*n*is

*n*

^{2}– 3

*n*+ 2 a prime number?

Interactive version of problem and solution.

For how many positive integers *n* is *n*^{2} – 3*n* + 2 a prime number?

Interactive version of problem and solution.

Interactive version of problem and solution.

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

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.

Interactive version of problem and solution.

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.

Interactive version of problem and solution.

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.

Interactive version of problem and solution.

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* ?

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?

Buy *Hexaflexagons, Probability, Paradoxes, and the Tower of Hanoi* from the MAA Bookstore

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.

Interactive version of problem and solution.

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?

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.

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

Interactive version of problem and solution.

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.

Interactive version of problem and solution.

A positive number *x* has the property that *x*% of* x* is 4. What is *x*?

Interactive version of problem and solution.

Interactive version of problem and solution.

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.

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

Interactive version of problem and solution.

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.

Interactive version of problem and solution.

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.

Interactive version of problem and solution.

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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