Small Hint:

Break the logo into congruent tilted squares

Big Hint:

Each tilted square is half of a $2$ by $2$ grid square

Solution:

The logo is made from $5$ congruent tilted squares. Each tilted square has both diagonals of length $2$, so its area is $\frac{2\cdot2}{2}=2$ square inches.

The total area is $5\cdot2=10$.

Thus, the correct answer is A.

Small Hint:

Evaluate $5\,\blacklozenge\,3$ first

Big Hint:

Substitute the value from the first operation into $\star 6$

Solution:

Using the definitions, $$\begin{align*} (5\,\blacklozenge\,3) \star 6 &= (5^2-3^2) \star 6 \\ &=16 \star 6 \\ &=(16-6)^2\\ &= 100\end{align*}$$

Thus, the correct answer is D.

Small Hint:

The smallest number must be a factor of $100$

Big Hint:

Try possible values of $a$ with $a^3<100$

Solution:

Since $a< b< c$ and $abc=100$, we have $a^3<100$, so $a\le4$. Also $a$ must divide $100$, so $a\in\{1,2,4\}$.

If $a=1$, the pairs $(b,c)$ are $(2,50),(4,25),(5,20)$. If $a=2$, the only possible pair is $(5,10)$. If $a=4$, then $bc=25$, and there is no integer $b$ with $4< b< c$.

There are $4$ possible triples.

Thus, the correct answer is E.

Small Hint:

Reflecting over $q$ moves the letter across the diagonal line

Big Hint:

After the first reflection, reflect the new image over the horizontal line $p$

Solution:

Reflecting over the diagonal line $q$ changes the orientation of the letter as if the diagonal were a mirror. Reflecting that image over the horizontal line $p$ then places the final letter below $p$, to the right of the vertical line, with the orientation shown in choice E.

Thus, the correct answer is E.

Small Hint:

Bella is $5$ years older now than when she turned $6$

Big Hint:

Find Bella’s current age and the kitten’s current age before finding Anna’s

Solution:

If Bella was $6$ five years ago, then she is $11$ right now.

If the kitten was a newborn five years ago, then it is $5$ right now.

Since the sum of all $3$ ages is $30,$ Anna’s age is $$30-5-11 = 14$$

Since Anna is $14$ and Bella is $11,$ she is $3$ years older than Bella.

Thus, the correct answer is C.

Small Hint:

The middle of three equally spaced numbers is their average

Big Hint:

If the smallest is $x$, then the largest is $4x$

Solution:

Let the smallest number be $x$. Then the largest is $4x$. Since the three numbers are equally spaced, the middle number is the average of the smallest and largest: $$\frac{x+4x}{2}=15.$$ Thus $5x=30$, so $x=6$.

Thus, the correct answer is C.

Small Hint:

Convert megabytes to kilobits first

Big Hint:

$4.2\cdot8000$ kilobits divided by $56$ kilobits per second gives seconds

Solution:

The song has $4.2\cdot8000=33{,}600$ kilobits. At $56$ kilobits per second, the download takes $$\frac{33{,}600}{56}=600$$ seconds, which is $10$ minutes.

Thus, the correct answer is B.

Small Hint:

Most numerator and denominator factors cancel

Big Hint:

After cancellation, only $1\cdot2$ remains on top

Solution:

Since every integer from $3$ to $20$ occurs once as a denominator and once as a numerator, they cancel each other out.

After canceling every number out, we have only $1$ and $2$ left as numerators and $21$ and $22$ left as denominators.

The remaining fraction is $\dfrac{1 \cdot 2}{21 \cdot 22} .$ This simplifies to $\dfrac{2}{462} = \dfrac{1}{231}$

Thus, the correct answer is B.

Small Hint:

Track the difference from room temperature

Big Hint:

In $15$ minutes, the difference is halved $3$ times

Solution:

The current difference is $212-68 = 144 .$

Since we have $15$ minutes while halving every $5$ minutes, we halve the difference $3$ times.

This means the difference is multiplied by $(\dfrac{1}{2})^3 = \dfrac{1}{8} ,$ so our new difference is $\dfrac{1}{8}\cdot144=18.$ This makes our final temperature $68+18=86.$

Thus, the correct answer is B.

Small Hint:

Find the distance to the trail from the first drive

Big Hint:

The return drive is faster, so it takes less time than the outward drive

Solution:

She drives $45$ miles per hour for $2$ hours, so the trail is $90$ miles from her house. The graph must rise from $0$ to $90$ miles between $8$ AM and $10$ AM, then stay flat for the $3$-hour hike.

She starts home at $1$ PM. Driving $90$ miles at $60$ miles per hour takes $1.5$ hours, so she gets home at $2{:}30$ PM. This matches choice E.

Thus, the correct answer is E.

Small Hint:

Each bite increases the number of pasta pieces by $1$

Big Hint:

To end with $10$ pieces, Henry made $9$ bites

Solution:

Since there are $10$ pieces, there were $9$ locations where a bite was made. Since we have $9$ bites and $3$ inches are removed per bite, a total of $27$ inches were removed.

With $27$ inches removed and $17$ inches remaining, we know we started with $27+17 = 44$ inches.

Thus, the correct answer is D.

Small Hint:

Spinner A gives the tens digit and spinner B gives the ones digit

Big Hint:

Check which two-digit squares have tens digit $5,6,7,8$ and ones digit $1,2,3,4$

Solution:

Spinner A gives the tens digit and Spinner B gives the ones digit. There are $4\cdot4=16$ equally likely two-digit numbers.

The possible perfect squares between $51$ and $84$ are $64$ and $81$, and both can occur. Thus the probability is $$\frac{2}{16}=\frac{1}{8}.$$

Thus, the correct answer is B.

Small Hint:

Let the smaller of the two positive integers be $x$

Big Hint:

If the blank is $c$, the two numbers can be $x$ and $2x+c$

Solution:

Let the smaller number be $x$, and let the blank be $c$. Then the other number is $2x+c$, where both $x$ and $c$ are positive integers.

The sum condition gives $x+(2x+c)=28$, so $c=28-3x$. For $c$ to be positive, $28-3x>0$, so $x<\frac{28}{3}$.

Thus $x$ can be any integer from $1$ through $9$, giving $9$ possible values of the blank.

Thus, the correct answer is D.

Small Hint:

There are $5$ E’s and $4$ other letters

Big Hint:

The four non-E letters must fill the four gaps between the E’s

Solution:

The word has $5$ E’s and $4$ other letters: B, K, P, and R. To keep no two E’s adjacent, the only possible pattern is $$E\_E\_E\_E\_E,$$ with the four non-E letters in the four gaps.

The letters B, K, P, and R can be arranged in those gaps in $4!=24$ ways.

Thus, the correct answer is D.

Small Hint:

Compare price divided by weight

Big Hint:

For each weight, only the lowest plotted price can be best

Solution:

For each weight, only the lowest-price point at that weight can give the lowest price per ounce.

At $1$ ounce, the lowest price is more than $1$ dollar, so the price per ounce is more than $1$.

At $2$ ounces, the lowest price is $2$ dollars, so the price per ounce is $1$.

At $3$ ounces, the lowest price is about $2.5$ dollars, so the price per ounce is about $\frac{2.5}{3}$, less than $1$.

At $4$ ounces, the lowest price is close to $3.9$ dollars, so the price per ounce is close to $\frac{3.9}{4}$, still larger than the $3$-ounce option.

At $5$ ounces, the lowest price is close to $4.5$ dollars, so the price per ounce is close to $\frac{4.5}{5}$, also larger than the $3$-ounce option.

The price per ounce is lowest at $3$ ounces.

Thus, the correct answer is C.

Small Hint:

Convert each average into a sum

Big Hint:

Find $a+b+c+d$, then subtract $b+c$

Solution:

Let the numbers be $a,b,c,d$ in order.

Since the average of $a$ and $b$ is $21,$ we know their sum is $21\cdot2 = 42.$
Since the average of $c$ and $d$ is $30,$ we know their sum is $30\cdot2 = 60.$
Since $a+b = 42$ and $c+d = 60 ,$ we know $$\begin{align*} a+b+c+d &= 42 + 60\\ &= 102 \end{align*}$$

Now, with the average of $b$ and $c$ being $26,$ we know their sum is $26\cdot2=52.$ This means $b+c = 52.$

Subtracting this result from the sum of all the terms yields $a+d =50.$
Since $\dfrac{a+d}{2}=25,$ our answer is $25.$

Thus, the correct answer is B.

Small Hint:

Any even double factorial at least $10!!$ has a factor of $10$

Big Hint:

Only $2!!,4!!,6!!,$ and $8!!$ can affect the units digit

Solution:

For every even $n\ge10,$ the product $n!!$ contains a factor of $10,$ so that term has units digit $0.$ Only the first four terms can affect the units digit.

They sum to $$\begin{gathered} 2!!+4!!+6!!+8!!\\ =2+8+48+384\\ =442. \end{gathered}$$ Therefore the full sum has units digit $2.$

Thus, the correct answer is B.

Small Hint:

The given midpoints form a parallelogram inside the rectangle

Big Hint:

The midpoint parallelogram has half the area of the original rectangle

Solution:

The four given midpoints form a parallelogram. Its horizontal base has length $2-(-3)=5$, and its height is $4$, so its area is $5\cdot4=20$.

For any rectangle, the parallelogram formed by joining the side midpoints has half the area of the rectangle. Therefore the rectangle’s area is $2\cdot20=40$.

Thus, the correct answer is C.

Small Hint:

The median is controlled by the $10$th and $11$th scores

Big Hint:

Count how many scores are at least $85$ before and after regrading

Solution:

All scores are multiples of $5,$ and adding $5$ preserves that fact. If the $10$th and $11$th scores average $85$ but are not both $85,$ then the $11$th score must be at least $90.$ That would require at least $10$ scores of $90$ or more.

However, even after regrading, only the original scores of $85$ or more can reach $90.$ The plot has only $7$ such scores, so the two middle scores must both be $85.$

The plot initially has $7$ scores of at least $85.$ To make the $10$th and $11$th scores both $85,$ there must be at least $11$ scores of at least $85.$ Raising four of the $80$’s to $85$ is both necessary and sufficient.

Thus, the correct answer is C.

Small Hint:

The completed row and column sums must match the top row

Big Hint:

Express the other missing entries in terms of $x$

Solution:

Adding the numbers in the top row shows that every row and column must have sum $12$.

In the first column, the missing number above $x$ is $14-x$. In the bottom row, the missing number to the right of $x$ is $4-x$. In the middle row, the remaining missing number is $x-1$.

Since $x$ is larger than the other missing numbers, we need $x>14-x$, $x>4-x$, and $x> x-1$. The strongest condition is $x>7$, so the smallest possible integer value is $8$.

Thus, the correct answer is D.

Small Hint:

Steph made $25$ baskets in $30$ attempts overall

Big Hint:

Candace must also have made $25$ baskets total

Solution:

Steph made $25$ baskets in $30$ attempts. Candace also took $12+18=30$ attempts, and their overall percentages were the same, so Candace also made $25$ baskets.

Let $f$ be Candace’s first-half baskets and $s$ be her second-half baskets. Then $f+s=25$.

Steph’s first-half percentage was $\frac{15}{20}=\frac34$, so $\frac{f}{12}<\frac34$, giving $f<9$. Steph’s second-half percentage was $1$, so $s<18$.

Thus $f\le8$ and $s\le17$. Since $f+s=25$, the only possibility is $f=8$ and $s=17$, so $s-f=9$.

Thus, the correct answer is C.