When three consecutive positive integers are multiplied, the product is 16 times the sum of the three integers. What is the difference of the product minus the sum?
Kathy Beckhardt weighs four of her sheep at the fair. She can weigh two of them at a time on the big scale. Sheep A and sheep B have an average weight of 150 pounds, sheep B and sheep C have an average weight of 127 pounds, and sheep C and sheep D have an average weight of 168 pounds. What is the average weight of sheep A and sheep D?
David’s optometrist sold him a bottle of eyeglass cleaner containing 30 mL of glasscleaning solution. Assuming there are 20 drops per milliliter, and assuming proper cleaning requires 3 drops of glass cleaner on each side of each lens, what is the maximum number of times David can properly clean his glasses before he must buy a new bottle of eyeglass cleaner?
How many positive integers in the set of numbers from 1 to 1000, inclusive, are multiples of 2, 3 and 5 but not 8?
Annette, Mary and Lynn team up to pick apples. Annette can pick 4 baskets of apples per hour, and Mary can pick 5 baskets of apples per hour. Annette, Mary and Lynn work together to pick 6 baskets of apples in half an hour. How many baskets of apples can Lynn pick by herself in 3 hours?
Pamela Wickham writes a sequence of four consecutive integers on a sheet of paper. The sum of three of these integers is 206. What is the other integer?
Alexander visits Sweet Dreams bakery, which sells three kinds of cookies. How many unique assortments of a dozen cookies can Alexander buy?
How many ordered pairs of prime numbers (a, b) are there such that a + b = 100?
What is the greatest possible absolute difference between the median and mean of a list of 10 positive integers that are at most 20? Express your answer as a decimal to the nearest tenth.
If three fair coins are simultaneously flipped, what is the probability that exactly two heads will be showing? Express your answer as a fraction in simplest form.
How many ordered triples of integers (m, n, p) exist such that mn = p, np = m and mp = n?
What is the sum of all prime numbers p less than 60 such that there exists a right triangle whose side lengths are all integers and whose hypotenuse has length p?
Edna enters a room with 1000 bottles lined up in a row left to right. One bottle contains a tasteless magic potion. All bottles to the left of the magic potion contain tasteless water. All bottles to the right of the magic potion contain a bitter poison. Edna can drink from no more than two bottles containing poison before becoming sick and being unable to drink anything else. She can take an unlimited number of drinks from any other bottle. What is the minimum number of bottles from which Edna may need to drink to ensure she can identify the bottle containing the magic potion no matter where it is in the lineup?
If n is the product of three consecutive positive integers and n = 22 × 14 × k, what is the least possible value of k?
How many pairs of numbers (a, b) satisfy rules I and II shown here?
I. a = 0 or b = −1 or b = 1
II. a = −1 or a = 1 or b = 0
What fraction of the positive integer factors of 1000\(^3\) are perfect squares? Express your answer as a common fraction.
Suppose N is a positive integer such that N − 1 is even, N − 2 is divisible by 3, N − 3 is divisible by 5, and N − 5 is divisible by 7. What is the least possible value of N?
Let K = 168 × 900 = 151,200. How many positive integer divisors does K have?
If p(x) = ax\(^2\) + bx + c is a quadratic polynomial satisfying p(0) = 4, p(1) =15, p(2) = 36, what is the value of the product abc?
How many integers between 100 and 1000 contain no digits other than 3, 4 or 5?