🎯 Complete AMC10 Strategy Session

First Pass: Problem #1, straightforward. Order of operations: $2 + 3 \times 4 = 2 + 12 = 14$.

Time Target: 30 seconds

Answer: A

Using order of operations: $2 + 3 \times 4 = 2 + 12 = 14$.

First Pass: Problem #2, manageable. Formula: $\frac{n(n+1)}{2} = \frac{10 \cdot 11}{2} = 55$.

Time Target: 45 seconds

Answer: C

Using sum formula: $\frac{n(n+1)}{2} = \frac{10 \cdot 11}{2} = 55$.

First Pass: Problem #3, straightforward. Area = length × width = 8 × 6 = 48.

Time Target: 30 seconds

Answer: D

Area = length × width = 8 × 6 = 48 square units.

First Pass: Problem #4, 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: B

Since $2^3 = 8 \equiv 1 \pmod{7}$, we have $2^{10} = 2^{3 \cdot 3 + 1} \equiv 2 \pmod{7}$.

First Pass: Problem #5, 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: 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 $\frac{3}{8}$.

First Pass: Problem #6, straightforward. $\sqrt{144} = 12$ since $12^2 = 144$.

Time Target: 30 seconds

Answer: C

Since $12^2 = 144$, we have $\sqrt{144} = 12$.

First Pass: Problem #7, manageable. $f(3) = 3^2 + 2(3) + 1 = 9 + 6 + 1 = 16$.

Time Target: 45 seconds

Answer: A

Substituting $x = 3$: $f(3) = 3^2 + 2(3) + 1 = 9 + 6 + 1 = 16$.

First Pass: Problem #8, straightforward. 25% of 80 = 0.25 × 80 = 20.

Time Target: 30 seconds

Answer: B

25% of 80 = 0.25 × 80 = 20.

First Pass: Problem #9, manageable. We need two numbers that multiply to 6 and add to -5. These are -2 and -3, so $(x-2)(x-3)$.

Time Target: 1 minute

Answer: A

We need two numbers that multiply to 6 and add to -5. These are -2 and -3, so $x^2 - 5x + 6 = (x-2)(x-3)$.

First Pass: Problem #10, manageable. Use distance formula: $\sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Time Target: 1 minute

Answer: C

Using distance formula: $\sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

First Pass: Problem #11, manageable. Adding the equations: $2x = 6$, so $x = 3$. Then $y = 5 - 3 = 2$.

Time Target: 1 minute

Answer: B

Adding the equations: $2x = 6$, so $x = 3$. Then $y = 5 - 3 = 2$.

First Pass: Problem #12, manageable. Area = $\pi r^2 = \pi \cdot 5^2 = 25\pi$.

Time Target: 45 seconds

Answer: D

Area = $\pi r^2 = \pi \cdot 5^2 = 25\pi$.

First Pass: Problem #13, manageable. This is $4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$.

Time Target: 45 seconds

Answer: D

This is $4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$ arrangements.

First Pass: Problem #14, manageable. Factor: $(x-2)(x-4) = 0$, so $x = 2$ or $x = 4$.

Time Target: 1 minute

Answer: A

Factor: $(x-2)(x-4) = 0$, so $x = 2$ or $x = 4$.

First Pass: Problem #15, manageable. Volume = side³ = 3³ = 27.

Time Target: 30 seconds

Answer: C

Volume = side³ = 3³ = 27 cubic units.

Second Pass: Problem #16, 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: 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.

Second Pass: Problem #17, time management. With 10 minutes left, try to eliminate some choices first, then guess. This maximizes expected value.

Time Target: 2 minutes

Answer: B

With 10 minutes left and on problem 17, try to eliminate some choices first, then guess. This maximizes expected value while managing time.

Second Pass: 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: D

The best verification strategy is to use multiple methods: solve again, use different approach, and check if answer makes sense.

Second Pass: Problem #19, elimination strategy. For area of a triangle, you can eliminate negative numbers (area must be positive), zero (unless degenerate), and very large numbers (unless reasonable).

Time Target: 1 minute

Answer: D

For area of a triangle, you can eliminate negative numbers (area must be positive), zero (unless degenerate), and very large numbers (unless reasonable).

Second Pass: Problem #20, 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: 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.

Second Pass: Problem #21, elimination strategy. For probability, you can eliminate negative numbers (probability must be non-negative) and numbers greater than 1 (probability must be at most 1).

Time Target: 1 minute

Answer: A

For probability, you can eliminate negative numbers (probability must be non-negative) and numbers greater than 1 (probability must be at most 1).

Second Pass: Problem #22, time management. With 3 minutes left, try to eliminate some choices first, then guess. This maximizes expected value.

Time Target: 2 minutes

Answer: B

With 3 minutes left and on problem 22, try to eliminate some choices first, then guess. This maximizes expected value while managing time.

Second Pass: Problem #23, strategic decision. 4 choices remaining: $\text{EV}_\text{guess} = \frac{6}{4} = 1.5$ vs. $\text{EV}_\text{blank} = 1.5$. Both have the same expected value.

Time Target: 1 minute

Answer: D

With 4 remaining choices: $\text{EV}_\text{guess} = \frac{6}{4} = 1.5$ vs. $\text{EV}_\text{blank} = 1.5$. Both have the same expected value.

Second Pass: Problem #24, time management. With 2 minutes left, try to eliminate some choices first, then guess. This maximizes expected value.

Time Target: 1 minute

Answer: B

With 2 minutes left and on problem 24, try to eliminate some choices first, then guess. This maximizes expected value while managing time.

Second 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: B

With 1 minute left and on problem 25, try to eliminate some choices first, then guess. This maximizes expected value while managing time.