Pengi Editor's Note
The Pengi editorial team curated this Think Academy 2023 AMC 10B breakdown. With representative problems, detailed solutions, and common mistake analysis, it provides the substance students need to learn from one of the most recent AMC 10 exams.
Source: Think Academy Blog
2023 AMC 10B Real Questions and Analysis
In this article, you’ll find:
- Representative real questions from each module with detailed solutions.
- The complete 2023 AMC 10B Answer Key.
- The best resources to prepare effectively for the AMC 10.
- A concise topic distribution chart showing which areas appeared most in the 2023 AMC 10B.
- A module-to-question mapping table highlighting the core concepts tested in each module for the 2023 AMC 10B.
Real Question and Solutions Explained
Algebra Example – Problem 6
Question:
Let \( L_1 = 1,\ L_2 = 3 \), and \( L_{n+2} = L_{n+1} + L_n \) for \( n \ge 1 \). How many terms in the sequence \( L_1, L_2, \ldots, L_{2023} \) are even?
(A) 673 (B) 1011 (C) 675 (D) 1010 (E) 674
Solution:
Work \( \bmod 2 \).
With \( L_1 = 1 \) (odd) and \( L_2 = 3 \) (odd), we get
\( L_3 \equiv 1 + 1 \equiv 0 \),
\( L_4 \equiv 0 + 1 \equiv 1 \),
\( L_5 \equiv 1 + 0 \equiv 1 \), …
Thus, the parity pattern repeats every 3 terms as O, O, E.
Among 2023 terms, \( \lfloor 2023 / 3 \rfloor = 674 \) complete blocks contribute 674 even terms,
and the extra one term (position 2023) is odd.
Answer:(E)
Common Mistakes:
- Assuming Fibonacci parity (starting O, E, …) instead of checking from given seeds.
- Forgetting to use modular arithmetic and attempting to compute large terms.
- Off-by-one error when handling the remainder after full 3-term cycles.
Number Theory Example – Problem 8
Question:
What is the units digit of \( 2022^{2023} + 2023^{2022} \)?
(A) 7 (B) 1 (C) 9 (D) 5 (E) 3
Solution:
Units digits depend only on the base’s units digit.
\( 2022^{2023} \) has the same units digit as \( 2^{2023} \).
The cycle for 2 is 2, 4, 8, 6 (length 4).
Since \( 2023 \equiv 3 \pmod{4} \Rightarrow 2^{2023} \) ends with 8.
\( 2023^{2022} \) behaves like \( 3^{2022} \).
The cycle for 3 is 3, 9, 7, 1 (length 4).
Since \( 2022 \equiv 2 \pmod{4} \Rightarrow 3^{2022} \) ends with 9.
Sum of units digits: \( 8 + 9 = 17 \Rightarrow 7 \).
Answer: (A)
Common Mistakes:
- Using Euler’s \( \varphi(10) = 4 \) without checking the remainder.
- Mixing up the cycle lengths for 2 and 3.
- Adding full numbers instead of focusing only on the units digits.
Geometry Example – Problem 3
Question:
A 3–4–5 right triangle is inscribed in circle A, and a 5–12–13 right triangle is inscribed in circle B.
What is the ratio of the area of circle A to the area of circle B?
(A) \( \frac{9}{25} \) (B) \( \frac{1}{9} \) (C) \( \frac{1}{5} \) (D) \( \frac{25}{169} \) (E) \( \frac{4}{25} \)
Solution:
For a right triangle, the circumradius \( R \) is half the hypotenuse.
Thus, \( R_A = \frac{5}{2}, \; R_B = \frac{13}{2}. \)
\[
\text{Area ratio} = \frac{\pi R_A^2}{\pi R_B^2} = \frac{(5/2)^2}{(13/2)^2} = \frac{25}{169}.
\]
Answer:(D)
Common Mistakes:
- Using the inradius formula instead of the circumradius relationship.
- Cancelling \( \pi \) incorrectly or squaring radii wrongly.
- Comparing perimeters instead of circle areas.
Combinatorics Example – Problem 11
Question:
Suzanne went to the bank and withdrew $800. The teller gave her this amount using $20 bills, $50 bills, and $100 bills, with at least one of each denomination. How many different collections of bills could Suzanne have received?
(A) 45 (B) 21 (C) 36 (D) 28 (E) 32
Solution:
Let counts be \( x, y, z \ge 1 \) for the numbers of \$20, \$50, and \$100 bills.
\[
20x + 50y + 100z = 800 \Rightarrow 2x + 5y + 10z = 80.
\]
Set \( x’ = x – 1, \; y’ = y – 1, \; z’ = z – 1 \ge 0 \):
\[
2x’ + 5y’ + 10z’ = 63.
\]
Since \( 2x’ \) is even, \( 5y’ + 10z’ \) must be odd, so \( y’ \) is odd.
Let \( y’ = 2k + 1 \; (k \ge 0) \). Then
\[
2x’ + 10(k + z’) = 58 \Rightarrow x’ + 5t = 29,
\]
where \( t = k + z’ \ge 0 \).
For \( t = 0, 1, 2, 3, 4, 5 \), we have \( x’ = 29 – 5t \ge 0 \).
For each fixed \( t \), \( k \) can be any integer from 0 to \( t \)
(making \( y’ = 2k + 1 \) odd and \( z’ = t – k \ge 0 \)).
\[
\text{Total solutions} = \sum_{t=0}^{5} (t + 1) = 21.
\]
Answer:(B)
Common Mistakes:
- Forgetting the “at least one of each” condition (not shifting to \( x’, y’, z’ \)).
- Ignoring the parity requirement that \( y’ \) must be odd.
- Treating denominations as indistinct when counting ordered triples.
2023 AMC 10B Answer Key
| Question | Number |
|---|---|
| 1 | C |
| 2 | B |
| 3 | D |
| 4 | C |
| 5 | A |
| 6 | E |
| 7 | B |
| 8 | A |
| 9 | B |
| 10 | C |
| 11 | B |
| 12 | C |
| 13 | B |
| 14 | C |
| 15 | C |
| 16 | E |
| 17 | D |
| 18 | E |
| 19 | B |
| 20 | A |
| 21 | E |
| 22 | B |
| 23 | B |
| 24 | E |
| 25 | B |
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 |
2023 AMC 10B Topic Distribution
The 2023 AMC 10B 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
Learn more about 2023 AMC 10B Strategies Here: 2023 AMC 10B Exam Analysis: Difficulty, Structure, and Strategies
Detailed Module Analysis
| Module | Question Numbers | What It Tests (Brief) |
|---|---|---|
| Algebra (+ Arithmetic Reasoning) | 1, 2, 5, 6, 9, 12, 13, 14, 22, 23 | Averages and percentages; linear and quadratic equations; number sequences and recurrences; absolute value and inequalities; functional and polynomial reasoning. |
| Number Theory | 8, 15, 18 | Modular arithmetic (remainders mod 10); perfect squares and integer conditions; greatest common divisor (gcd) and divisibility logic. |
| Geometry | 3, 4, 7, 17, 20, 24, 25 | Right triangles and circumradius relationships; unit, length, and area conversions; rotations and similarity; 3-D geometry and parametric shapes; polygon area reasoning. |
| Counting & Probability / Combinatorics | 10, 11, 16, 19, 21 | Grid coverings and strategic counting; integer partitions (money or value problems); digit-pattern constraints (rising/falling digits); geometric and classical probability. |
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