Friday, February 27, 2009

Thursday, February 26, 2009

February 26, 2009

Real numbers a and b satisfy the equations 3a = 81b + 2 and 125b = 5a – 3. What is ab?




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Wednesday, February 25, 2009

February 25, 2008

Integers a, b, c, and d, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that ad–bc is even?

Interactive version of problem and solution.


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Tuesday, February 24, 2009

February 24, 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?

Interactive version of problem and solution.


Monday, February 23, 2009

February 23, 2009

Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?

Interactive version of problem and solution.




Thursday, February 19, 2009

February 19, 2009

The set {3; 6; 9; 10} is augmented by a fifth element n, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of n ?


Wednesday, February 18, 2009

February 18, 2008

Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League?




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Tuesday, February 17, 2009

February 17, 2009

How many integers between 1000 and 2000 have all three of the numbers 15, 20 and 25 as factors?

Interactive version of problem and solution.



Wednesday, February 11, 2009

February 11, 2009

The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is 20 cents. If she had one more quarter, the average value would be 21 cents. How many dimes does she have in her purse?


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Tuesday, February 10, 2009

February 10, 2009

The positive integers A, B, A – B, and A + B are all prime numbers. The sum of these four primes is...

(A) even (B) divisible by 3 (C) divisible by 5 (D) divisible by 7 (E) prime

Interactive version of problem and solution.


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Monday, February 9, 2009

February 9, 2009

A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white?

Interactive version of problem and solution.


Friday, February 6, 2009

Thursday, February 5, 2009

February 5, 2009

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?

(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

Interactive version of problem and solution.


Wednesday, February 4, 2009

February 4, 2009

An envelope contains eight bills: 2 ones, 2 fives, 2 tens, and 2 twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $20 or more?

Interactive version of problem and solution.


Tuesday, February 3, 2009

February 3, 2009

Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with a radius of 50 meters. Kershaw runs counterclockwise at 300 m/min and uses the outer lane with a radius of 60 meters, starting on the same radial line as Odell. How many times after the start do they pass each other?

Interactive version of problem and solution.



Monday, February 2, 2009

February 2, 2009

Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have?


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