Question 1

One ticket to a show costs at full price. Susan buys 4 tickets using a coupon that gives her a discount. Pam buys 5 tickets using a coupon that gives her a discount. How many more dollars does Pam pay than Susan?

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Question 2

Define and . What is ?

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Question 3

An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise?

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Question 4

The larger of two consecutive odd integers is three times the smaller. What is their sum?

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Question 5

A school store sells 7 pencils and 8 notebooks for . It also sells 5 pencils and 3 notebooks for . How much do 16 pencils and 10 notebooks cost?

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Question 6

At Euclid High School, the number of students taking the AMC 10 was in 2002, in 2003, in 2004, in 2005, in 2006, and is in 2007. Between what two consecutive years was there the largest percentage increase?

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Question 7

Last year Mr. Jon Q. Public received an inheritance. He paid in federal taxes on the inheritance, and paid of what he had left in state taxes. He paid a total of for both taxes. How many dollars was his inheritance?

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Question 8

Triangles and are isosceles with and . Point is inside triangle , angle measures 40 degrees, and angle measures 140 degrees. What is the degree measure of angle ?

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Question 9

Real numbers and satisfy the equations and . What is ?

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Question 10

The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is , the father is years old, and the average age of the mother and children is . How many children are in the family?

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Question 11

The numbers from to are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?

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Question 12

Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?

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Question 13

Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?

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Question 14

A triangle with side lengths in the ratio is inscribed in a circle with radius . What is the area of the triangle?

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Question 15

Four circles of radius are each tangent to two sides of a square and externally tangent to a circle of radius , as shown. What is the area of the square?

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Question 16

Integers and , not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that is even?

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Question 17

Suppose that and are positive integers such that . What is the minimum possible value of ?

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Question 18

Consider the -sided polygon , as shown. Each of its sides has length , and each two consecutive sides form a right angle. Suppose that and meet at . What is the area of quadrilateral ?

Solution

Extend AB and CH to meet at N.
Triangles CBN and CHG are similar with ratio of 1:2. BN=2, BC=4, so area of triangle CBN=2*4/2 = 4
Triangles MAN and MHG are similar with ratio of 6:4=3:2.
Draw a vertical line from M to AN at M1, to HG at M2. Line M1M2= 12. Ratio of MM1:MM2=3:2.
So MM1=3/5 * 12; So area of MAN=1/2 * 6 * 3/5 * 12.
So area of ABCM = area of MAN - area of CBN = 1/2 * 6 * 3/5 * 12 - 4 = 88/5

Question 19

A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width?

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Question 20

Suppose that the number satisfies the equation . What is the value of ?

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Question 21

A sphere is inscribed in a cube that has a surface area of square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?

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Question 22

A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let be the sum of all the terms in the sequence. What is the largest prime factor that always divides ?

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Question 23

How many ordered pairs of positive integers, with , have the property that their squares differ by ?

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Question 24

Circles centered at and each have radius , as shown. Point is the midpoint of , and . Segments and are tangent to the circles centered at and , respectively, and is a common tangent. What is the area of the shaded region ?

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Question 25

For each positive integer , let denote the sum of the digits of For how many values of is

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