The 40 Most Important AMC Problems: Boost Your AMC Score with GLeaM!

Hello everybody! I've been receiving a lot of requests to write more competition-related articles, especially because the AMC is approaching pretty quickly: the exams will be hosted on February 4 and February 10, 2021. Today, I thought I'd take the opportunity to curate a set of my 40 favorite problems to serve as a resource for your AMC prep, grouped by category. I aimed this compilation at the intersection of AMC 10 and AMC 12, so they should cover both exams effectively. I carefully chose these to cover the widest range of topics possible, and they'll serve as a roadmap to figuring out what content you understand and what you might want to learn or review before the AMC. This is coming from what I have found personally successful in prepping both myself and my math team for the exams, so I hope you find it helpful!


There will be an answer key below to check all your answers! You'll have to search each problem individually on Art of Problem Solving's database for more expansive solutions; it's simply too much content to fit in this article. You are always welcome to discuss individual problems and solutions through the forum, the comment section, emailing me... or any other way you choose to reach me or our community!


Here are the links to AoPS's database:

AMC 10: https://artofproblemsolving.com/wiki/index.php/AMC_10_Problems_and_Solutions

AMC 12: https://artofproblemsolving.com/wiki/index.php/AMC_12_Problems_and_Solutions


Also, because Wix, which hosts this website, still does not allow LaTeX or other mathematical typesetting, the formatting below may be less than ideal, so I took the time to create a second version.


I created this better-formatted version here, which can also be downloaded and printed as a PDF: http://bit.ly/gleamamcproblems

This took a lot of effort, so make sure you check it out!


COMBINATORICS

This section includes Casework, Complimentary Counting, Venn Diagrams, Stars and Bars, Properties of Combinations and Permutations, Factorials, Path Counting, and Probability.


In order to not prematurely tip you off as to how to solve a problem, I won't reveal the topic for each problem, but for a challenge, see if you can match the topics to the problems for this combinatorics section and the other three sections! Feel free to email me to discuss this. (These lists are helpful to show you what you need to study for the AMC as well!)


1. 2002 AMC 10B Problem 18; 12B Problem 14: Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?

A) 8 B) 9 C) 10 D) 12 E) 16


2. 2002 AMC 12B Problem 10: How many different integers can be expressed as the sum of three distinct members of the set {1,4,7,10,13,16,19}?

A) 13 B) 16 C) 24 D) 30 E) 35


3. 2019 AMC 8 Problem 25: Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?

A) 105 B) 114 C) 190 D) 210 E) 380

Note: Though this problem is from the AMC 8, it resembles the level of a mid-AMC 10 problem.


4. 2020 AMC 10B Problem 5: How many distinguishable arrangements are there of 1 brown tile, 2 green tiles, and 3 yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.)

A) 210 B) 420 C) 630 D) 840 E) 1050


5. 2006 AMC 10A Problem 21: How many four-digit positive integers have at least one digit that is a 2 or a 3?

A) 2439 B) 4096 C) 4903 D) 4904 E) 5416


6. 2017 AMC 10B Problem 13: There are 20 students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are 10 students taking yoga, 13 taking bridge, and 9 taking painting. There are 9 students taking at least two classes. How many students are taking all three classes?

A) 1 B) 2 C) 3 D) 4 E) 5


7. 2004 AMC 10A Problem 10: Coin A is flipped three times and coin B is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same?

A) 29/128 B) 23/128 C) 1/4 D) 35/128 E) 1/2


8. 2004 AMC 10A Problem 16: The 5x5 grid shown contains a collection of squares with sizes from 1x1 to 5x5. How many of these squares contain the black center square?

A) 12 B) 15 C) 17 D) 19 E) 20


9. 2010 AMC 12A Problem 18: A 16-step path is to go from (-4, -4) to (4,4) with each step increasing either the x-coordinate or the y-coordinate by 1. How many such paths stay outside or on the boundary of the square -2 <= x <= 2, -2 <= y <= 2 at each step?

A) 92 B) 144 C) 1568 D) 1698 E) 12800

Note: <= is less than or equal to.


10. 2016 AMC 10A Problem 20: For some particular value of N, when (a+b+c+d+1)^N is expanded and like terms are combined, the resulting expression contains exactly 1001 terms that include all four variables a, b, c, and d, each to some positive power. What is N?

A) 9 B) 14 C) 16 D) 17 E) 19

Note: This is where algebra and combinatorics come together!


ALGEBRA

This section includes Sequences & Series, Distance = Rate*Time Problems, Numerical Reasoning, Median/Mean/Mode, Functional Equations, Polynomials, Logarithms, and Trigonometry.


1. 2010 AMC 12A Problem 5: Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot, a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next n shots are bullseyes she will be guaranteed victory. What is the minimum value for n?

A) 38 B) 40 C) 42 D) 44 E) 46


2. 2017 AMC 10B Problem 7; 12B Problem 4: Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all, it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?

A) 2.0 B) 2.2 C) 2.8 D) 3.4 E) 4.4


3. 2014 AMC 10A Problem 10; 12A Problem 9: Five positive consecutive integers starting with a have average b. What is the average of 5 consecutive integers that start with b?

A) a+3 B) a+4 C) a+5 D) a+6 E) a+7


4. 2018 AMC 10B Problem 20; 12B Problem 18: A function f is defined recursively by f(1) = f(2) = 1 and f(n) = f(n-1) - f(n-2) + n for all integers n > 2. What is f(2018)?

A) 2016 B) 2017 C) 2018 D) 2019 E) 2020


5. 2006 AMC 10A Problem 19: How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression? A) 0 B) 1 C) 59 D) 89 E) 178

Note: It's algebra in disguise!


6. 2013 AMC 12A Problem 14: The sequence log_12(162), log_12(x), log_12(y), log_12(z), log_12(1250) is an arithmetic progression. What is x?

A) 125 sqrt(3) B) 270 C) 162 sqrt(5) D) 434 E) 225 sqrt(6)

Note: "sqrt" represents square root and log_a(b) represents a logarithm with base a and argument b.