Problem of the Week Day 3: Week of 7/15/13 - 7/19/13

Today is day three of the problem of the week. Before I begin the problems, I would like to explain one thing, which will be used throughout the medium and hard problems. You might not have learned it in school, but the concept is very simple.

A floor function is the largest integer less than or equal to a given number. For example, the floor function of 6.7 is 6. The floor function of 4.973 is 4. The floor function of π is 3. Basically, if you round the number down to the nearest whole, you will have its floor function.

Similarly, a ceiling function is the smallest integer greater than or equal to a given number. So, the ceiling function of 6.7 is 7. The ceiling function of 4.973 is 5. The ceiling function of π is 4. While rounding down yields the floor function, rounding up gives the ceiling function.

Normally, this is denoted by a special sort of bracket. Since I don’t know how to type these brackets into the computer, I will use the following notation:

floor(x) = the floor function of x

ceiling(x) = the ceiling function of x

This is an easy way to eliminate fractions and decimals from numbers to make the problems slightly easier and more realistic.

Good luck!

Easy:

What is the Least Common Multiple of f, g, and h? Use the letter j to denote the answer.

j = ____

Medium:

What is the number of dots in a regular (f₄ - f₂)-gon array whose sides are of length (f₁ - f₃)?

n = ____

Hard:

Find the explicit formula for the following sequence:

v, t, s, floor((g + f)/10), (g - 60)/2, d - 2, ...

The formula should be of the form ax² + bx + c. So, write your answer in terms of the value of coefficients a, b, and c.

a = ____

b = ____

c = ____