I present some hacks for newbies to get up to speed fast creating math diagrams with Asymptote.
Here I introduce 2 time-saving tips and simplified notation for tracking the sides of a cube after rotation.
A cube is constructed from white unit cubes and blue unit cubes. How many different ways are there to construct the cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
In a particular game, each of players rolls a standard -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 , given that he won the game?
I like how this problem nicely combines properties of quadratic equations with solving inequalities.
How many ordered pairs of positive integers exist where both and do not have distinct, real solutions?
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.
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?
I have been enjoying the podcast Tests and the Rest, where hosts Amy Seeley and Mike Bergin go deep on testing, from the College Board to IQ to AP exams. They just dropped an episode where they interviewed me about math contests like AMC and MathCounts. Here’s a link: https://gettestbright.com/competitive-math-and-testing/
Define a sequence recursively by and for integers . Find the least value of such that the sum of the zeros of exceeds .
While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.