馃殌 15-Minute Strategy Sprint
鈴憋笍 Time Allocation
- Problems 1-5: 3 minutes (easy problems, high accuracy)
- Problems 6-10: 4 minutes (medium problems, strategic approach)
- Problems 11-15: 5 minutes (difficult problems, elimination focus)
- Review & Guessing: 3 minutes (final decisions)
馃 Practice Problems
Problem 1
Tags: Sprint 路 source: AMC10 2020 #2
What is the value of $3 + 4 \times 5$?
A) $23$
B) $24$
C) $25$
D) $26$
E) $27$
Strategic Analysis
Quick Triage: Problem #2, straightforward. Order of operations: $3 + 4 \times 5 = 3 + 20 = 23$.
Time Target: 30 seconds
Answer & Solution
Answer: A
Using order of operations: $3 + 4 \times 5 = 3 + 20 = 23$.
Problem 2
Tags: Sprint 路 source: AMC10 2020 #4
What is the value of $1 + 2 + 3 + \cdots + 10$?
A) $45$
B) $50$
C) $55$
D) $60$
E) $65$
Strategic Analysis
Quick Triage: Problem #4, 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: Sprint 路 source: AMC10 2020 #6
A rectangle has length 8 and width 6. What is its area?
A) $14$
B) $24$
C) $28$
D) $48$
E) $64$
Strategic Analysis
Quick Triage: Problem #6, straightforward. Area = length 脳 width = 8 脳 6 = 48.
Time Target: 30 seconds
Answer & Solution
Answer: D
Area = length 脳 width = 8 脳 6 = 48 square units.
Problem 4
Tags: Sprint 路 source: AMC10 2020 #8
What is the remainder when $2^{10}$ is divided by 7?
A) $1$
B) $2$
C) $3$
D) $4$
E) $5$
Strategic Analysis
Quick Triage: Problem #8, manageable. Use modular arithmetic: $2^3 = 8 \equiv 1 \pmod{7}$, so $2^{10} = 2^{3 \cdot 3 + 1} \equiv 2 \pmod{7}$.
Time Target: 1 minute
Answer & Solution
Answer: B
Since $2^3 = 8 \equiv 1 \pmod{7}$, we have $2^{10} = 2^{3 \cdot 3 + 1} \equiv 2 \pmod{7}$.
Problem 5
Tags: Sprint 路 source: AMC10 2020 #10
A fair coin is flipped 3 times. What is the probability of getting exactly 2 heads?
A) $\frac{1}{8}$
B) $\frac{1}{4}$
C) $\frac{3}{8}$
D) $\frac{1}{2}$
E) $\frac{3}{4}$
Strategic Analysis
Quick Triage: Problem #10, manageable. There are $2^3 = 8$ total outcomes. Getting exactly 2 heads can happen in $\binom{3}{2} = 3$ ways: HHT, HTH, THH. So the probability is $\frac{3}{8}$.
Time Target: 1 minute
Answer & Solution
Answer: C
There are $2^3 = 8$ total outcomes. Getting exactly 2 heads can happen in $\binom{3}{2} = 3$ ways: HHT, HTH, THH. So the probability is $\frac{3}{8}$.
Problem 6
Tags: Sprint 路 source: AMC10 2020 #12
What is the value of $\sqrt{144}$?
A) $10$
B) $11$
C) $12$
D) $13$
E) $14$
Strategic Analysis
Quick Triage: Problem #12, 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 7
Tags: Sprint 路 source: AMC10 2020 #14
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
Quick Triage: Problem #14, 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: 45 seconds
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 8
Tags: Sprint 路 source: AMC10 2020 #16
You have 5 minutes left and are on problem 16. What should you do?
A) Try to solve problem 16 completely
B) Try to eliminate some choices, then guess
C) Skip problem 16 and move on
D) Go back to check previous answers
E) Need more information
Strategic Analysis
Quick Triage: Problem #16, time management. With 5 minutes left, try to eliminate some choices first, then guess. This maximizes expected value.
Time Target: 1 minute
Answer & Solution
Answer: B
With 5 minutes left and on problem 16, try to eliminate some choices first, then guess. This maximizes expected value while managing time.
Problem 9
Tags: Sprint 路 source: AMC10 2020 #18
You solved a problem and got answer C. How should you verify?
A) Solve it again the same way
B) Solve it using a different method
C) Check if the answer makes sense
D) All of the above
E) None of the above
Strategic Analysis
Quick Triage: Problem #18, verification strategy. The best verification strategy is to use multiple methods: solve again, use different approach, and check if answer makes sense.
Time Target: 1 minute
Answer & Solution
Answer: D
The best verification strategy is to use multiple methods: solve again, use different approach, and check if answer makes sense.
Problem 10
Tags: Sprint 路 source: AMC10 2020 #20
You have 1 minute 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
Quick Triage: Problem #20, final decision. With 1 minute 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: 30 seconds
Answer & Solution
Answer: B
With 1 minute 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 11
Tags: Sprint 路 source: AMC10 2020 #22
What is the value of $2^{10}$?
A) $512$
B) $1024$
C) $2048$
D) $4096$
E) $8192$
Strategic Analysis
Quick Triage: Problem #22, strategic range. $2^{10} = 1024$. This is a standard power of 2 that should be memorized.
Time Target: 45 seconds
Answer & Solution
Answer: B
$2^{10} = 1024$.
Problem 12
Tags: Sprint 路 source: AMC10 2020 #24
What is the value of $\log_2(1000)$?
A) $8$
B) $9$
C) $10$
D) $11$
E) $12$
Strategic Analysis
Quick Triage: Problem #24, strategic range. $2^{10} = 1024$, so $\log_2(1000) \approx 10$. Since $1000 < 1024$, the answer is slightly less than 10.
Time Target: 1 minute
Answer & Solution
Answer: C
Since $2^{10} = 1024$ and $1000 < 1024$, we have $\log_2(1000) \approx 10$.
Problem 13
Tags: Sprint 路 source: AMC10 2020 #25
You’re stuck on a problem with 5 answer choices. You can eliminate 3 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
Quick Triage: Problem #25, strategic decision. 2 choices remaining: $\text{EV}_\text{guess} = \frac{6}{2} = 3$ vs. $\text{EV}_\text{blank} = 1.5$. Since $3 > 1.5$, guess.
Time Target: 45 seconds
Answer & Solution
Answer: A
With 2 remaining choices: $\text{EV}_\text{guess} = \frac{6}{2} = 3$ vs. $\text{EV}_\text{blank} = 1.5$. Since $3 > 1.5$, guess.
Problem 14
Tags: Sprint 路 source: AMC10 2020 #25
You have 30 seconds 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
Quick Triage: Problem #25, final decision. With 30 seconds left, try to eliminate some choices first, then guess. This maximizes expected value while managing time.
Time Target: 30 seconds
Answer & Solution
Answer: B
With 30 seconds left and on problem 25, try to eliminate some choices first, then guess. This maximizes expected value while managing time.
Problem 15
Tags: Sprint 路 source: AMC10 2020 #25
You have 10 seconds left and 1 problem remaining. You can eliminate 2 choices. What should you do?
A) Try to solve the problem completely
B) Guess on the problem
C) Skip the problem
D) Go back to check previous answers
E) Need more information
Strategic Analysis
Quick Triage: Problem #25, final decision. With 10 seconds left, guess immediately. With 2 eliminations: $\text{EV}_\text{guess} = \frac{6}{3} = 2$ vs. $\text{EV}_\text{blank} = 1.5$. Since $2 > 1.5$, guess.
Time Target: 10 seconds
Answer & Solution
Answer: B
With 10 seconds left, guess immediately. With 2 eliminations, guessing gives higher expected value than skipping.
馃搳 Sprint Success Metrics
Time Targets
- Problems 1-5: 30-60 seconds each
- Problems 6-10: 45-90 seconds each
- Problems 11-15: 60-120 seconds each
- Review & Guessing: 3 minutes total
Strategic Goals
- Elimination accuracy: 80%+ of eliminations should be correct
- Expected value decisions: Make optimal guessing vs. skipping choices
- Time management: Stay within time targets
- Error prevention: Catch mistakes before submission
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