2022 AMC 10B Real Questions and Analysis

In this article, you’ll find:

• Representative real questions from each module with detailed solutions.
• The complete 2022 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 2022 AMC 10B.
• A module-to-question mapping table highlighting the core concepts tested in each module for the 2022 AMC 10B.

Real Question and Solutions Explained

Algebra Example – Problem 1

Question:

Define \(x\diamond y=|x-y|\) for all real numbers \(x,y\). What is the value of \((1\diamond(2\diamond3))-((1\diamond2)\diamond3)\)?

(A) -2 (B) -1 (C) 0 (D) 1 (E) 2

Solution:

Compute \(2\diamond3=|2-3|=1.\)
Then \(1\diamond(2\diamond3)=1\diamond1=|1-1|=0.\)
Compute \(1\diamond2=|1-2|=1.\)
Then \((1\diamond2)\diamond3=1\diamond3=|1-3|=2.\)
Finally, \((1\diamond(2\diamond3))-((1\diamond2)\diamond3)=0-2=-2.\)

Answer:(A)

Common Mistakes:

• Assuming \(\diamond\) is associative (e.g., \(1\diamond(2\diamond3)=(1\diamond2)\diamond3\)).
• Dropping absolute values or signs.

Number Theory Example – Problem 8

Question:

Consider the 100 sets \(\{1,2,\ldots,10\},\{11,12,\ldots,20\},\ldots,\{991,992,\ldots,1000\}\). How many of these sets contain exactly two multiples of 7?

(A) 40 (B) 42 (C) 43 (D) 49 (E) 50

Solution:

Each set is \([10k+1,10k+10]\).
The number of multiples of 7 in each block repeats every 7 sets (periodicity 70).
Pattern across one cycle of 7 sets: \(1,1,2,1,2,1,2\) — three of them have two multiples.
There are 100 sets = 14 complete cycles (98 sets) + 2 remaining sets (both with 1 multiple).
Total = \(14\times3=42.\)

Answer:(B)

Common Mistakes:

• Ignoring the periodic repetition by 70.
• Counting endpoints incorrectly when both are included.

Geometry Example – Problem 2

Question:

In rhombus ????????????????, point ???? lies on segment \(\overline{AD}\) so that \(\overline{BP}\perp\overline{AD}\), \(AP=3\), and \(PD=2\). What is the area of ?????????????????

(A) 3.5 (B) 10 (C) 6.5 (D) 20 (E) 25

Solution:

Place \(A(0,0)\) and \(D(5,0)\Rightarrow AD=5.\)
Point \(P\) is at \(x=3\); let \(B=(3,h)\) since \(BP\perp AD.\)
In a rhombus, \(|AB|=|AD|\Rightarrow\sqrt{(3-0)^2+h^2}=5\Rightarrow9+h^2=25\Rightarrow h=4.\)
Height \(BP=4\); base \(AD=5.\)
Area \(=AD\times BP=5\times4=20.\)

Answer: (D)

Common Mistakes:

• Treating the rhombus as a rectangle or using diagonal formulas unnecessarily.
• Forgetting that area = base × perpendicular height.

Combinatorics Example – Problem 12

Question:

A pair of fair 6-sided dice is rolled \(n\) times. What is the least \(n\) such that the probability the sum equals 7 at least once is greater than \(\frac{1}{2}\)?

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

Solution:

For one roll, \(P(\text{sum}=7)=\frac{1}{6}.\)
Probability of no 7 in \(n\) rolls is \((\frac{5}{6})^n.\)
We need \(1-(\frac{5}{6})^n>\frac{1}{2}\iff(\frac{5}{6})^n<\frac{1}{2}.\) Step 4: Take logarithms: \(n>\frac{\ln(\frac{1}{2})}{\ln(\frac{5}{6})}\approx3.80.\)
Smallest integer \(n=4.\)

Answer:(C)

Common Mistakes:

• Using \((\frac{1}{6})^n>\frac{1}{2}\) instead of complement.
• Rounding \(3.80\) down to 3 instead of up to 4.

2022 AMC 10B Answer Key

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

Download AMC 10 Mock Exam & Practice Pack

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• Mock Exams: Timed practice with full explanations to build test readiness and review mistakes.
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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

2022 AMC 10B Topic Distribution

The 2022 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

Detailed Module Analysis

Module	Question Numbers	What It Tests (Brief)
Algebra (+ Arithmetic Reasoning)	1, 4, 5, 7, 9, 10, 13, 15, 18, 21, 24	Quadratic equations, linear and nonlinear equations, inequalities, rational expressions, and function manipulation
Number Theory	6, 8, 17, 25	Integer properties, prime factorization, divisibility, and modular arithmetic
Geometry	2, 16, 20, 22	Plane and coordinate geometry, emphasizing similarity, circle geometry, and quadrilateral properties (especially rhombi and parallelograms)
Counting & Probability / Combinatorics	3, 11, 12, 14, 19, 23	Counting principles, logical reasoning, and probability-based problem solving.

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