Thursday, April 30, 2009

April 30, 2009

Suppose cos x = 0 and cos(x + z) = 1/2. What is the smallest possible positive value of z?


Wednesday, April 29, 2009

April 29, 2009

What is the average (mean) of all 5-digit numbers that can be formed by using each of the digits 1, 3, 5, 7, and 8 exactly once?

Interactive version of problem and solution.

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Monday, April 27, 2009

April 27, 2009

On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning.

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Friday, April 24, 2009

Thursday, April 23, 2009

April 23, 2009

When 15 is appended to a list of integers, the mean is increased by 2. When 1 is appended to the enlarged list, the mean of the enlarged list is decreased by 1. How many integers were in the original list?


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Wednesday, April 22, 2009

April 22, 2009

Let n be a positive integer such that 1/2 + 1/3 + 1/7 + 1/n is an integer. Which of the following statements is not true:

(A) 2 divides n
(B) 3 divides n
(C) 6 divides n
(D) 7 divides n
(E) n > 84

Interactive version of problem and solution.



Tuesday, April 21, 2009

April 21, 2009

Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner?


Monday, April 20, 2009

April 20, 2009

Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. Lance, Sally, Joy and Fred are chosen for the team. In how many ways can the three starters be chosen?

Interactive version of problem and solution.


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Friday, April 17, 2009

April 17, 2009

Twelve friends met for dinner at Oscar's Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for 18 people. If they share, how many meals should they have ordered to have just enough food for the 12 of them?

Interactive version of problem and solution.



Thursday, April 16, 2009

April 16, 2009

On a map, a 12-centimeter length represents 72 kilometers. How many kilometers does a 17-centimeter length represent?

Interactive version of problem and solution.




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Wednesday, April 15, 2009

April 15, 2009

How many different four-digit numbers can be formed by rearranging the four digits in 2004?

Interactive version of problem and solution.


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Tuesday, April 14, 2009

April 14, 2009

Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers 1 through 10. Each throw hits the target in a region with a different value. The scores are: Alice 16 points, Ben 4 points, Cindy 7 points, Dave 11 points, and Ellen 17 points. Who hits the region worth 6 points?


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Monday, April 13, 2009

April 13, 2009

You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of $1.02, with at least one coin of each type. How many dimes must you have?

Interactive version of problem and solution.



Friday, April 10, 2009

April 10, 2009

Let S be the set of the 2005 smallest positive multiples of 4, and let T be the set of the 2005 smallest positive multiples of 6. How many elements are common to S and T?


Thursday, April 9, 2009

Wednesday, April 8, 2009

April 8, 2009

How many positive integers n satisfy the following condition:

(130n)50 > n100 > 2200 ?

Interactive version of problem and solution.



Tuesday, April 7, 2009

April 7, 2009

A merchant offers a large group of items at 30% off. Later, the merchant takes 20% off these sale prices and claims that the final price of these items is 50% off the original price. The total discount is actually...

(A) 35% (B) 44% (C) 50% (D) 56% (E) 60%

Interactive version of problem and solution.




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Friday, April 3, 2009

April 3, 2009

A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 3. What is the area of the triangle?

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Thursday, April 2, 2009

April 2, 2009

Triangles ABC and ADE have areas 2007 and 7002, respectively, with B = (0, 0), C = (223, 0), D = (680, 380), and E = (689, 389). What is the sum of all possible x-coordinates of A?

Interactive version of problem and solution.



Wednesday, April 1, 2009

April 1, 2009

The 2007 AMC 10 will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave only the last 3 unanswered. How many of the first 22 problems must she solve correctly in order to score at least 100 points?