Question 1

What is the value of when ?

Solution

Question 2

The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of and is closest to which integer?

Solution

Question 3

Let . What is the value of ?

Solution

Question 4

The ratio of the measures of two acute angles is , and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?

Solution

Question 5

The War of started with a declaration of war on Thursday, June , . The peace treaty to end the war was signed days later, on December , . On what day of the week was the treaty signed?

Solution

Question 6

All three vertices of lie on the parabola defined by , with at the origin and parallel to the -axis. The area of the triangle is . What is the length of ?

Solution

Question 7

Josh writes the numbers . He marks out , skips the next number , marks out , and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number , skips the next number , marks out , skips , marks out , and so on to the end. Josh continues in this manner until only one number remains. What is that number?

Solution

Question 8

A thin piece of wood of uniform density in the shape of an equilateral triangle with side length inches weighs ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of inches. Which of the following is closest to the weight, in ounces, of the second piece?

Solution

Question 9

Carl decided to fence in his rectangular garden. He bought fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl???s garden?

Solution

Question 10

A quadrilateral has vertices , , , and , where and are integers with . The area of is . What is ?

Solution

Question 11

How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line , the line and the line

Solution 1

Solution 2

I am posting this diagram so it will be easier for you to visualize the solution.

Question 12

All the numbers are written in a array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to . What is the number in the center?

Solution

Question 13

Alice and Bob live miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is from Alice's position and from Bob's position. Which of the following is closest to the airplane's altitude, in miles?

Solution

Question 14

The sum of an infinite geometric series is a positive number , and the second term in the series is . What is the smallest possible value of

Solution

Question 15

All the numbers are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?

Solution

Question 16

In how many ways can be written as the sum of an increasing sequence of two or more consecutive positive integers?

Solution

Question 17

In shown in the figure, , , , and is an altitude. Points and lie on sides and , respectively, so that and are angle bisectors, intersecting at and , respectively. What is ?

Solution

Question 18

What is the area of the region enclosed by the graph of the equation

Solution

Question 19

Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times?

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Question 20

A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won games and lost games; there were no ties. How many sets of three teams were there in which beat , beat , and beat

Solution

Question 21

Let be a unit square. Let be the midpoint of . For let be the intersection of and , and let be the foot of the perpendicular from to . What is

Solution

Question 22

For a certain positive integer less than , the decimal equivalent of is , a repeating decimal of period of , and the decimal equivalent of is , a repeating decimal of period . In which interval does lie?

Solution

Question 23

What is the volume of the region in three-dimensional space defined by the inequalities and ?

Solution

Question 24

There are exactly ordered quadruplets such that and . What is the smallest possible value for ?

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Question 25

The sequence is defined recursively by , , and for . What is the smallest positive integer such that the product is an integer?

Solution

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