# 2021 MathCounts National Target Round #7 (video solution)

This one uses a rarely used formula for the area of a triangle.

What is the largest possible perimeter of a triangle whose sides have integer side lengths and that can fit inside a circle of radius 20 cm?

# Math Contests and the Spacing Effect

The Spacing Effect is the idea that students are better able to recall information if they study and restudy the same ideas over multiple sessions in time. This is the opposite of cramming for a test, where that knowledge quickly degrades and becomes unavailable to our brains. Retrieving the same information trains our brains to assign greater importance to those facts.

Math generally lends itself well to the Spacing Effect since topics build on each other. Algebra is used to solve geometry problems, and geometry can be used to solve probability problems. If you need to find the area of square to solve a probability problem, then boom, you are recalling facts you learned last year. And now your understanding is deeper and you are less likely to forget.

Math contests are a great way to leverage the Spacing Effect. Typically students learn a math algorithm or property or theorem, complete a bunch of homework problems, take a test and move on to the next topic. Math contests include problems from all different subject areas: algebra, probability, geometry, number theory, etc. In order to answer a question, you may need to retrieve information you studied even 1-2 years ago! Take enough practice tests and you are using that information repeatedly in new situations.

I like to say that students who participate in math contests aren’t allowed to forget the math they’ve learned. They are recycling their knowledge and using it in novel problems.

These are not students who need to prepare for the math section of the SAT because they’ve been preparing for it their whole academic lives. Without putting a label on it, they’ve been using the Spacing Effect to their advantage.

# Learning from AMC Statistics

I never waste an opportunity to teach using data, and a perfect example is this histogram of scores from the latest AMC 10. Notice the spikes at regular intervals among the lower scores to the left. What is going on here?

These students do not understand how the AMC is scored, in particular that you receive 1.5 points for each answer left blank. While you receive 6 points for each correct answer, you receive no points for each wrong answer.

Take a look at the spike occurring at 30 points. You can earn 30 points by answering 5 questions correctly and the rest are wrong, with no questions left unanswered. A student can choose the correct answer to 5 problems on a multiple choice exam with 25 questions and 5 answer choices, by selecting answers at random.

As an exercise, try to generate scores 24, 36, and 42 as a naive student who does not leave any question blank. There is no excuse to not know this scoring rubric, as it is explained on the front of each exam booklet:

I coach all my AMC students to write a “clean exam.” That is one where all answered questions are correct and no questions are answered incorrectly. All other questions are left unanswered to achieve the highest possible score. (Notice the higher scoring students to the right are savvy to this strategy.)

# How to use Asymptote at AoPS (video)

I present some hacks for newbies to get up to speed fast creating math diagrams with Asymptote.

# Fall 2021 AMC 10 B #24 video solution

Here I introduce 2 time-saving tips and simplified notation for tracking the sides of a cube after rotation.

A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

# Fall 2021 AMC 10 B #20 video solution

In a particular game, each of $4$ players rolls a standard $6$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo’s first roll was a $5$, given that he won the game?

# Fall 2021 AMC 10 A #20 video solution

I like how this problem nicely combines properties of quadratic equations with solving inequalities.

How many ordered pairs of positive integers $(b,c)$ exist where both $x^2+bx+c=0$ and $x^2+cx+b=0$ do not have distinct, real solutions?

# 5 Steps to an AIME Qualification

I counsel my students to earn a “respectable” score on the AMC, and that means within about 20 points of qualifying for the AIME. For the AMC 10 that is at least a score of 80 and for the AMC 12, about 60 points, preferably more.

However, there is an important binary for the AMC: qualifying for AIME. You either did or you didn’t, and being an AIME qualifier is a great feather in your cap when it comes to college applications to technical schools. If you are within 20 points of qualifying, then that accomplishment is within reach.

With limited time, focus on the following for the greatest ROI on improving your AMC scores and earning an AIME qualification:

• Know your cutoffs. For the AMC 10, that’s 100 – 120 points (15-18 correct answers, and the rest blank) depending on the year. For the AMC 12, that’s 80 – 100 points (11-13 correct, and the rest blank.)
• Know how to calculate your score. It’s 6 points for each correct answer, and 1.5 points for each answer left blank. Zero points for each wrong answer. Don’t answer a question unless you are very certain you are correct.
• Take old exams. Score them. How close are you to qualifying for that year? Are you making silly errors and losing 1.5 points each time?
• Strive to write a “clean exam.” A clean exam means every single problem is either answered correctly or left blank. You’ve checked your work, there are no silly errors, and you’ve earned your qualifying score.
• Focus your effort on the first 10-20 problems. Problems generally increase in difficulty. Don’t waste too much time on the so-called Final Five, the most difficult problems on the exam. Do take a quick read to see if any are doable, but otherwise focus on getting all the earlier, easier problems correct.

# 2013 MathCounts Chapter Target #3

This problem is a good example of finding a bijection or a 1:1 correspondence between a set that is difficult to count and another that is easier to count, in this case using binary numbers.

A circular spinner has seven sections of equal size, each of which is colored either red or blue. Two colorings are considered the same if one can be rotated to yield the other. In how many ways can the spinner by colored?