Question 1
Each morning of her five-day workweek, Jane bought either a -cent muffin or a -cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?
Solution
Question 2
Paula the painter had just enough paint for identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for rooms. How many cans of paint did she use for the rooms?
Solution
Question 3
Twenty percent off is one-third more than what number?
Solution
Question 4
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths and meters. What fraction of the yard is occupied by the flower beds?
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Question 5
Kiana has two older twin brothers. The product of their ages is . What is the sum of their three ages?
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Question 6
By inserting parentheses, it is possible to give the expression several values. How many different values can be obtained?
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Question 7
In a certain year the price of gasoline rose by during January, fell by during February, rose by during March, and fell by during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is ?
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Question 8
When a bucket is two-thirds full of water, the bucket and water weigh kilograms. When the bucket is one-half full of water the total weight is kilograms. In terms of and , what is the total weight in kilograms when the bucket is full of water?
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Question 9
Triangle has vertices , , and , where is on the line . What is the area of ?
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Question 10
A particular -hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a , it mistakenly displays a . For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?
Solution
Question 11
On Monday, Millie puts a quart of seeds, of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
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Question 12
The fifth and eighth terms of a geometric sequence of real numbers are and respectively. What is the first term?
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Question 13
Triangle has and , and the altitude to has length . What is the sum of the two possible values of ?
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Question 14
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from to , divides the entire region into two regions of equal area. What is ?
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Question 15
Assume . Below are five equations for . Which equation has the largest solution ?
Solution 1
Solution 2
Question 16
Trapezoid has , , , and . The ratio is . What is ?
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Question 17
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?
Solution
Question 18
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every seconds, and Robert runs clockwise and completes a lap every seconds. Both start from the start line at the same time. At some random time between minutes and minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?
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Question 19
For each positive integer , let . What is the sum of all values of that are prime numbers?
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Question 20
A convex polyhedron has vertices , and edges. The polyhedron is cut by planes in such a way that plane cuts only those edges that meet at vertex . In addition, no two planes intersect inside or on . The cuts produce pyramids and a new polyhedron . How many edges does have?
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Question 21
Ten women sit in seats in a line. All of the get up and then reseat themselves using all seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated?
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Question 22
Parallelogram has area . Vertex is at and all other vertices are in the first quadrant. Vertices and are lattice points on the lines and for some integer , respectively. How many such parallelograms are there?
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Question 23
A region in the complex plane is defined by A complex number is chosen uniformly at random from . What is the probability that is also in ?
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Question 24
For how many values of in is ? Note: The functions and denote inverse trigonometric functions.
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Question 25
The set is defined by the points with integer coordinates, , . How many squares of side at least have their four vertices in ?
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Answer Keys
Question 1: B
Question 2: C
Question 3: D
Question 4: C
Question 5: D
Question 6: C
Question 7: B
Question 8: E
Question 9: A
Question 10: A
Question 11: D
Question 12: E
Question 13: D
Question 14: C
Question 15: B
Question 16: B
Question 17: B
Question 18: C
Question 19: E
Question 20: C
Question 21: A
Question 22: C
Question 23: D
Question 24: B
Question 25: E