2020 AMC 10A Real Questions and Analysis

In this article, you’ll find:

• Representative real questions from each module with detailed solutions.
• The complete 2020 AMC 10A Answer Key.
• The best resources to prepare effectively for the AMC 10.
• A concise topic distribution chart showing which areas appeared most in the 2020 AMC 10A.
• A module-to-question mapping table highlighting the core concepts tested in each module for the 2020 AMC 10A.

Real Question and Solutions Explained

Algebra Example – Problem 3

Question:

Assuming \(a\neq3\), \(b\neq4\), and \(c\neq5\), what is the value in simplest form of
\[
\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}
\quad (2020\ \text{AMC 10A, Question #3})
\]

(A) \(-1\) (B) \(1\) (C) \(\frac{abc}{60}\) (D) \(\frac{1}{abc}-\frac{1}{60}\) (E) \(\frac{1}{60}-\frac{1}{abc}\)

Solution:

Pair terms to simplify sign changes:
\[
\frac{a-3}{3-a}=-1,\quad \frac{b-4}{4-b}=-1,\quad \frac{c-5}{5-c}=-1
\]
Thus,
\[
(-1)\times(-1)\times(-1)=-1
\]

Answer (A)

Common Mistakes:

• Canceling mismatched terms like \(a-3\) with \(5-c\).
• Missing the negative sign when flipping \(x-y\) and \(y-x\).
• Assuming remaining factors of \(a,b,c\) do not cancel.

Number Theory Example – Problem 6

Question:

How many 4-digit positive integers (from 1000 to 9999) having only even digits are divisible by 5?

(A) 80 (B) 100 (C) 123 (D) 200 (E) 500

Solution:

Divisibility by 5 with even digits means the last digit must be 0.

• Thousands digit: \({2,4,6,8}\) → 4 choices
• Hundreds and tens digits: each \({0,2,4,6,8}\) → \(5\times5\) choices
• Total numbers: \(4\times5\times5=100\)

Answer (B)

Common Mistakes:

• Using 5 choices for the first digit (cannot be 0).
• Allowing the last digit 5 (not even).
• Forgetting each middle digit is independent.

Geometry Example – Problem 10

Question:

Seven cubes, whose volumes are \(1,8,27,64,125,216,\text{ and }343\) cubic units, are stacked vertically with volumes decreasing from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?

(A) 644 (B) 658 (C) 664 (D) 720 (E) 749

Solution:

Side lengths: \(1,2,3,4,5,6,7\).
Total surface area without overlaps:
\[
6\sum_{n=1}^{7}n^2=6(140)=840
\]
Subtract overlapped faces between stacked cubes (each hidden pair removes \(2n^2\) for \(n=1,2,\dots,6\)):
\[
2(1^2+2^2+3^2+4^2+5^2+6^2)=2(91)=182
\]
Net exposed area: \(840-182=658\).

Answer (B)

Common Mistakes:

• Subtracting the entire top face of the larger cube instead of only the covered portion.
• Forgetting to include the bottom face of the lowest cube.
• Using cube volumes instead of side areas.

Combinatorics Example – Problem 9

Question:

A single bench section at a school event can hold either 7 adults or 11 children. When \(N\) bench sections are connected end to end, an equal number of adults and children together will occupy all the bench space. What is the least possible positive integer value of \(N\)?

(A) 9 (B) 18 (C) 27 (D) 36 (E) 77

Solution:

Let \(k\) be the number of adults (and also \(k\) children).
Each adult occupies \(\frac{1}{7}\) of a bench, each child \(\frac{1}{11}\).
Total benches used:
\[
\frac{k}{7}+\frac{k}{11}=N=\frac{18k}{77}
\]
For \(N\) to be an integer, \(77\mid18k\). Since \(\gcd(18,77)=1\), the smallest \(k=77\), giving
\[
N=\frac{18\times77}{77}=18
\]

Answer (B)

Common Mistakes:

• Setting \(7k=11k\) or comparing capacities directly.
• Taking \(\text{lcm}(7,11)=77\) as the answer without converting to bench count.
• Forgetting adults and children counts are equal, not sections.

2020 AMC 10A Answer Key

Question	Answer
1	E
2	C
3	A
4	E
5	C
6	B
7	C
8	B
9	B
10	B
11	C
12	C
13	B
14	D
15	E
16	B
17	E
18	C
19	E
20	D
21	C
22	A
23	A
24	C
25	A

Free Download: 2020 AMC 10A Problems & Solution

Download AMC 10 Mock Exam & Practice Pack

Download AMC 10 mock exams with full solutions, key AR/HR knowledge reviews, and daily practice sets — perfect for focused and efficient prep.

• Mock Exams: Timed practice with full explanations to build test readiness and review mistakes.
• Knowledge Points: Covers must-know Algebra, Geometry, and Number Theory concepts for AMC 10.
• Daily Practice: Sharpen accuracy and speed with high-frequency AMC-style problems.

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Last 10 Years AMC 10 Real Questions and Analysis

Think Academy provides in-depth breakdowns of the past decade of AMC 10 exams. Click below to explore:

• Year-by-year topic trend insights and concept distributions

• Real AMC 10 exams from the last 10 years

• Official answer keys

• Representative questions, detailed solutions, and common mistakes

AMC 10A	AMC 10B
2024 AMC 10A	2024 AMC10B
2023 AMC 10A	2023 AMC10B
2022 AMC 10A	2022 AMC10B
2021 AMC 10A	2021 AMC10B
2020 AMC 10A	2020 AMC10B
2019 AMC 10A	2019 AMC10B
2018 AMC 10A	2018 AMC10B
2017 AMC 10A	2017 AMC10B
2016 AMC 10A	2016 AMC10B

2020 AMC 10A Topic Distribution

The 2020 AMC 10A featured 25 questions to be completed in 75 minutes, emphasizing advanced problem-solving and proof-based reasoning skills.

Learn more about AMC 10 Format and Scoring Here: AMC 10 FAQ and Resources: Your Ultimate Guide

Detailed Module Analysis

Module	Question Numbers	What It Tests (Brief)
Algebra	1, 3, 4, 5, 7, 8, 14, 17, 21	Linear equations, ratios, functions
Number Theory	6, 9, 22, 24	Divisibility, GCD/LCM
Geometry	10, 12, 19, 20, 23	Triangles, solids
Combinatorics / Probability	2, 11, 13, 15, 16, 18, 25	Counting and probability

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