Wednesday, July 17, 2013

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!

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

j = ____

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

n = ____

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 = ____

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