AMC 8 Daily Practice Round 11

Dividing both the numerator and the denominator by , we get:

Thus, the correct answer is .

The ratio of the areas of the unshaded region to the shaded region is . Since the difference in their areas is square meters, the area of the shaded region is calculated as:

.

The minimum total cost is calculated by taking 3 of the least expensive items and 1 of the most expensive item:

The maximum total cost is calculated by taking 3 of the most expensive items and 1 of the least expensive item

Thus, the total cost must be between and . Since the option is outside this range, it cannot be the total cost.

, there are possibilities.

Select one red-painted edge for length, width, and height, ensuring that the chosen red edges are not in the same plane.

For example, as shown in the figure:

Thus, the correct answer is .

The amount of salt in the solution remains unchanged before and after evaporation. Let kilograms of water be evaporated. Then, the equation is:

Solving for , we get:

Thus, the amount of water that needs to be evaporated is .

To solve the problem, we first list all possible three-digit numbers that can be formed using the digits , , and : . There are such numbers in total. A number is a multiple of if its last digit is . From the list, and meet this condition, so there are favorable outcomes. Using the probability formula, the probability of forming a number that is a multiple of is:

Observing the new operation, it calculates the average of two numbers.

First, treating the parentheses as a whole, the average of the number inside the parentheses and equals , meaning:

Thus, the average of and equals , leading to:

Thus, the correct answer is .

, , which is a perfect number, then .

If each bench seats more students (i.e., students per bench), and there are unused benches, this means there are empty seats in total. The total number of benches is therefore

The total number of students is then

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By analyzing the problem, we know that point satisfies . Since , we calculate . Therefore, is translated 1 unit to the right along the -axis to form . Point is also shifted 1 unit to the right, resulting in point having coordinates . Thus, the correct answer is .

From the problem statement, the maximum possible value of is .

Since

and

the maximum value of is:

Thus, the correct answer is .

Since the digits of the four-digit number must follow a strict order (units digit thousands digit hundreds digit tens digit), each selection of 4 distinct digits from to corresponds to exactly one valid number. This is because the order of the digits is uniquely determined by their sizes. Note that can be included in the selection, as it will occupy the smallest position (tens place) if chosen, satisfying the requirement that the leading digit (thousands place) cannot be . The total number of ways to choose 4 digits from to is:

Thus, there are such four-digit numbers.

The sequence of pentagonal numbers is formed as follows:

Continuing the pattern, we calculate:

Thus, the 7th pentagonal number is .

We can solve this problem using place value .

Step : Considering the Units Digit

If no borrowing occurs, we get:

Solving these equations:

.

However, if the minuend’s hundreds digit is , the difference cannot be , so borrowing must have occurred.

Step : Considering Borrowing

With borrowing, we modify the equation:

Rearranging:

Since , solving for and :

Step : Finding

By analyzing the tens and hundreds digits, we determine:

Thus, the three-digit number is .

Final Answer:

Thus, the correct answer is , which is .

To find all possible passwords, we need to determine combinations of four nonzero digits that sum to . The valid combinations are ; ; ; ; ; and , resulting in six groups. For the combination , the digit can occupy any of the 4 positions, giving possibilities. For , can occupy 4 positions, can occupy 3 remaining positions, and fills the rest, yielding possibilities. Similarly, each of , , and also has possibilities. For , can occupy any of the 4 positions, giving possibilities. By the addition principle, the total number of passwords is:

Thus, Vivi needs to try at least attempts to guarantee opening the safe.

Connect ，

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Then, ,

,

,

,

,

,

,

,

Since .

When , there are exactly points lie inside the circle.

Since the two sectors have the same radius, the ratio of their areas is equal to the ratio of their central angles. Given that the difference in their central angles is and the sum of their central angles is . We can solve for and . Hence, the ratio of their central angles is . It follows that the area ratio is also . Thus, the area of is .

The product can be simplified as follows

Since , we have

Thus, the expression becomes

This represents a number with digits (the digits contribute 2 digits, and adds 32 zeros). Therefore, the product is a 34-digit number.

Let the total number of club members be represented as unit. Using the inclusion-exclusion principle, the total number of members can be calculated as:

Simplify the expression:

The number of members involved only in academic competitions is:

Thus, the number of members involved only in academic competitions is .

Represent the students as points , and use a line connecting two points to indicate that the corresponding pair of students shares exactly one common book. Since each book is owned by exactly two students, there is exactly one line between any two points.

To find the total number of books, we count the total number of lines in a complete graph with points. The number of lines (or edges) in a complete graph with points is given by:

For , the total number of lines is:

Thus, the students share a total of different books. The answer is .

Let the maximum one-way distance the plane can travel be kilometers. Based on the problem, the equation is

Simplifying, we find

After verification, satisfies the original equation. Therefore, the plane can travel a maximum of one way.