Answer: B

Solution:

Since Connie multiplied by $2,$ her original number was $60 \div 2 = 30.$ To get the correct answer, she must divide by $2$ to get $30 \div 2 = 15.$

Thus, B is the correct answer.

Answer: C

Solution:

Karl paid 5 \times $ 2.5 = $ 12.5 for his folders. $20 \%$ off of this would have been .2 \times $ 12.5 = $ 2.5, which means Karl would have saved $ 2.5.

Thus, C is the correct answer.

Answer: D

Solution:

Note that every dark square not along the diagonal must have a corresponding dark square in the other half.

Each of the $4$ dark squares not on the diagonal do not have their corresponding square colored, so $4$ additional squares must be colored dark.

Thus, D is the correct answer.

Answer: C

Solution:

The perimeter of the triangle is $$6.1 + 8.2 + 9.7 = 24 \text{ cm.}$$ This means that the side length of the square is $$24 \div 4 = 6 \text{ cm.}$$ Therefore, the area of the square is $$6^2 = 36 \text{ cm}^2.$$

Thus, C is the correct answer.

Answer: B

Solution:

To minimize the number of packs, we can use the largest packs first.

We can use three $24$-packs to have $$90 - 3 \cdot 24$$$$= 90 - 72$$$$= 18$$ cans left.

From this, we can see that we need one $12$-pack and one $6$-pack.

Therefore, we need $3 + 1 + 1 = 5$ packs.

Thus, B is the correct answer.

Answer: C

Solution:

We can rearrange the inequality as follows. $$\begin{align*} 2.00d5 &\gt 2.005 \\ .00d5 &\gt .005 \\ d.5 &\gt 5 \\ d &\gt 4.5 \end{align*}$$

From this, we can see that $d$ must be greater than $5,$ yielding $5$ values ($5$ through $9$).

Thus, C is the correct answer.

Answer: B

Solution:

Note that what we want is the length of the diagonal. We can use the Pythagorean theorem to find this. $$\sqrt{\left(\dfrac{1}{2} + \dfrac{1}{2}\right)^2 + \dfrac{3}{4}^2}$$ $$= \sqrt{1 + \dfrac{9}{16}} = \sqrt{\dfrac{25}{16}} = \dfrac{5}{4}$$

Thus, B is the correct answer.

Answer: E

Solution:

Recall the four following rules:

• odd plus odd and even plus even is even

• even plus odd is odd

• even times anything is even

• odd times odd is odd

These rules can be easily verified by representing arbitrary odd numbers as $2m+1$ and arbitrary even numbers as $2n$ respectively, for integers $m,n.$

With this in mind, let's examine each answer choice individually:

A:

Note that $3$ is odd. This gives us $$\text{odd} + \text{odd} \cdot \text{odd}.$$ From our above rules, we know that this is even.

B:$$\text{odd} \cdot \text{odd} - \text{odd}.$$ Once again, this is even.

C: $$\text{odd} \cdot \text{odd}^2 + \text{odd} \cdot \text{odd}^2.$$ This is also even.

D: $$(\text{odd} \cdot \text{odd} + \text{odd})^2.$$ Unfortunately, this is also even.

E: $$\text{odd} \cdot \text{odd} \cdot \text{odd}.$$ This is odd.

Therefore, E is the only answer choice that is an odd integer.

Thus, E is the correct answer.

Answer: D

Solution:

Note that $\triangle ADC$ is isosceles. This means that $\angle DAC = \angle DCA.$

We get that $$\begin{gather*} \angle DAC + \angle DCA + \angle ADC = 180^{\circ} \\ \angle DAC + \angle DCA = 120^{\circ} \\ \angle DAC = \angle DCA = 60^{\circ}. \end{gather*}$$

This shows that $\triangle ADC$ is equilateral. This gives us that $$AC = CD = 17.$$

Thus, D is the correct answer.

Answer: D

Solution:

If Joe runs $3$ times as fast as he walks, it would take him a third of the time to run the same distance versus walking.

This means that Joe will take $6 \div 3 = 2$ minutes to walk half way to school, for a total time of $6 + 2 = 8$ minutes.

Thus, D is the correct answer.

Answer: C

Solution:

Jack's total is $$90.00 \times 1.06 \times .8$$ dollars. Jill's total is $$90.00 \times .8 \times 1.06$$ dollars. Note that both these values are the same.

This means that the difference between the totals is $ 0.

Thus, C is the correct answer.

Answer: D

Solution:

Let $x$ be the number of bananas Big Al ate on May $3.$ Then Big Al ate the following amounts starting from May $1:$ $$x - 12, x - 6, x, x + 6, x + 12.$$

The sum of this is $5x,$ which equals $100.$ This gives us that $x = 20.$ Then on May $5,$ Big Al ate $20 + 12 = 32$ bananas.

Thus, D is the correct answer.

Answer: C

Solution:

Extend $\overline{AF}$ and $\overline{CD}$ to meet at $P.$ Then $ABCP$ is a rectangle with area $8 \cdot 9 = 72.$

Then the area of $$FEDP = 72 - 52 = 20.$$ We know that $$ED = BC - AF = 9 - 5 = 4.$$ Therefore, $$FE = [FEDP] \div DE$$$$= 20 \div 4 = 5.$$ The desired answer is therefore $4 + 5 = 9.$

Thus, C is the correct answer.

Answer: B

Solution:

Let us focus on one team. This team plays against $5$ other teams twice in its own division, for a total of $2 \cdot 5 = 10$ games.

This team also plays $6$ games with teams from the other division. Therefore, they play a total of $10 + 6 = 16$ games.

There are $12$ teams total, so there are $12 \cdot 16 = 192$ games conducted. Each game, however, involves $2$ teams, so we have to divide by $2.$

This gives us the actual total number of games to be $192 \div 2 = 96.$

Thus, B is the correct answer.

Answer: C

Solution:

Let $x$ be the length of the base of the triangle and $y$ be the length of the legs.

We get that $$2y + x = 23.$$ The triangle inequality tells us that $2a \gt b.$ Plugging in the first equation, we get $$2a \gt 23 - 2a$$and therefore$$a \gt 5.75.$$

This tells us that the minimum value of $a$ is $6.$ We get the maximum value when we plug in $b = 1$ to get $a = 11.$

This gives us $6$ values for $a,$ yielding $6$ different triangles.

Thus, C is the correct answer.

Answer: D

Solution:

Note that the Martian can pull out $4$ socks of each color without drawing $5$ of the same color. The $13$th sock, however, must be the $5$th sock of some color.

Thus, D is the correct answer.

Answer: E

Solution:

Average speed is distance over time, which is given by the slope of the line through the point and the origin.

Evelyn has the steepest slope, telling us that she had the greatest average speed.

Thus, E is the correct answer.

Answer: C

Solution:

We want to find how many $k$ exist such that $13k$ is a three-digit number.

The smallest possible $k$ such that $13k \gt 99$ is $k = 8.$

The largest possible $k$ such that $13k \lt 1000$ is $k = 76.$

This tells us that $k$ can range from $8$ to $76,$ which gives us $69$ values for $k.$

Thus, C is the correct answer.

Answer: A

Solution:

Drop the altitude at $C$ to intersect $\overline{AD}$ at $F.$

Then we can find $\overline{AE}$ and $overline{FD}$ using the Pythagorean theorem. We get that $$AE = \sqrt{30^2 - 24^2}$$$$= \sqrt{324}$$$$= 18$$ and $$DF = \sqrt{25^2 - 24^2}$$$$= \sqrt{49}$$$$= 7.$$

We also know that $$EF = BC = 50.$$ The perimeter is therefore $$50 + 30 + 25 + 18 + 7 + 50$$$$= 180.$$

Thus, A is the correct answer.

Answer: A

Solution:

Let $k$ be the number of turns. Then Alice moves $5k$ points and bob moves $9k$ points.

They coincide when combined, they have made a whole number of rotations. Namely, when $5k + 9k = 14k$ is a multiple of $12.$

$14$ is a multiple of $2,$ so $k$ must have must have a multiple of $2$ and of $3.$ Therefore the smallest value of $k$ that works is $2 \cdot 3 = 6.$

Thus, A is the correct answer.

Answer: C

Solution:

Notice that choosing any of these $3$ dots forms a triangle, except for the $2$ triples that form a straight line.

There are $\binom{6}{3} = 20$ ways to choose the points, and then we subtract $2$ to get $20 - 2 = 18.$

Thus, C is the correct answer.

Answer: E

Solution:

WLOG, let the price of the small be $ 1 and the size of the large be $10$ oz.

Using these values, we get that the medium costs $ 1.50 and contains $8$ oz. The small's size is $5$ oz and the large costs $ 1.95.

Calculating the unit prices, we get the small, medium, and large are $$0.200, 0,188, \text{ and } 0.195$$ dollars per oz.

Thus, E is the correct answer.

Answer: B

Solution:

Note that the area of the full circle would be $2 \cdot 2 \pi = 4 \pi.$ Then the radius $r$ of the semicircle is $$\sqrt{\dfrac{4 \pi}{\pi}} = \sqrt{4} = 2.$$ We can also see that each leg of $\triangle ABC$ is two times $r,$ which is $2 \cdot 2 = 4.$

Then the area of the triangle is $$\dfrac{1}{2} \cdot 4 \cdot 4 = 8.$$

Thus, B is the correct answer.

Answer: B

Solution:

We can work down from $200$ to $1.$ If we are at even number, we can just divide by $2,$ and if it is odd, we subtract $1$ before dividing.

$$\begin{align*} 200 \div 2 &= 100 \\ 100 \div 2 &= 50 \\ 50 \div 2 &= 25 \\ 25 - 1 &= 24 \\ 24 \div 2 &= 12 \\ 12 \div 2 &= 6 \\ 6 \div 2 &= 3 \\ 3 - 1 &= 2 \\ 2 \div 2 &= 1 \end{align*}$$

From this, we can see that we need $9$ key strokes.

Thus, B is the correct answer.

Answer: A

Solution:

Let the area of the region inside the circle and outside the square be $x.$ Then $$4 - x = \pi r^2 - x.$$ This gives us that $\dfrac{4}{\pi} = r^2.$ Then $r = \dfrac{2}{\sqrt{\pi}}.$

Thus, A is the correct answer.