Pengi Editor's Note
This article presents four worked problems from the 2020 AMC 10A alongside the complete answer key and topic distribution breakdown. The Pengi editorial team selected this resource for students who want hands-on practice with real 2020 AMC 10A problems and a clear understanding of which topics to prioritize in preparation.
Source: Think Academy Blog
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}
]
(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 |
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.
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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