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 2*x* 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?

## Set 08

## Present and Future Ages 3

Alfie is 5 years old. His mother is 28. How old will Alfie be when he is exactly half as old as his mother?

## 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?

## Yorkshire to London

Marla walked from Yorkshire toward London at a constant rate. If she had gone 1/2 mph faster she would have walked the distance in 4/5 of the time. If she had gone 1/2 mph slower she would have taken 2.5 hours longer. How many miles did she walk?

## Three Valves

Each valve A, B, and C, when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour; with only valves A and C open, it takes 1.5 hours; and with only valves B and C open, it takes 2 hours. The number of hours required with only valves A and B open is:

- 1.1
- 1.15
- 1.2
- 1.25
- 1.75

## Progression to Quadratic

Consider the following interesting list of integers:

*P*

_{0}= 41,

*P*

_{1}= 43 = 41 + 2,

*P*

_{2}= 47 = 43 + 4,

*P*

_{3}= 53 = 47 + 6, … ,

in which *P*_{n} is obtained by adding 2*n* to *P*_{n-1}.

It so happens that there is a quadratic function *F*(*x*) with the property that *P*_{n} = *F*(*n*)*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*.

## Inequality with Radicals

Find all real solutions to the inequality √12 – *x* – √1 + *x* < 1.

## Inequalities, Several

Which of the following inequalities are satisfied for all real numbers *a*, *b*, *c*, *x*, *y*, *z* that satisfy the conditions *x* < *a*, *y* < *b*, and *z* < *c*?

*xy*+*yz*+*zx*<*ab*+*bc*+*ca**x*^{2}+*y*^{2}+*z*^{2}<*a*+*b*+*c**xyz*<*abc*

- None is satisfied
- I only
- II only
- III only
- All are satisfied

## 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?

*x*<*z*/2*y*<*x*+*z*/2*y*<*z*/2

## Plywood, Circular Holes

A circular piece of plywood has a radius of *x* inches. Four holes, each having a radius of *y* inches, are removed from the piece. Write a formula for the area of the remaining wood. (These holes don’t overlap each other at all.) Evaluate your formula if *x* = 10 inches and *y* = 2 inches. (Use ≈ 3.14).