# Number Theory

## 2020 Chapter Competition Review

*During the month of February, the 2020 Chapter Competitions took place across the country. Let’s look back at a few of the problems our competitors encountered. *

## State Competitions

*State Competitions are coming up fast! Are you ready to compete? Let’s try a few 2019 State Competition problems to get ready.*

__2019 State Sprint Round, #18__

If C is a digit such that the product of the three-digit numbers 2C8 and 3C1 is the five-digit number 90C58, what is the value of C?

## Let the Competitions Begin!

Chapter Competitions officially started this past weekend! Are you ready to compete? Let’s try a few 2019 Chapter Competition problems to get warmed up.

__2019 Chapter Sprint Round, #20__

Jones is chasing a car 800 meters ahead of him. He is on a horse moving at 50 km/h. If Jones catches up to the car in 4 minutes, how fast was the car moving?

*Jones catches up 800 m in 4 min and thus, at a rate of:*

## Camsie's New Job

Camsie is starting a new job where she earns *n* dollars on her* n*th day of work. What is the total amount that Camsie will earn for the first ten days of work at her new job?

## School's Out for Summer

At Walter Whitman Middle School, to celebrate the last day of school, Mrs. Hanks brought in treats for her class. She has chocolate bars, lollipops and gumballs. Mrs. Hanks tells each student they can choose any combination of three treats from what she has. How many different combinations of three treats can one of her students chose?

## Memorial Day

The distinct letters from the word MEMORIAL are used to create set Z, such that Z = {M, E, O, R, I, A, L}. The letters from the word DAY are used to create set Y, such that Y = {D, A, Y}. If set X is the intersection of sets Z and Y, what are the letters in set X?

*The intersection of two sets includes all the elements (or members) that are in both sets. So set X = ***{A}***.*

## 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?