⏱️ Three-Pass Method
🎯 Time Allocation
- First Pass (Problems 1-15): 45 minutes
- Second Pass (Problems 16-25): 25 minutes
- Third Pass (Review & Guessing): 5 minutes
🧠 Practice Problems
Problem 1
Tags: First Pass · source: AMC10 2020 #3
What is the value of $2 + 3 \times 4$?
A) $14$
B) $15$
C) $16$
D) $17$
E) $18$
Strategic Analysis
First Pass: Problem #3, straightforward. Order of operations: $2 + 3 \times 4 = 2 + 12 = 14$.
Time Target: 30 seconds
Answer & Solution
Answer: A
Using order of operations: $2 + 3 \times 4 = 2 + 12 = 14$.
Problem 2
Tags: First Pass · source: AMC10 2020 #8
What is the value of $1 + 2 + 3 + \cdots + 10$?
A) $45$
B) $50$
C) $55$
D) $60$
E) $65$
Strategic Analysis
First Pass: Problem #8, manageable. Formula: $\frac{n(n+1)}{2} = \frac{10 \cdot 11}{2} = 55$.
Time Target: 45 seconds
Answer & Solution
Answer: C
Using sum formula: $\frac{n(n+1)}{2} = \frac{10 \cdot 11}{2} = 55$.
Problem 3
Tags: First Pass · source: AMC10 2020 #12
A rectangle has length 8 and width 6. What is its area?
A) $14$
B) $24$
C) $28$
D) $48$
E) $64$
Strategic Analysis
First Pass: Problem #12, straightforward. Area = length × width = 8 × 6 = 48.
Time Target: 30 seconds
Answer & Solution
Answer: D
Area = length × width = 8 × 6 = 48 square units.
Problem 4
Tags: First Pass · source: AMC10 2020 #15
What is the value of $\sqrt{144}$?
A) $10$
B) $11$
C) $12$
D) $13$
E) $14$
Strategic Analysis
First Pass: Problem #15, straightforward. $\sqrt{144} = 12$ since $12^2 = 144$.
Time Target: 30 seconds
Answer & Solution
Answer: C
Since $12^2 = 144$, we have $\sqrt{144} = 12$.
Problem 5
Tags: Second Pass · source: AMC10 2020 #18
A function $f$ is defined by $f(x) = x^2 + 2x + 1$. What is $f(3)$?
A) $16$
B) $17$
C) $18$
D) $19$
E) $20$
Strategic Analysis
Second Pass: Problem #18, strategic range. $f(3) = 3^2 + 2(3) + 1 = 9 + 6 + 1 = 16$.
Time Target: 1-2 minutes
Answer & Solution
Answer: A
Substituting $x = 3$: $f(3) = 3^2 + 2(3) + 1 = 9 + 6 + 1 = 16$.
Problem 6
Tags: Second Pass · source: AMC10 2020 #20
You’re stuck on a problem with 5 answer choices. You can eliminate 2 choices. Should you guess or skip?
A) Guess
B) Skip
C) Need more information
D) Both have same expected value
E) Cannot be determined
Strategic Analysis
Second Pass: Problem #20, strategic decision. 3 choices remaining: $\text{EV}_\text{guess} = \frac{6}{3} = 2$ vs. $\text{EV}_\text{blank} = 1.5$. Since $2 > 1.5$, guess.
Time Target: 1 minute
Answer & Solution
Answer: A
With 3 remaining choices: $\text{EV}_\text{guess} = \frac{6}{3} = 2$ vs. $\text{EV}_\text{blank} = 1.5$. Since $2 > 1.5$, guess.
Problem 7
Tags: Second Pass · source: AMC10 2020 #22
You have 10 minutes left and are on problem 22. What should you do?
A) Try to solve problem 22 completely
B) Try to eliminate some choices, then guess
C) Skip problem 22 and move on
D) Go back to check previous answers
E) Need more information
Strategic Analysis
Second Pass: Problem #22, time management. With 10 minutes left, try to eliminate some choices first, then guess. This maximizes expected value.
Time Target: 2 minutes
Answer & Solution
Answer: B
With 10 minutes left and on problem 22, try to eliminate some choices first, then guess. This maximizes expected value while managing time.
Problem 8
Tags: Second Pass · source: AMC10 2020 #24
You’re stuck on a problem with 5 answer choices. You can eliminate 1 choice. Should you guess or skip?
A) Guess
B) Skip
C) Need more information
D) Both have same expected value
E) Cannot be determined
Strategic Analysis
Second Pass: Problem #24, strategic decision. 4 choices remaining: $\text{EV}_\text{guess} = \frac{6}{4} = 1.5$ vs. $\text{EV}_\text{blank} = 1.5$. Both have same expected value.
Time Target: 1 minute
Answer & Solution
Answer: D
With 4 remaining choices: $\text{EV}_\text{guess} = \frac{6}{4} = 1.5$ vs. $\text{EV}_\text{blank} = 1.5$. Both have same expected value.
Problem 9
Tags: Third Pass · source: AMC10 2020 #25
You have 5 minutes left and 2 problems remaining. You can eliminate 1 choice on each. What should you do?
A) Try to solve both problems
B) Guess on both problems
C) Try to solve 1, guess on 1
D) Skip both problems
E) Need more information
Strategic Analysis
Third Pass: Problem #25, final decision. With 5 minutes left and 2 problems, guess on both. With 1 elimination each: $\text{EV}_\text{guess} = \frac{6}{4} = 1.5$ points per problem, which equals the blank value.
Time Target: 2 minutes
Answer & Solution
Answer: B
With 5 minutes left and 2 problems, guess on both. With 1 elimination each, guessing gives the same expected value as skipping, but guessing gives you a chance to get points.
Problem 10
Tags: Third Pass · source: AMC10 2020 #25
You have 1 minute left and are on problem 25. What should you do?
A) Try to solve problem 25 completely
B) Try to eliminate some choices, then guess
C) Skip problem 25 and move on
D) Go back to check previous answers
E) Need more information
Strategic Analysis
Third Pass: Problem #25, final decision. With 1 minute left, try to eliminate some choices first, then guess. This maximizes expected value while managing time.
Time Target: 1 minute
Answer & Solution
Answer: B
With 1 minute left and on problem 25, try to eliminate some choices first, then guess. This maximizes expected value while managing time.
📊 Time Checkpoints
| Time | Should Be On | First Pass | Second Pass | Third Pass |
|---|---|---|---|---|
| 15 min | Problem 5-6 | ✓ | ||
| 30 min | Problem 10-12 | ✓ | ||
| 45 min | Problem 15-17 | ✓ | ✓ | |
| 60 min | Problem 20-22 | ✓ | ||
| 75 min | Contest End | ✓ |
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