Mixed Deck — All AMC 12 Topics Flashcards

How should differentiation be applied in educational technology?

Answer: Instruction is adapted based on student readiness, interest, and learning profile

Effective differentiation adapts instruction based on each student's readiness level, interests, and learning profile.

What role does data play in professional development decisions?

Answer: Data from multiple sources informs instructional planning and intervention

Using data from multiple sources ensures educational decisions are evidence-based and targeted to student needs.

What is the sum of the first 10 triangular numbers (1, 3, 6, 10, ...)?

Answer: 220

The sum of the first n triangular numbers is n(n+1)(n+2)/6, giving 10·11·12/6 = 220.

What is the value of Σ(k=1 to ∞) of (1/2)^k?

Answer: 1

This geometric series has first term a = 1/2 and ratio r = 1/2, so S = (1/2)/(1 − 1/2) = 1.

Research shows that taking timed practice AMC tests improves long-term retention more than rereading notes. What is this called?

Answer: Testing effect (retrieval practice)

The testing effect shows that actively retrieving information strengthens memory more than restudying.

What is the real part of (1 + 2i)/(1 − i)?

Answer: −1/2

Multiplying by the conjugate: (1+2i)(1+i)/2 = (−1+3i)/2, giving real part −1/2.

According to Deci and Ryan's self-determination theory, which three needs support intrinsic motivation to study math contests?

Answer: Autonomy, competence, relatedness

Self-determination theory names autonomy, competence, and relatedness as basic psychological needs.

What is the principal argument of the complex number i?

Answer: π/2

The number i has real part 0 and imaginary part 1, placing it at angle π/2 on the unit circle.

Which time-management habit should a coach model for AMC 12 students during practice tests?

Answer: Skip hard problems and return if time remains

Skipping and returning maximizes points within the 75-minute limit.

A population of 1000 grows by 10% per year, compounded annually. What is it after 2 years?

Answer: 1210

1000·1.1^2 = 1210.

What is the final stage of the U.S. competition pathway that starts with the AMC 12?

Answer: Representing the U.S. at the International Mathematical Olympiad

The pathway goes AMC, AIME, USAMO, then the Olympiad program, which selects the U.S. IMO team.

What is the probability of getting at least one head in 4 fair coin flips?

Answer: 15/16

Complementary counting gives 1 − (1/2)⁴ = 15/16.

What is the primary goal of learning theories in educational settings?

Answer: Supporting diverse learner needs and maximizing student achievement

Learning Theories in education aims to support diverse learner needs and maximize achievement for all students.

What is the greatest common divisor of 84 and 126?

Answer: 42

84 = 2^2 x 3 x 7 and 126 = 2 x 3^2 x 7, so gcd = 2 x 3 x 7 = 42.

The Fibonacci sequence is defined by F₁ = 1, F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂. What is F₇?

Answer: 13

Computing in order: F₃=2, F₄=3, F₅=5, F₆=8, F₇=13.

What is tan(π/4)?

Answer: 1

tan(π/4) = sin(π/4)/cos(π/4) = (√2/2)/(√2/2) = 1.

How many integers from 1 to 100 are both perfect squares and perfect cubes?

Answer: 2

Such numbers are sixth powers: 1 and 64.

What format do AIME answers take?

Answer: An integer from 000 to 999

Every AIME answer is an integer between 0 and 999 inclusive.

Which resource is the official source for past AMC 12 problems that students can use for practice?

Answer: The Mathematical Association of America (MAA)

The MAA runs the AMC and publishes past problems and solutions.

After eliminating exactly two of the five answer choices, what is a random guess worth on average compared with leaving the question blank?

Answer: Better than a blank (2 vs 1.5)

With 3 choices left, a guess is worth 6/3 = 2 points on average, which beats the 1.5 for a blank.