Solution:

To make the sum small, the two smallest digits go in the hundreds places, the next two in the tens places, and the two largest in the units places.

One such arrangement is $468 + 579 = 1047$, and every arrangement of this type gives the same sum.

Thus, the correct answer is C .

Solution:

Changing a digit in the tenths place changes the number more than changing any digit farther to the right. The digit in the tenths place is $1$.

Changing it gives $.92345$, which is the largest possible result.

Thus, the correct answer is A .

Solution:

The diagonal from one corner to the opposite corner splits the square into two equal halves. Every shaded piece on one side of the diagonal is the mirror image of an equal unshaded piece on the other side.

So the shaded and unshaded areas are equal, and exactly $\dfrac12$ of the square is shaded.

Thus, the correct answer is E .

Solution:

The units digit of a square is determined by the units digit of the number squared. Squaring $0$ through $9$ gives units digits $0, 1, 4, 9, 6, 5, 6, 9, 4, 1$.

So a square can only end in $0, 1, 4, 5, 6,$ or $9$; it can never end in $2, 3, 7,$ or $8$. Among the choices, only $8$ is impossible.

Thus, the correct answer is E .

Solution:

Since $.48017$ is close to $\dfrac12$, the product is close to

$$\left(\dfrac12\right)^3 = \dfrac18 = 0.125.$$

The choice closest to $0.125$ is $0.110$.

Thus, the correct answer is B .

Solution:

Choices A, B, and E are all very close to $13579$, and choice C multiplies $13579$ by a tiny number, making it much smaller.

Choice D divides by $\dfrac{1}{2468}$, which is the same as multiplying by $2468$. This makes it thousands of times larger than $13579$, so it is the largest.

Thus, the correct answer is D .

Solution:

For the product of three numbers to be positive, either all three are positive or exactly two are negative. There are only two positive numbers, so we must use two negatives and one positive.

To maximize, take the two most negative numbers and the largest positive: $(-3)(-2)(5) = 30$.

Thus, the correct answer is C .

Solution:

The sale price is $\dfrac34 \cdot \$80 = \$60.$

The tax is $10\%$ of $\$60,$ which is $\$6,$ so the total is $\$60 + \$6 = \$66.$

Thus, the correct answer is D .

Solution:

A grade of C corresponds to scores from $75$ to $84$. The qualifying scores are $77, 75, 84, 78, 80$, which is $5$ students.

So the percent is $\dfrac{5}{15} = \dfrac13 = 33\dfrac13\%.$

Thus, the correct answer is D .

Solution:

Moving one column right adds $1$ to the date, and moving one row down adds $7$. Let $C$ have date $n$. Then $A$, one column to the right, is $n+1$. The letter $P$ sits two rows directly below $C$, so $P = n+14$, and $B$, one column left of $P$ in the bottom row, is $n+13$.

We need a letter whose date $d$ satisfies $d + n = (n+1) + (n+13)$, so $d = n + 14$. That date is behind $P$.

Thus, the correct answer is A .

Solution:

The six consecutive numbers include $11, 14, 15$, so five of the faces are $11, 12, 13, 14, 15$ and the sixth is $10$ or $16$.

If the sixth were $10$, equal opposite sums would force the pairs $(10,15), (11,14), (12,13)$, making $11$ and $14$ opposite. But the figure shows $11, 14, 15$ meeting at one corner, so no two of them are opposite. Hence the sixth number is $16$, with pairs $(11,16), (12,15), (13,14)$, each summing to $27$.

The total is $3 \cdot 27 = 81$.

Thus, the correct answer is E .

Solution:

Each leading digit accounts for $6$ of the $24$ numbers. Positions $1$–$6$ begin with $2$, positions $7$–$12$ begin with $4$, and positions $13$–$18$ begin with $5$.

So the $17$th number is the $5$th one beginning with $5$: $5247, 5274, 5427, 5472, 5724$. The fifth is $5724$.

Thus, the correct answer is B .

Solution:

The first ounce costs $30$¢. The remaining $3.5$ ounces are charged as $4$ additional ounces, since any fraction rounds up to a full ounce.

That is $4 \cdot 22$¢ $= 88$¢, so the total is $30$¢ $+ 88$¢ $= 118$¢ $= \$1.18.$

Thus, the correct answer is C .

Solution:

Since blue balls make up $\dfrac14$ of the bag and there are $6$ of them, the total is $6 \cdot 4 = 24$ balls.

So the number of green balls is $24 - 6 = 18$.

Thus, the correct answer is B .

Solution:

Each of the four squares has area $\dfrac{100}{4} = 25 \text{ cm}^2$, so each side is $5 \text{ cm}.$

The outline of this staircase shape is made up of $10$ such sides, so the perimeter is $10 \cdot 5 = 50 \text{ cm}.$

Thus, the correct answer is E .

Solution:

Group the terms as $(1990 - 1980) + (1970 - 1960) + \cdots + (30 - 20) + 10.$ Each parenthesized pair equals $10$.

The first elements $1990, 1970, \ldots, 30$ number $99$, so there are $99$ pairs, giving $990$, plus the leftover $+10$ at the end for a total of $1000$.

Thus, the correct answer is D .

Solution:

The thickness is $3$ inches $= \dfrac14$ foot, so the volume is $3 \cdot 60 \cdot \dfrac14 = 45$ cubic feet.

Since $1$ cubic yard $= 27$ cubic feet, this is $\dfrac{45}{27} = 1\dfrac23$ cubic yards. Rounding up to a whole number, the contractor must order $2$ cubic yards.

Thus, the correct answer is A .

Solution:

A rectangular prism starts with $12$ edges, and cutting corners only shortens them without removing any.

Each of the $8$ corner cuts creates a small triangular face with $3$ new edges, adding $8 \cdot 3 = 24$ edges. The total is $12 + 24 = 36$.

Thus, the correct answer is C .

Solution:

To force the next person next to someone, every empty seat must be adjacent to an occupied one. The most efficient way is to seat people in a repeating pattern of occupied-empty-empty, filling the middle seat of every group of three.

With $120$ seats, this uses $\dfrac{120}{3} = 40$ occupied seats.

Thus, the correct answer is B .

Solution:

Only one entry changed. The incorrect total exceeds the actual total by $\$980{,}000 - \$98{,}000 = \$882{,}000.$

Since this extra amount is spread over $1000$ families, the means differ by $\dfrac{\$882{,}000}{1000} = \$882.$

Thus, the correct answer is A .

Solution:

Since each term is the product of the two before it, dividing any term by the term just before it recovers the term two positions earlier. Working backwards from the last three terms $16, 64, 1024$:

$$64 \div 16 = 4, \quad 16 \div 4 = 4, \quad 4 \div 4 = 1, \quad 4 \div 1 = 4, \quad 1 \div 4 = \dfrac14.$$

These fill in the earlier terms, giving the full list $\dfrac14, 4, 1, 4, 4, 16, 64, 1024$. The first number is $\dfrac14$.

Thus, the correct answer is A .

Solution:

Chris takes the first piece, and then the bag goes around until Chris takes the last ($100$th) piece. So the $99$ pieces after Chris's first must be exactly a whole number of trips around the table back to Chris.

This means the number of students divides $99$. Among the choices, only $11$ divides $99$.

Thus, the correct answer is B .

Solution:

The average speed during any single hour equals the distance the airplane covers in that hour, which on the graph is the vertical rise of the curve over that one-hour interval. So the largest average speed happens during the hour where the curve is steepest.

From $t=1$ to $t=2$ the curve climbs from about $500$ miles to about $1000$ miles, a rise of roughly $500$ miles, giving an average speed near $500$ mph. During every other hour the curve rises by less than $350$ miles, so those average speeds are all smaller. The steepest climb, and hence the largest average speed, is during the second hour.

Thus, the correct answer is B .

Solution:

Let a triangle, diamond, and dot weigh $t$, $d$, and $1$. The conditions give $3t + d = 9$ and $t = d + 1$.

Substituting, $3(d+1) + d = 9$, so $4d + 3 = 9$ and $d = \dfrac32$. Then two diamonds weigh $2d = 3$ dots.

Thus, the correct answer is C .

Solution:

Classify the two shaded squares up to rotations and reflections. Patterns that include a corner: two adjacent corners (same edge), two diagonally opposite corners, corner with the center, corner with an adjacent edge-middle, and corner with a far edge-middle. That is $5$ patterns.

Patterns with no corner: two adjacent edge-middles, two opposite edge-middles, and an edge-middle with the center. That is $3$ more.

In total there are $5 + 3 = 8$ distinct patterns.

Thus, the correct answer is C .