August 31, 2009

Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for 24 hours. If she is using it constantly, the battery will last for only 3 hours. Since the last recharge, her phone has been on 9 hours, and during that time she has used it for 60 minutes. If she doesn't talk any more but leaves the phone on, how many more hours will the battery last?

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August 28, 2009

Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of 5?

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August 27, 2009

If an arc of 45 degrees on circle A has the same length as an arc of 30 degrees on circle B, then the ratio of the area of circle A to the area of circle B is what?

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August 26, 2009

Let {a_k} be a sequence of integers such that a₁ = 1 and a_m+n = a_m + a_n + mn, for all positive integers m and n. Then a₁₂ is what?

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August 25, 2009

A wooden cube n units on a side is painted red on all six faces and then cut into n³ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is n?

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August 24, 2009

Figure ABCD is a square. Inside this square three smaller squares are drawn with side lengths as labeled. the area of the shaded L-shaped region is

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August 21, 2009

A rectangle with a diagonal of length x is twice as long as it is wide. What is the area of the rectangle?

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August 20, 2009

Two is 10% of x and 20% of y. What is x-y?

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August 19, 2009

A player chooses one of the numbers 1 through 4. After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered 1 through 4. If the number chosen appears on the bottom of exactly one die after it is rolled, then the player wins $1. If the number chosen appears on the bottom of both of the dice, then the player wins $2. If the number chosen does not appear on the bottom of either of the dice, the player loses $1. What is the expected return to the player, in dollars, for one roll of the dice?

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August 18, 2009

A singles tournament had six players. Each player played every other player only once, with no ties. If Helen won 4 games, Ines won 3 games, Janet won 2 games, Kendra won 2 games and Lara won 2 games, how many games did Monica win?

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August 17, 2009

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is 6. How many two-digit numbers have this property?

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August 14, 2009

Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?

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August 13, 2009

Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee ?

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August 12, 2009

Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?

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August 11, 2009

Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet?

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August 10, 2009

Elmo makes N sandwiches for a fundraiser. For each sandwich he uses B globs of peanut butter at 4¢ per glob and J blobs of jam at 5¢ per blob. The cost of the peanut butter and jam to make all the sandwiches is $2.53. Assume that B, J, and N are positive integers with N > 1. What is the cost of the jam Elmo uses to make the sandwiches?

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August 7, 2009

How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?

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August 6, 2009

The sum of the two 5-digit numbers AMC10 and AMC12 is 123422. What is A +M + C?

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August 5, 2009

In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have?

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August 4, 2009

Two dice are thrown. What is the probability that the product of the two numbers is a multiple of 5?

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August 3, 2009

A collector offers to buy state quarters for 2000% of their face value. At that rate how much will Bryden get for his four state quarters?

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