# The Hardest Problems on the 2017 AMC 8 are Extremely Similar to Previous Problems on the AMC 8, 10, 12, Kangaroo, and MathCounts

Henry Wan, Ph.D.

The detailed article is in pdf format and can be viewed and downloaded HERE.

We developed a comprehensive, integrated, well-annotated database “CMP” consisting of various competitive math problems, including all previous problems on the AMC 8, 10, 12, AIME, MATHCOUNTSMath Kangaroo ContestMath Olympiads for Elementary and Middle Schools (MOEMS), ARML, HMMT, Math League, PUMaC, Stanford Math Tournament (SMT), Berkeley Math Tournament, the Carnegie Mellon Informatics and Mathematics Competition (CMIMC). The CPM is an invaluable “big data” system we use for our research and development, and is a golden resource for our students, who are the ultimate beneficiaries.

Based on artificial intelligence (AI), machine learning, and deep learning, we also devised a data mining and predictive analytics tool for math problem similarity searching. Using this powerful tool, we can align query math problems against those present in the target database “CPM,” and then detect those similar problems in the CMP database.

The AMC 8 is a 25-question, 40-minute, multiple choice examination in middle school mathematics designed to promote the development and enhancement of problem solving skills. The problems generally increase in difficulty as the exam progresses. Usually the last 5 problems are the hardest ones.

Among the final 5 problems on the 2017 AMC 8 contest, there is one algebra problem: Problem 21; there are 2 discrete math problems (which contains number theory and counting): Problems 23 and 24; and there are 2 geometry problems: Problems 22 and 25.

For those hardest problems on the 2017 AMC 8, based on the database searching, we found:

• 2017 AMC 8 Problem 21 is almost the same as 1977 AHSME Problem 8
• 2017 AMC 8 Problem 22 is exactly the same as Problem 15 on the International Kangaroo Mathematics Contest 2012 — Junior Level (Class 9 & 10) (click HERE to find this problem), and is very similar to the following 6 problems:
• 1950 AHSME Problem 35
• 1967 AHSME Problem 5
• 1970 AHSME Problem 27
• 2017 MathCounts State Sprint Problem 24
• 2015 MathCounts State Sprint Problem 16
• 2012 MathCounts State Sprint Problem 21
• In my AMC 8/MathCounts Prep Class, I ever used Problem 15 on the International Kangaroo Mathematics Contest 2012 — Junior Level (Class 9 & 10), as a typical example, to show the art of solving problems with semicircles inscribed in a right triangle. When my students attended the AMC 8 on Nov. 14, 2017, they already knew how to solve this problem and its answer. So they took one second to bubble the correct answer (D) and then got 1 point easily!
• 2017 AMC 8 Problem 24 is almost the same as the following 2 problems:
• 2005 AMC 12A Problem 18
• 2001 AMC 10 Problem 25/2001 AMC 12 Problem 12
• 2017 AMC 8 Problem 25 is very similar to the following 4 problems:
• 2012 AMC 10B Problem 16
• 2014 AMC 10A Problem 12
• 2012 AMC 8 Problem 24
• 1992 AJHSME Problem 24

We can see that Problem 23 is the only problem that is new and original. Every other problem has strong similarities to previous problems.

This year’s AMC 8 was more difficult than the last year’s AMC 8. Some hard problems were even AMC 10 level. For example, Problem 23 and Problem 24 on the 2017 AMC 8 are two typical AMC 10 hard problems.

Problem 23 is involved in detecting a sequence of four factors of 60 that forms an arithmetic progression with a common difference of 5.

Problem 24 is equivalent to finding the number of integers among the first 365 positive integers that are not divisible by 3, 4, or 5. We should use the principle of inclusion and exclusion (for 3 sets) to solve this problem.

Because the AMC 8 problems are getting harder, we must practice not only previous AMC 8 problems but also easy, medium, and even high difficulty level problems from previous AMC 10 to do well on the AMC 8.

The detailed article is in pdf format and can be viewed and downloaded HERE.

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