AMC 数学竞赛练习

Subtract 5 from both sides: x = 12 - 5 = 7. The other values do not satisfy the equation.

First compute 4 + 2 = 6, then 3 × 6 = 18, and finally 18 - 5 = 13.

Combine like terms: 2x + 3x = 5x, then 5x - x = 4x.

Add 6 to both sides: 2x = 10, then divide by 2: x = 5.

The differences are 4, 6, 8, so the next difference is 10. Therefore 20 + 10 = 30.

In slope-intercept form y = mx + b, m is the slope. Here m = 3 and b = 2.

In y = mx + b, b is the y-intercept. Here b = 5.

Substitute 3: 3^2 + 1 = 9 + 1 = 10.

The positive square root of 49 is 7 because 7 × 7 = 49.

The quadratic factors as (x - 2)(x - 3) = 0, so the roots are 2 and 3. Their sum is 5, which also equals the negative coefficient of x.

Add exponents when multiplying like bases: 2^(3+2) = 2^5 = 32.

When raising a power to a power, multiply exponents: 2 × 3 = 6, so the result is x^6.

Area = length × width = 6 × 4 = 24. Perimeter would be 20, but the question asks for area.

Perimeter = 4 × side = 4 × 5 = 20. Area would be 25.

Area = (base × height) / 2 = (8 × 5) / 2 = 40 / 2 = 20.

Circumference = 2πr = 2π × 7 = 14π. The area would be 49π.

Area = πr^2 = π × 3^2 = 9π. Circumference would be 6π.

By the Pythagorean theorem: 3^2 + 4^2 = 9 + 16 = 25, so the hypotenuse is 5.

The interior angles of any triangle sum to 180 degrees. A quadrilateral sums to 360 degrees.

A quadrilateral can be divided into two triangles, so 2 × 180° = 360°.

Volume = edge^3 = 3 × 3 × 3 = 27. Surface area of the same cube would be 54.

Each face has area 2 × 2 = 4, and a cube has 6 faces, so 6 × 4 = 24. Volume would be 8.

Diameter = 2 × radius = 2 × 9 = 18.

Area = ((b1 + b2) / 2) × height = ((4 + 6) / 2) × 5 = 5 × 5 = 25.

The divisors of 12 are 1, 2, 3, 4, 6, 12 and of 18 are 1, 2, 3, 6, 9, 18. The largest shared divisor is 6.

Multiples of 4: 4, 8, 12; multiples of 6: 6, 12. The smallest common multiple is 12.

15% = 0.15, and 0.15 × 200 = 30.

36 = 4 × 9 = 2^2 × 3^2. The other options are not prime factorizations.

A prime number is a positive integer greater than 1 with exactly two divisors. Since 1 has only one divisor, it is not prime.

The sum is 10 × 11 / 2 = 55 using the formula n(n+1)/2.

7! / 5! = (7 × 6 × 5!) / 5! = 7 × 6 = 42.

7^2 = 49, so the units digit is 9.

0.25 = 25/100, which simplifies to 1/4.

Rewrite 2/3 as 4/6, then 4/6 + 1/6 = 5/6.

The sum is 40, and 40 / 4 = 10.

The values are already ordered, and the middle value is 9.

The number of arrangements is 3! = 3 × 2 × 1 = 6.

There are 3 choices for the first letter and 2 for the second: 3 × 2 = 6.

4 choose 2 = 4! / (2! × 2!) = 24 / 4 = 6.

4 choose 2 = 4! / (2! × 2!) = 24 / 4 = 6.

Each flip has 2 outcomes, so 2 × 2 × 2 = 8 total outcomes.

There is one favorable outcome out of six equally likely outcomes.

The probability is (1/2) × (1/2) = 1/4.

Each die has 6 outcomes, so 6 × 6 = 36 total outcomes.

The favorable pairs are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), which is 6 out of 36, or 1/6.

The formula is 3! / 2! = 6 / 2 = 3 because the two A letters are identical.

A set with n elements has 2^n subsets. For n = 3, that is 2^3 = 8.

A standard deck has 26 red cards out of 52, so the probability is 26/52 = 1/2.

Understanding and restating the problem helps you identify what is given and what is asked. Guessing before understanding is not a reliable strategy.

Substituting the answer back verifies that both sides match. Choosing randomly or altering the problem does not check the solution.

Testing smaller cases, drawing diagrams, and looking for patterns help you understand the structure. Guessing randomly is not productive.

The AMC 8 is 40 minutes long. AMC 10 and AMC 12 are 75 minutes.

The AMC 8 has 25 multiple-choice questions.

AMC contests use multiple-choice questions with choices A through E. They are not free response, true/false, or essays.

Moving past hard questions and returning later protects time for problems you can solve. Fixed order or quitting early loses points.

Integers include ..., -2, -1, 0, 1, 2, ... . Fractions, decimals, and irrational numbers are not integers.

A prime has exactly two positive divisors: 1 and itself. Odd numbers and numbers ending in 1 can be composite.

Even integers are divisible by 2, such as -4, 0, and 8. Ending in 5 or being negative does not define even.

Diagrams, divisibility checks, and pattern recognition are strong strategies. Wild guessing is not reliable.

Integers are whole numbers, primes have exactly two divisors, even numbers are divisible by 2, and the median is the middle value.