Pregunta 1
If x + 5 = 12, what is x?
Subtract 5 from both sides: x = 12 - 5 = 7. The other values do not satisfy the equation.
Práctica de AMC
Practica 60 preguntas originales de AMC sobre álgebra, geometría, teoría de números, combinatoria y estrategia.
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Pregunta 2
What is the value of 3(4 + 2) - 5?
First compute 4 + 2 = 6, then 3 × 6 = 18, and finally 18 - 5 = 13.
Pregunta 3
Simplify 2x + 3x - x.
Combine like terms: 2x + 3x = 5x, then 5x - x = 4x.
Pregunta 4
If 2x - 6 = 4, what is x?
Add 6 to both sides: 2x = 10, then divide by 2: x = 5.
Pregunta 5
What is the next term in the sequence 2, 6, 12, 20, ...?
The differences are 4, 6, 8, so the next difference is 10. Therefore 20 + 10 = 30.
Pregunta 6
What is the slope of the line y = 3x + 2?
In slope-intercept form y = mx + b, m is the slope. Here m = 3 and b = 2.
Pregunta 7
What is the y-intercept of the line y = -2x + 5?
In y = mx + b, b is the y-intercept. Here b = 5.
Pregunta 8
If f(x) = x^2 + 1, what is f(3)?
Substitute 3: 3^2 + 1 = 9 + 1 = 10.
Pregunta 9
Solve x^2 = 49 for the positive value of x.
The positive square root of 49 is 7 because 7 × 7 = 49.
Pregunta 10
What is the sum of the solutions of x^2 - 5x + 6 = 0?
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.
Pregunta 11
What is the value of 2^3 × 2^2?
Add exponents when multiplying like bases: 2^(3+2) = 2^5 = 32.
Pregunta 12
Simplify (x^2)^3.
When raising a power to a power, multiply exponents: 2 × 3 = 6, so the result is x^6.
Pregunta 13
What is the area of a rectangle with length 6 and width 4?
Area = length × width = 6 × 4 = 24. Perimeter would be 20, but the question asks for area.
Pregunta 14
What is the perimeter of a square with side length 5?
Perimeter = 4 × side = 4 × 5 = 20. Area would be 25.
Pregunta 15
What is the area of a triangle with base 8 and height 5?
Area = (base × height) / 2 = (8 × 5) / 2 = 40 / 2 = 20.
Pregunta 16
What is the circumference of a circle with radius 7?
Circumference = 2πr = 2π × 7 = 14π. The area would be 49π.
Pregunta 17
What is the area of a circle with radius 3?
Area = πr^2 = π × 3^2 = 9π. Circumference would be 6π.
Pregunta 18
What is the hypotenuse of a right triangle with legs 3 and 4?
By the Pythagorean theorem: 3^2 + 4^2 = 9 + 16 = 25, so the hypotenuse is 5.
Pregunta 19
What is the sum of the interior angles of a triangle?
The interior angles of any triangle sum to 180 degrees. A quadrilateral sums to 360 degrees.
Pregunta 20
What is the sum of the interior angles of a quadrilateral?
A quadrilateral can be divided into two triangles, so 2 × 180° = 360°.
Pregunta 21
What is the volume of a cube with edge length 3?
Volume = edge^3 = 3 × 3 × 3 = 27. Surface area of the same cube would be 54.
Pregunta 22
What is the surface area of a cube with edge length 2?
Each face has area 2 × 2 = 4, and a cube has 6 faces, so 6 × 4 = 24. Volume would be 8.
Pregunta 23
What is the diameter of a circle with radius 9?
Diameter = 2 × radius = 2 × 9 = 18.
Pregunta 24
What is the area of a trapezoid with bases 4 and 6 and height 5?
Area = ((b1 + b2) / 2) × height = ((4 + 6) / 2) × 5 = 5 × 5 = 25.
Pregunta 25
What is the greatest common divisor of 12 and 18?
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.
Pregunta 26
What is the least common multiple of 4 and 6?
Multiples of 4: 4, 8, 12; multiples of 6: 6, 12. The smallest common multiple is 12.
Pregunta 27
What is 15% of 200?
15% = 0.15, and 0.15 × 200 = 30.
Pregunta 28
What is the prime factorization of 36?
36 = 4 × 9 = 2^2 × 3^2. The other options are not prime factorizations.
Pregunta 29
1 is a prime number.
A prime number is a positive integer greater than 1 with exactly two divisors. Since 1 has only one divisor, it is not prime.
Pregunta 30
What is the sum of the first 10 positive integers?
The sum is 10 × 11 / 2 = 55 using the formula n(n+1)/2.
Pregunta 31
What is 7! / 5!?
7! / 5! = (7 × 6 × 5!) / 5! = 7 × 6 = 42.
Pregunta 32
What is the units digit of 7^2?
7^2 = 49, so the units digit is 9.
Pregunta 33
What is 0.25 as a fraction in simplest form?
0.25 = 25/100, which simplifies to 1/4.
Pregunta 34
What is 2/3 + 1/6?
Rewrite 2/3 as 4/6, then 4/6 + 1/6 = 5/6.
Pregunta 35
What is the average of 4, 8, 12, and 16?
The sum is 40, and 40 / 4 = 10.
Pregunta 36
What is the median of 3, 7, 9, 12, 15?
The values are already ordered, and the middle value is 9.
Pregunta 37
How many ways can 3 distinct books be arranged?
The number of arrangements is 3! = 3 × 2 × 1 = 6.
Pregunta 38
How many 2-letter arrangements can be made from A, B, C without repetition?
There are 3 choices for the first letter and 2 for the second: 3 × 2 = 6.
Pregunta 39
How many ways are there to choose 2 items from a set of 4?
4 choose 2 = 4! / (2! × 2!) = 24 / 4 = 6.
Pregunta 40
What is the value of 4 choose 2?
4 choose 2 = 4! / (2! × 2!) = 24 / 4 = 6.
Pregunta 41
A fair coin is flipped 3 times. How many possible outcomes are there?
Each flip has 2 outcomes, so 2 × 2 × 2 = 8 total outcomes.
Pregunta 42
What is the probability of rolling a 4 on a fair six-sided die?
There is one favorable outcome out of six equally likely outcomes.
Pregunta 43
What is the probability of getting heads on two fair coin flips?
The probability is (1/2) × (1/2) = 1/4.
Pregunta 44
How many outcomes are there when rolling two fair six-sided dice?
Each die has 6 outcomes, so 6 × 6 = 36 total outcomes.
Pregunta 45
What is the probability of rolling a sum of 7 with two fair dice?
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.
Pregunta 46
How many distinct arrangements are there of the letters AAB?
The formula is 3! / 2! = 6 / 2 = 3 because the two A letters are identical.
Pregunta 47
What is the number of subsets of a set with 3 elements?
A set with n elements has 2^n subsets. For n = 3, that is 2^3 = 8.
Pregunta 48
What is the probability of drawing a red card from a standard deck?
A standard deck has 26 red cards out of 52, so the probability is 26/52 = 1/2.
Pregunta 49
What should you do first when you see a hard AMC problem?
Understanding and restating the problem helps you identify what is given and what is asked. Guessing before understanding is not a reliable strategy.
Pregunta 50
What is a good way to check an answer to an algebra problem?
Substituting the answer back verifies that both sides match. Choosing randomly or altering the problem does not check the solution.
Pregunta 51
What should you do when you are stuck on an AMC problem?
Testing smaller cases, drawing diagrams, and looking for patterns help you understand the structure. Guessing randomly is not productive.
Pregunta 52
How long is the AMC 8?
The AMC 8 is 40 minutes long. AMC 10 and AMC 12 are 75 minutes.
Pregunta 53
How many questions are on the AMC 8?
The AMC 8 has 25 multiple-choice questions.
Pregunta 54
What is the answer format of AMC 8?
AMC contests use multiple-choice questions with choices A through E. They are not free response, true/false, or essays.
Pregunta 55
What is a good time-management strategy for AMC?
Moving past hard questions and returning later protects time for problems you can solve. Fixed order or quitting early loses points.
Pregunta 56
What is an integer?
Integers include ..., -2, -1, 0, 1, 2, ... . Fractions, decimals, and irrational numbers are not integers.
Pregunta 57
What is a prime number?
A prime has exactly two positive divisors: 1 and itself. Odd numbers and numbers ending in 1 can be composite.
Pregunta 58
What is an even integer?
Even integers are divisible by 2, such as -4, 0, and 8. Ending in 5 or being negative does not define even.
Pregunta 59
Which of the following are useful AMC problem-solving strategies? Select all that apply.
Diagrams, divisibility checks, and pattern recognition are strong strategies. Wild guessing is not reliable.
Pregunta 60
Match each math term to its meaning.
Integers are whole numbers, primes have exactly two divisors, even numbers are divisible by 2, and the median is the middle value.