# Problem Solving (Misc.)

## A Look Back at Chapter

This week, the 2018 State Competitions will wrap up, and 224 state competitors will begin preparing to compete at the Raytheon MATHCOUNTS National Competition. Let’s take a look back at a few of the chapter problems they encountered on their way to state.

## Mathopolis Snowfall

The snowfall in Mathopolis totaled 68.1 inches in 2015 and 11.2 inches in 2017. If the average yearly snowfall for 2015, 2016 and 2017 was 35.6 inches, how many total inches of snow fell in Mathopolis in 2016? Express your answer as a decimal to the nearest tenth.

## A Mathy Valentine

On a coordinate grid, graph the following segments:

*y* = *x*, for 0 ≤ *x* ≤ 2*y* = 2*x* − 2, for 2 ≤ *x* ≤ 3*x* = 3, for 4 ≤ *y* ≤ 6*y* = −*x* + 9, for 2 ≤ *x* ≤ 3*y* = 7, for 1 ≤ *x* ≤ 2*y* = *x* + 6, for 0 ≤ *x* ≤ 1

Now reflect each of the segments over the *y*-axis. What popular shape have you drawn?

## Sequences and Patterns

This Saturday’s date, 02-10-18, forms an arithmetic sequence since **2** + 8 = **10** and 10 + 8 = **18**. In honor of this occurrence, try your hand at solving these problems involving sequences and patterns.

What is the tenth term of the sequence −50, −49, −47, −44, −40, …?

## Let the Competitions Begin!

Chapter competitions officially start this week! Are you ready to compete? Let’s try a few 2017 Chapter Competition problems to get warmed up.

If the sum of the positive integer *a* and 5 is less than 12, what is the sum of all possible values of *a*?

*[Sprint #16] *

## One for the Ages

Cara was born on January 1, 2010, and her mother, Sydney, was born on January 1, 1982. In what year will Sydney’s age be twice Cara’s age?

If we let Cara’s age be C, then Sydney’s age is S = C + 28, since Cara was born 28 years after Sydney. We are interested in determining when Sydney’s age is twice Cara’s age, in other words, when S = 2C. Substituting, we have C + 28 = 2C. Solving for C we get C = 28. Therefore, Sydney’s age will be twice Cara’s age in the year 2010 + 28 =.2038

## First-Class Postage

Civil Rights leader and Nobel Prize winner Martin Luther King, Jr. was born January 15, 1929. On January 13, 1979 a 15-cent stamp was issued by the United States Postal Service as part of the Black Heritage Series of commemorative stamps. In 1979 it cost 15 cents to mail a first-class letter weighing up to 1 ounce, and the current cost to mail a first-class letter weighing no more than 1 ounce is 49 cents. Suppose $24,965,250 worth of the Black Heritage King stamps have been issued by the USPS.

## Game Show Probability

Troy, Elliott and Shaun are three contestants on a game show. In one of the rounds of the game show the 3 contestants have a chance to spin a wheel up to two times in a row to try to get the highest possible sum without going over 100. The wheel that they spin is divided into 21 equal portions, each containing one element from the set of numbers containing 1 and all the multiples of 5 less than or equal to 100.

## 'Tis the Season

Each year Desmond and his family take part in the cultural celebration of Kwanzaa. The observance centers around the seven principles of Kwanzaa: Umoja, Kujichagulia, Ujima, Ujamaa, Nia, Kuumba and Imani. Suppose Desmond writes each of the five letters of the seventh principle, Imani (Swahili for Faith) on its own index card. If Desmond places the five cards in a bag and randomly chooses three cards, what is the probability that he chooses the three letters of the fifth principle, Nia (Swahili for Purpose)? Express your answer as a common fraction.

## O, Christmas Tree

Although, Franklin Pierce, the 14th U.S. President, originally brought the Christmas tree tradition to the White House, it was Calvin Coolidge, the 30th U.S. President, who, in 1923, started the National Christmas Tree Lighting Ceremony now held every year on the White House lawn. Last week, the president and first family continued that tradition by lighting a tree on the White House lawn. And speaking of trees, try to solve the following Christmas tree-themed problems.