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2010 AMC 12A Problems and Solution

Problem 1

What is

?

Solution

.

Problem 2

A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?

Solution

It is easy to see that the ferry boat takes trips total. The total number of people taken to the island is

Problem 3

Rectangle Square

, pictured below, shares shares

of its area with square

. What is

?

.

of its area with rectangle

Solution Solution 1

Let

, let

, and let

.

Solution 2

The answer does not change if we shift horizontal lines to divide

to coincide with

, and add new

into five equal parts:

This helps us to see that

.

and

, where

. Hence

Problem 4

If

, then which of the following must be positive?

Solution

is negative, so we can just place a negative value into each expression and find the one that is positive. Suppose we use

.

Obviously only

is positive.

Problem 5

Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next shots are bullseyes she will be guaranteed victory. What is the minimum value for ?

Solution

Let be the number of points Chelsea currently has. In order to guarantee victory, we must consider the possibility that the opponent scores the maximum amount of points by getting only bullseyes.

The lowest integer value that satisfies the inequality is

.

Problem 6

A

, such as 83438, is a number that remains the same when its digits

are three-digit and four-digit palindromes,

are reversed. The numbers and

respectively. What is the sum of the digits of ?

Solution

is at most

must be

It follows that is

, so

.

, so the sum of the digits is

.

is at most

. The minimum value of and

is

is

.

However, the only palindrome between , which means that

Problem 7

Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?

Solution

The water tower holds

times more water than Logan's

times shorter .

miniature. Therefore, Logan should make his tower than the actual tower. This is Also, the fact that

meters high, or choice

doesn't matter since only the ratios are important.

Problem 8

Triangle such that

and suppose that

has

. Let . Let

and

be on

and ?

, respectively,

and

,

be the intersection of segments

is equilateral. What is

Solution

Let

.

Since , triangle is a triangle, so

Problem 9

A solid cube has side length inches. A -inch by -inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?

Solution Solution 1

Imagine making the cuts one at a time. The first cut removes a box second cut removes two boxes, each of dimensions cuts is

.

. . The

, and the third cut

does the same as the second cut, on the last two faces. Hence the total volume of all

Therefore the volume of the rest of the cube is