# Problem Solving (Misc.)

## Final Countdown

On Sunday, May 12th, 224 middle-school math students participated in the written rounds of the 2019 Raytheon MATHCOUNTS National Competition. On Monday, May 13th, the top 12 competitors will go head to head in the National Countdown Round to determine the 2019 MATHCOUNTS National Champion. Let’s solve a few problems from the 2018 National Countdown Round.

**2018 National Countdown #21**

In square units, what is the area of the triangle with vertices P(−2, 1), Q(3, 8) and R(9, 3)? Express your answer as a decimal to the nearest tenth.

## Follow the Rules

Define *x* @ *y* as (*x*^{3} − *y*)/*x*, for distinct positive integers* x* and *y*. What is the value of 5 @ 10?

*Evaluating 5 @ 10 yields (5 ^{3} − 10)/5 = (125 − 10)/5 = 115/5 = *

**23***.*

## The Pythagorean Triple Threat

The Pythagorean Triple Threat is a coed softball team. League rules require the batting order to alternate boy-girl-boy-girl… (or girl-boy-girl-boy…). If there are five girls and five boys on the team, how many different batting orders are possible?

## It's Moving Day

Jessie and Marcus are moving from their apartment to a new condominium. Marcus is responsible for moving all his belongings and the living room furniture, while Jessie is responsible for moving all the kitchen items, including the dining table and chairs, and his personal belongings.

## The Day That Goes on Forever

March 14 has been dubbed Pi Day. That’s because March 14 is often written 3/14, and a decimal approximation of pi is 3.14. But that’s only if you use decimals and our style of writing dates. In some countries, it’s customary to list the date first followed by the month; for example, March 14 would be 14-3 in some places. Consequently, some places celebrate Pi Approximation Day on July 22, because that date can be written as 22-7, and a fractional approximation of pi is 22/7. What percentage of the year occurs between Pi Day and Pi Approximation Day?

## Odds and Evens

The sum of any four consecutive odd integers is a multiple of eight. For example, 1 + 3 + 5 + 7 = 16 and 5 + 7 + 9 + 11 = 32. What is the product of the greatest and least of four consecutive odd integers whose sum is 64?

## Preparing for State

In just a few days, the 2019 MATHCOUNTS State Competitions will begin, so let’s look back at some of the 2018 State Sprint round problems and solve them in preparation.

## MATHCOUNTS Valentine

On some graph paper, 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?

## Martin Luther King, Jr. Day

Dr. Martin Luther King, Jr., a civil rights activist whose passion and leadership led to great cultural changes in America, was born on January 15, 1929. On November 2, 1983, President Ronald Reagan signed a bill to create a federal holiday honoring Dr. King. The national holiday celebrating the birth of Dr. King was first observed on January 20, 1986. Every year since, Martin Luther King, Jr. Day has been observed on the third Monday of January.

What is the earliest date in January that Martin Luther King, Jr. Day can occur?

## Shaping Up

Jayden started a new exercise program that includes running, cycling, yoga and aerobics. The chart shows how many calories Jayden burns by doing these exercises. Use this data to answer the questions that follow.