Cubic to Quadratic to Linear

One root of a certain third-degree equation is 1. When the cubic term of the equation is crossed off, the resulting quadratic equation has a root of 2. When the squared term is also crossed off, the resulting linear equation has a root of 3. Reconstruct the original third-degree equation, expressing it in the form ax3 + bx2 + cx = d, with all coefficients as integers.

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Newton’s Cows

This problem has been attributed to Sir Isaac Newton. Three cows eat in two weeks all the grass on two acres of land, together with all the grass that grows there in the two weeks. Two cows eat in four weeks all the grass on two acres of land, together with all the grass that grows there in the four weeks. How many cows, then, will eat in six weeks all the grass on six acres of land together with all the grass that grows there in the six weeks? Assume that the quantity of grass on each acre is the same when the cows begin to graze, that the rate of growth is uniform during the time of grazing, and that the cows eat the same amount of grass each week.

Kumquat House Numbers

A new housing development, Mango Manors, is being built outside of town. It costs five cents to buy a single digit house number, but two digits can be bought for nine cents. If each home owner on Kumquat Street, a street with only single digit and 2-digit house numbers, buys the numbers for his own house, the total amount paid is $2.15. If one person buys them all, then the total amount paid is $2.12. How many single digit house numbers are on Kumquat Street?

  1. 3
  2. 4
  3. 5
  4. 6
  5. 7

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Euler Fortune

A father leaves at his death several children who share in his property in the following manner:

  • The first child receives 100 crowns and a tenth part of what remains.
  • The second child receives 200 crowns and a tenth part of what remains.
  • The third child receives 300 crowns and a tenth part of what remains.
  • And so on.

Now, it is found in the end that the property has been divided equally among all the children. How much was the fortune, how many children were there, and how much did each child receive?

Cows, Horses, Sheep in Pasture

Kurt, Chris, and David paid a total of $100 to buy a pasture. They split up the cost according to the amount of grass their animals would eat. Kurt put in 9 horses; David put in 12 cows for twice the time; and Chris put in some sheep for 2.5 times as long as David’s cows. If Chris paid half the cost of the pasture, how many sheep did Chris have, and how much did Kurt and David each pay, if 6 cows eat as much as 4 horses and 10 sheep eat as much as 3 cows in a given amount of time?

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Supermarket, Apples

The Superduper Supermarket has 128 crates of apples that have just arrived at the loading dock. Each crate contains at least 120 apples and at most 144 apples. What is the largest integer n you can find such that there must be at least n crates containing the same number of apples? (This means that, even if you make the number of apples in each crate as different as you can possibly make them, there are still going to be n crates with the same number of apples.)

  1. 4
  2. 5
  3. 6
  4. 24
  5. 25

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Shawn and Rhonda bought identical boxes of stationery. Shawn used hers to write 1-sheet letters and Rhonda used hers to write 3-sheet letters. Shawn used all of the envelopes and had 50 sheets of paper left, while Rhonda used all of the paper and had 50 envelopes left. The number of sheets of paper in each box was:

  1. 150
  2. 125
  3. 120
  4. 100
  5. 80

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