring around the middle tile has $6$ tiles. The second ring has $12$ tiles.

Following this pattern, we get that the third ring has $18$ tiles. The total number of dark tiles is $1 + 12 = 13$ and bright tiles is $6 + 18 = 24.$

The difference is therefore $24 - 13 = 11.$

Thus, C is the correct answer.

Answer: C

Solution(s):

The first pitcher contains $600 \cdot \dfrac{1}{3} = 200$ mL of orange juice. The second one has $600 \cdot \dfrac{2}{5} = 240$ mL.

The large container then has $200 + 240 = 440$ mL of orange juice. The total amount of mixture is $2 \cdot 600 = 1200$ mL.

Then the fraction of orange juice is $\dfrac{440}{1200} = \dfrac{11}{30}.$

Thus, C is the correct answer.

Answer: D

Solution(s):

Recall that stars and bars gives us the number of ways to put $n$ objects into $k$ bins, where each bin must have at least one object, by $$\binom{n - 1}{k - 1}.$$

In our scenario, we must split $6$ pencils amongst $3$ people, where each person must have at least one pencil. Therefore, the number of ways is $$\binom{6 - 1}{3 - 1} = \binom{5}{2} = 10.$$

Thus, D is the correct answer.

Answer: A

Solution(s):

The only way to get Ben's score is with a $1$ and $3$ since he can't hit $2$ twice.

Cindy can achieve her score with $$1 + 6, 2 + 5, \text{ or } 3 + 4.$$ Ben already hit $1$ and $3,$ so Cindy must have hit $2$ and $5.$

Similarly, Dave must have hit $4$ and $7.$ Finally, since $7$ is already used, Alice is forced to have hit $6$ and $10$ with Ellen hitting $8$ and $9.$

Thus, A is the correct answer.

Answer: B

Solution(s):

Let $x$ be the number. Then $x - 2$ is divisible by $3, 4, 5,$ and $6.$

The least common multiple of these numbers is $60,$ which makes $x$ $62.$

Thus, B is the correct answer.

Answer: D

Solution(s):

Note that $\dfrac{1}{4}$ of the chairs are empty. Since this is equal to $6,$ there are $3 \cdot 4 = 18$ taken chairs.

If $18$ is $\dfrac{2}{3}$ the number of people, then the total number of people is $18 \div \dfrac{2}{3} = 27.$

Thus, D is the correct answer.

Answer: D

Solution(s):

For the product to be even, then at least one of the spinners must land on an even number.

We can use complementary counting and calculate the probability of both spinners landing on odds.

This happens with a probability of $$\dfrac{1}{2} \cdot \dfrac{2}{3} = \dfrac{1}{3}.$$

Then the probability of landing on at least one even is $$1 - \dfrac{1}{3} = \dfrac{2}{3}.$$

Thus, D is the correct answer.

Answer: B

Solution(s):

WLOG, let there be $5$ women in the room. Then there are $5 \cdot \dfrac{2}{5} = 2$ single women.

This means that there are $5 - 2 = 3$ married women, which is also the number of married men.

There are a total of $5 + 3 = 8$ people in the room. The fraction of married men is $\dfrac{3}{8}.$

Thus, B is the correct answer.

Answer: D

Solution(s):

Tess cannot be represented by a constant horizontal line because this would imply that she is running in a circle centered at her starting point.

This means that B and E are incorrect as they involve straight lines.

A is incorrect because the maximum distance is at the end of Tess's journey, and C is wrong because it has $2$ maximums.

Thus, D is the correct answer.

Answer: C

Solution(s):

First, we can calculate the area of the parallelogram. This can be done be subtracting out the areas of the $4$ triangles to get $$10 \cdot 8 - 2 \cdot \dfrac{1}{2}(3 \cdot 4 + 6 \cdot 5)$$$$=80 - (12 + 30)$$$$= 38.$$

We can also calculate the area of the parallelogram by multiplying the base by the altitude, $d.$ Therefore $$38 = 5d$$$$d = \dfrac{38}{5} = 7.6$$ (we get the $5$ from the Pythagorean theorem).

Thus, C is the correct answer.

Answer: D

Solution(s):

Note that the union of the two squares is a square with side length $2.$ This makes the area of the overlapping squares $$4^2 + 4^2 - 2^2 = 28.$$

We need to remove the area of the circle, which has radius $\sqrt{2}$ (we get this since the diameter is the diagonal of a square with side length $2$).

Therefore, the area of the shaded region is $$28 - \pi \left(\sqrt{2}\right)^2 = 28 - 2 \pi.$$

Thus, D is the correct answer.