Importing Truffles

Amy is in France buying truffles, which she will import to the US. At one quaint farm she buys x truffles at 3 for 10 euros, and at the next quaint farm she buys 2x truffles at 5 for 21 euros. What is the price per dozen at which she would need to sell the truffles she has bought so far in order to break even?

Continue reading

Harry the Hobo

Harry, the high school teacher, lost his marbles and became the world’s best-educated hobo. He was hiking briskly across a high railroad overpass on a sunny autumn morning, when he heard a train approaching in the distance behind him. Noting that he was 3/7 of the way across the bridge, he ran ahead to the far end and hopped off just as the train caught up with him. Later, when he had time to think about it, he wondered whether he could have run back, instead, toward the train and still gotten off in time. Analyze this. Under what circumstances could he have escaped the train by running either way?

Continue reading

Progression to Quadratic

Consider the following interesting list of integers:

P0 = 41,
P1 = 43 = 41 + 2,
P2 = 47 = 43 + 4,
P3 = 53 = 47 + 6, … ,

in which Pn is obtained by adding 2n to Pn-1.

It so happens that there is a quadratic function F(x) with the property that Pn = F(n) for all nonnegative n. Find a formula for F(x).

Once you have the formula, consider the question of whether or not F(n) is a prime number for every nonnegative value of n.

Continue reading

Points on Line Making Triangle

Points A, B, C, and D are distinct and lie, in given order, on a straight line. Line segments AB, AC, and AD have lengths x, y, and z, respectively. If line segments AB and CD may be rotated about points B and C, respectively, so that points A and D coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied?

  1. x < z/2
  2. y < x + z/2
  3. y < z/2
  4. Continue reading