# Number Theory

## 2019 National Champion

Last week the national competition concluded, and Daniel Mai from Massachusetts earned the title of MATHCOUNTS National Champion. Let’s look at some of the problems he had to solve on the way to the top!

**Sprint #14**
Two opposite vertices of a certain square are located at (1, 6) and (−3, 1). If the line *y* = *mx* divides this square into two regions of equal area, what is the value of *m*? Express your answer as a common fraction.

## 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.

## National Competition

On Sunday, May 12th, 224 of the nation’s most talented middle-school math minds will be in Orlando, FL for the 2019 Raytheon MATHCOUNTS National Competition. The stakes are high, and the problems will be tough. Here are a few problems national competitors solved in 2018.

## Counting Down to Nationals

The countdown round is often the most anticipated and exciting part of competition for Mathletes^{®} and spectators alike. Here are some of our favorite countdown round problems from the 2019 competitions

__School #16__

On a certain farm, each chicken has two feet and each rabbit has four feet. If the combined number of chickens and rabbits on the farm is 100 and there are a total of 260 feet on these animals, how many chickens are there?

## Counting Down to Nationals

The countdown round is often the most anticipated and exciting part of competition for Mathletes^{®} and spectators alike. Here are some of our favorite countdown round problems from the 2019 competitions

** School #16**
On a certain farm, each chicken has two feet and each rabbit has four feet. If the combined number of chickens and rabbits on the farm is 100 and there are a total of 260 feet on these animals, how many chickens are there?

## 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 Road to Nationals

“Job well done!” to all the Mathletes^{®} who participated in the MATHCOUNTS Competition Series at the school, chapter and state levels. And congratulations to the 224 students who are headed to Orlando next month to compete in the 2019 Raytheon MATHCOUNTS National Competition! Let’s look at some of the challenging problems competitors faced on the road to Nationals!

__School Target #8__

## 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?

## Happy New Year!

Happy New Year! Now that 2019 is here, let’s have some fun solving this collection of problems that all involve the number 2019.

What is the greatest prime factor of 2019?

*The factors of 2019 are 1, 3, 673, 2019. The prime factors of 2019 are 3 and 673, the greatest being ***673***.*

## Best of 2018

As 2018 comes to an end, let’s go back and solve some of our favorite problems of year.

__School Handbook__

**Problem 237: **Kendra starts at a positive integer *k *and counts up by 4s until she hits exactly 200. Mason starts at a positive integer *m *and counts up by 6s until he hits exactly 200. If it takes Kendra half as many steps to reach 200 as it takes Mason, what is the greatest possible value of *k *− *m*?