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AMC 12 2009 B

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

  
  2020-07-09 06:38:37

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

  
  2020-07-09 06:38:37

Question 3

Twenty percent off is one-third more than what number?

Solution

  
  2020-07-09 06:38:37

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?

Solution

  
  2020-07-09 06:38:37

Question 5

Kiana has two older twin brothers. The product of their ages is . What is the sum of their three ages?

Solution

  
  2020-07-09 06:38:37

Question 6

By inserting parentheses, it is possible to give the expression several values. How many different values can be obtained?

Solution

  
  2020-07-09 06:38:37

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 ?

Solution

  
  2020-07-09 06:38:37

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?

Solution

  
  2020-07-09 06:38:37

Question 9

Triangle has vertices , , and , where is on the line . What is the area of ?

Solution

  
  2020-07-09 06:38:37

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

  
  2020-07-09 06:38:37

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?

Solution

  
  2020-07-09 06:38:37

Question 12

The fifth and eighth terms of a geometric sequence of real numbers are and respectively. What is the first term?

Solution

  
  2020-07-09 06:38:37

Question 13

Triangle has and , and the altitude to has length . What is the sum of the two possible values of ?

Solution

  
  2020-07-09 06:38:37

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 ?

Solution

  
  2020-07-09 06:38:37

Question 15

Assume . Below are five equations for . Which equation has the largest solution ?

Solution 1

Intuitively, x will be largest for that option for which the value in the parentheses is smallest.
  
jimmy  2016-10-01 22:01:40

Solution 2

Intuitively, x will be largest for that option for which the value in the parentheses is smallest.
  
jimmy  2016-09-27 17:18:23

Question 16

Trapezoid has , , , and . The ratio is . What is ?

Solution

  
  2020-07-09 06:38:37

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

  
  2020-07-09 06:38:37

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?

Solution

  
  2020-07-09 06:38:37

Question 19

For each positive integer , let . What is the sum of all values of that are prime numbers?

Solution

  
  2020-07-09 06:38:37

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?

Solution

  
  2020-07-09 06:38:37

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?

Solution

  
  2020-07-09 06:38:37

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?

Solution

  
  2020-07-09 06:38:37

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 ?

Solution

  
  2020-07-09 06:38:37

Question 24

For how many values of in is ? Note: The functions and denote inverse trigonometric functions.

Solution

  
  2020-07-09 06:38:37

Question 25

The set is defined by the points with integer coordinates, , . How many squares of side at least have their four vertices in ?

Solution

  
  2020-07-09 06:38:37

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