Halloween Theme

On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.

"Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes.

"Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes.

"Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes.

How many pieces of candy in all did the principal give to the children who always tell the truth?

• 7

• 12

• 21

• 27

Answer explanation

• Truth-tellers would answer yes-no-no to the three questions in this order.

• Liars would answer yes-yes-no to the three questions in this order.

• Alternaters who responded truth-lie-truth would answer no-no-no to the three questions in this order.

• Alternaters who responded lie-truth-lie would answer yes-yes-yes to the three questions in this order.

Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of 3:2:1, respectively. Due to some confusion they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed?

• 1/18

• 5/18

• 1/6

• 5/12

At the annual Miramonte Halloween Event, of the children play soccer, of the children swim, and of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?

• 30%

• 40%

• 49%

• 51%

Tags

CCSS.6.RP.A.3C

On an exhilarating Halloween night, the diligent honor student, Leo Ding, found himself in a challenging predicament. Although he was eager to join his friends for a night of trick-or-treating, the looming Calc AB test scheduled for the next day demanded his full attention. Resigned to hitting the books, he encountered a fascinating question:

In the world of mathematics, where the functions sin(x) and cos(x) are periodic with least period (smallest period) 2pi, what is the least period of the function cos(sin(x))?

• pi/2

• pi

• 2pi

• The function is not periodic

Answer explanation

Start by noting that cos(-x) = cos(x). Then realize that under the function cos(sin(x)), the negative sine values yield the same as their positive value, so take the absolute value of the sine function to get the new period. This has period pi

Wikipedia's "Halloween" article contains some number X occurrences of the word "Halloween". What is the sum of all factors for the mystery number X? (Go to the Wikipedia Halloween page to find X)

• 431

• 432

• 110

• 111