Suppose cos

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

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.

Interactive version of problem and solution.

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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.

How many sets of two or more consecutive positive integers have a sum of 15?

Interactive version of problem and solution.

Interactive version of problem and solution.

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.

(A) 2 divides

(B) 3 divides

(C) 6 divides

(D) 7 divides

(E)

Interactive version of problem and solution.

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?

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.

Interactive version of problem and solution.

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

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.

Interactive version of problem and solution.

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.

Interactive version of problem and solution.

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

Interactive version of problem and solution.

Interactive version of problem and solution.

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?

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.

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*?

The ratio of Mary's age to Alice's age is 3 : 5. Alice is 30 years old. How many years old is Mary?

Interactive version of problem and solution.

Interactive version of problem and solution.

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

How many positive integers n satisfy the following condition:

(130*n*)^{50} > *n*^{100} > 2^{200} ?

Interactive version of problem and solution.

(130

Interactive version of problem and solution.

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.

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

Interactive version of problem and solution.

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.

Interactive version of problem and solution.

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?

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