Question 1

Isabella's house has bedrooms. Each bedroom is feet long, feet wide, and feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy square feet in each bedroom. How many square feet of walls must be painted?

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Question 2

Define the operation by What is

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Question 3

A college student drove his compact car miles home for the weekend and averaged miles per gallon. On the return trip the student drove his parents' SUV and averaged only miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?

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Question 4

The point is the center of the circle circumscribed about with and . What is the degree measure of

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Question 5

In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?

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Question 6

The will be scored by awarding points for each correct response, points for each incorrect response, and points for each problem left unanswered. After looking over the problems, Sarah has decided to attempt the first and leave only the last unanswered. How many of the first problems must she solve correctly in order to score at least points?

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Question 7

All sides of the convex pentagon are of equal length, and What is the degree measure of

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Question 8

On the trip home from the meeting where this AMC10 was constructed, the Contest Chair noted that his airport parking receipt had digits of the form where and was the average of and How many different five-digit numbers satisfy all these properties?

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Question 9

A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is place to its right in the alphabet (assuming that the letter is one place to the right of the letter ). The second time this same letter appears in the given message, it is replaced by the letter that is places to the right, the third time it is replaced by the letter that is places to the right, and so on. For example, with this code the word "banana" becomes "cbodqg". What letter will replace the last letter in the message

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Question 10

Two points and are in a plane. Let be the set of all points in the plane for which has area Which of the following describes

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Question 11

A circle passes through the three vertices of an isosceles triangle that has two sides of length and a base of length What is the area of this circle?

Solution

In the diagram below:

r^2 = 1^2 + h^2
3^2 = 1^2 + (h+r)^2

Solving the above, we get r = 9/(4*sqrt(2)), from which you can get the area of the circle.

Question 12

Tom's age is years, which is also the sum of the ages of his three children. His age years ago was twice the sum of their ages then. What is

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Question 13

Two circles of radius are centered at and at What is the area of the intersection of the interiors of the two circles?

Solution

We can try to find half of the intersected area, which is the sector area - triangle area (see diagram below). The sector area is 1/4 of the circle area, which is 1/4 * 2 * pi = pi. The triangle area is 1/2 * 2 * 2 = 2. So the intersected area is 2*(pi-2)

Question 14

Some boys and girls are having a car wash to raise money for a class trip to China. Initially of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then of the group are girls. How many girls were initially in the group?

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Question 15

The angles of quadrilateral satisfy What is the degree measure of rounded to the nearest whole number?

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Question 16

A teacher gave a test to a class in which of the students are juniors and are seniors. The average score on the test was The juniors all received the same score, and the average score of the seniors was What score did each of the juniors receive on the test?

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Question 17

Point is inside equilateral Points and are the feet of the perpendiculars from to and respectively. Given that and what is

Solution

Based on Heron's formula, a regular triangle with side s will have area

sqrt(3s/2 * (3s/2 - s) * (3s/2 - s) * (3s/2 - s)) =
(s/2)^2 * sqrt(3)

The perpendicular lines are height of 3 small triangles whose area sum is same as area of regular triangle:

s*1/2 + s*2/2 + s*3/2 = (s/2)^2 * sqrt(3)

Solving s you will get the answer.

Question 18

A circle of radius is surrounded by circles of radius as shown. What is ?

Solution

See diagram below. The red triangle is a 45-45-90 triangle. Why? Because I can draw the same 4 triangles with other outer circles. And 360/4=90.

(2r)^2 = (r+1)^2 + (r+1)^2

We have r=1+ sqrt(2)

Question 19

The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by and the second number is divided by The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square?

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Question 20

A set of square blocks is arranged into a square. How many different combinations of blocks can be selected from that set so that no two are in the same row or column?

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Question 21

Right has and Square is inscribed in with and on on and on What is the side length of the square?

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Question 22

A player chooses one of the numbers through . After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered through If the number chosen appears on the bottom of exactly one die after it has been rolled, then the player wins dollar. If the number chosen appears on the bottom of both of the dice, then the player wins dollars. If the number chosen does not appear on the bottom of either of the dice, the player loses dollar. What is the expected return to the player, in dollars, for one roll of the dice?

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Question 23

A pyramid with a square base is cut by a plane that is parallel to its base and units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?

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Question 24

Let denote the smallest positive integer that is divisible by both and and whose base- representation consists of only 's and 's, with at least one of each. What are the last four digits of

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Question 25

How many pairs of positive integers are there such that and have no common factors greater than and is an integer?

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