AMC 12 Practice Problems

AMC 12 Practice Problems: 25 Original Questions With Solutions.

Preparing for the AMC 12 requires more than simply reviewing formulas. You need to recognize patterns, choose efficient methods, manage time, and solve unfamiliar problems accurately. The best way to develop these skills is through consistent practice with problems covering different areas of mathematics.

This collection of AMC 12 practice problems includes 25 original questions designed to resemble the style and difficulty of mathematical contest problems. The problems cover algebra, geometry, number theory, counting, probability, sequences, and other topics commonly useful for AMC 12 preparation.

Try solving each problem yourself before looking at the solution. If you get stuck, use the explanation to understand the method rather than simply memorizing the answer.

How to Use These AMC 12 Practice Problems.

For effective AMC 12 preparation, treat this set like a mini practice test.

A useful approach is:

1. Read each problem carefully.

2. Try to solve it without checking the answer.

3. Give yourself a reasonable time limit.

4. Mark difficult problems and return to them later.

5. Study the solution after finishing your attempt.

6. Write down any technique you did not recognize.

7. Repeat similar problems until the method becomes familiar.

The goal is not only to get the correct answer. You should also learn how to identify the shortest path to that answer.

AMC 12 Practice Problems.

Problem 1: Algebra

If

$$x+\frac{1}{x}=5,$$

where 

$x\neq0$

, what is

$$x^2+\frac{1}{x^2}?$$

A) 21

B) 22

C) 23

D) 24

E) 25

Solution:

Square the original equation:

$$\left(x+\frac1x\right)^2=25.$$

Therefore,

$$x^2+2+\frac1{x^2}=25.$$

So,

$$x^2+\frac1{x^2}=23.$$

Answer: C) 23

Problem 2: Number Theory

What is the remainder when

$$3^{2026}$$

is divided by 10?

A) 1

B) 3

C) 5

D) 7

E) 9

Solution:

The last digits of powers of 3 repeat:

$$3,9,7,1.$$

The cycle has length 4.

Since

$$2026\equiv2\pmod4,$$

we need the second number in the cycle.

Thus the last digit is:

$$9.$$

Answer: E) 9

Problem 3: Geometry

A rectangle has length 12 and width 5. What is the distance between two opposite corners?

A) 11

B) 12

C) 13

D) 14

E) 17

Solution:

The diagonal forms a right triangle.

Using the Pythagorean theorem:

$$d^2=12^2+5^2$$ $$d^2=144+25=169.$$

Therefore,

$$d=13.$$

Answer: C) 13

Problem 4: Counting

How many three-digit positive integers have digits that are all different?

A) 648

B) 720

C) 729

D) 810

E) 900

Solution:

The hundreds digit has 9 choices because it cannot be zero.

The tens digit then has 9 choices because it can be zero but cannot equal the hundreds digit.

The units digit has 8 remaining choices.

Therefore,

$$9\times9\times8=648.$$

Answer: A) 648

Problem 5: Sequences

A sequence begins

$$2,5,10,17,26,\ldots$$

What is the 10th term?

A) 91

B) 99

C) 101

D) 109

E) 111

Solution:

Notice that the terms follow

$$1^2+1,\quad2^2+1,\quad3^2+1,\ldots$$

Thus

$$a_n=n^2+1.$$

For 

$n=10$

,

$$a_{10}=10^2+1=101.$$

Answer: C) 101

Problem 6: Probability

A fair six-sided die is rolled twice. What is the probability that the sum is 8?

A)

$\frac{1}{9}$

B)

$\frac{5}{36}$

C)

$\frac16$

D)

$\frac{7}{36}$

E)

$\frac29$

Solution:

The possible pairs that sum to 8 are:

$$(2,6),(3,5),(4,4),(5,3),(6,2).$$

There are 5 favorable outcomes out of

$$6\times6=36$$

total outcomes.

Therefore,

$$P=\frac5{36}.$$

Answer: B) 

$\frac{5}{36}$

Problem 7: Algebra

If

$$2x+3=17,$$

what is 

$x^2$

?

A) 36

B) 42

C) 49

D) 56

E) 64

Solution:

Subtract 3:

$$2x=14.$$

Thus,

$$x=7.$$

Therefore,

$$x^2=49.$$

Answer: C) 49

Problem 8: Geometry

A square has area 49. What is the perimeter of the square?

A) 21

B) 24

C) 28

D) 32

E) 49

Solution:

If the area is 49, the side length is

$$\sqrt{49}=7.$$

The perimeter is

$$4(7)=28.$$

Answer: C) 28

Problem 9: Divisibility

What is the smallest positive integer divisible by both 12 and 18?

A) 24

B) 30

C) 36

D) 48

E) 54

Solution:

The prime factorizations are

$$12=2^2\cdot3$$

and

$$18=2\cdot3^2.$$

The least common multiple is

$$2^2\cdot3^2=36.$$

Answer: C) 36

Problem 10: Exponents

What is

$$\frac{2^8\cdot2^5}{2^{10}}?$$

A) 4

B) 6

C) 8

D) 16

E) 32

Solution:

Use exponent rules:

$$\frac{2^8\cdot2^5}{2^{10}} =2^{8+5-10} =2^3.$$

Therefore,

$$2^3=8.$$

Answer: C) 8

More Challenging AMC 12 Practice Problems

The following questions require more than direct formula substitution. They are useful for developing the reasoning skills needed for harder AMC 12 questions.

Problem 11: Integer Solutions

How many positive integer pairs 

$(x,y)$

satisfy

$$x+y=10?$$

A) 8

B) 9

C) 10

D) 11

E) 12

Solution:

The positive possibilities are:

$$(1,9),(2,8),(3,7),(4,6),(5,5),$$

and their reverses.

There are 9 total pairs.

Answer: B) 9

Problem 12: Quadratic Expressions

If

$$x^2-6x+5=0,$$

what is the sum of the reciprocals of the roots?

A)

$\frac15$

B)

$\frac65$

C)

$\frac56$

D) 5

E) 6

Solution:

For

$$ax^2+bx+c=0,$$

the sum of the roots is

$$-\frac ba$$

and the product is

$$\frac ca.$$

Here,

$$x_1+x_2=6$$

and

$$x_1x_2=5.$$

Therefore,

$$\frac1{x_1}+\frac1{x_2} = \frac{x_1+x_2}{x_1x_2} = \frac65.$$

Answer: B) 

$\frac65$

Problem 13: Area

A triangle has base 10 and height 12. What is its area?

A) 50

B) 60

C) 72

D) 100

E) 120

Solution:

Use

$$A=\frac12bh.$$

Thus,

$$A=\frac12(10)(12)=60.$$

Answer: B) 60

Problem 14: Modular Arithmetic

What is the remainder when

$$7^{100}$$

is divided by 6?

A) 0

B) 1

C) 2

D) 3

E) 5

Solution:

Since

$$7\equiv1\pmod6,$$

we have

$$7^{100}\equiv1^{100}\equiv1\pmod6.$$

Answer: B) 1

Problem 15: Average

The average of five numbers is 18. Four of the numbers are 12, 15, 20, and 25. What is the fifth number?

A) 16

B) 18

C) 20

D) 22

E) 24

Solution:

The total of all five numbers is

$$5(18)=90.$$

The known numbers total

$$12+15+20+25=72.$$

Therefore, the missing number is

$$90-72=18.$$

Answer: B) 18

Problem 16: Counting Paths

A student can move from one point to another by taking exactly 3 steps right and 2 steps up. How many different arrangements of these steps are possible?

A) 5

B) 6

C) 8

D) 10

E) 12

Solution:

There are 5 total steps.

Choose the 2 positions occupied by the upward steps:

$$\binom52=10.$$

Answer: D) 10

Problem 17: Geometry

A circle has radius 6. What is its area?

A)

$6\pi$

B)

$12\pi$

C)

$18\pi$

D)

$36\pi$

E)

$72\pi$

Solution:

The area of a circle is

$$A=\pi r^2.$$

Therefore,

$$A=\pi(6^2)=36\pi.$$

Answer: D) 

$36\pi$

Problem 18: Inequality

If

$$3x-4<11,$$

what is the largest integer value of 

$x$

?

A) 4

B) 5

C) 6

D) 7

E) 8

Solution:

Add 4:

$$3x<15.$$

Divide by 3:

$$x<5.$$

The largest integer less than 5 is 4.

Answer: A) 4

Problem 19: Factorials

What is

$$\frac{7!}{5!}?$$

A) 12

B) 14

C) 35

D) 42

E) 49

Solution:

Cancel 

$5!$

:

$$\frac{7!}{5!} = 7\times6 = 42.$$

Answer: D) 42

Problem 20: Ratio

The ratio of boys to girls in a class is 

$3:5$

. If there are 24 students total, how many are girls?

A) 9

B) 12

C) 15

D) 18

E) 20

Solution:

The ratio contains

$$3+5=8$$

parts.

Each part represents

$$24/8=3$$

students.

Girls represent 5 parts:

$$5(3)=15.$$

Answer: C) 15

Advanced AMC 12 Practice Problems

Problem 21: Algebraic Substitution

If

$$a+b=10$$

and

$$ab=21,$$

what is

$$a^2+b^2?$$

A) 42

B) 58

C) 62

D) 79

E) 100

Solution:

Use

$$(a+b)^2=a^2+2ab+b^2.$$

Thus,

$$100=a^2+b^2+42.$$

Therefore,

$$a^2+b^2=58.$$

Answer: B) 58

Problem 22: Probability

A bag contains 4 red balls and 6 blue balls. Two balls are selected without replacement. What is the probability that both are red?

A)

$\frac{1}{10}$

B)

$\frac{2}{15}$

C)

$\frac15$

D)

$\frac29$

E)

$\frac25$

Solution:

The probability of selecting a red ball first is

$$\frac4{10}.$$

After one red ball is selected, 3 red balls remain among 9 balls:

$$\frac39.$$

Therefore,

$$P=\frac4{10}\cdot\frac39 =\frac{12}{90} =\frac2{15}.$$

Answer: B) 

$\frac{2}{15}$

Problem 23: Number Patterns

What is the sum of the first 20 positive integers?

A) 190

B) 200

C) 210

D) 220

E) 230

Solution:

Use the formula

$$1+2+\cdots+n=\frac{n(n+1)}2.$$

For 

$n=20$

,

$$\frac{20(21)}2=210.$$

Answer: C) 210

Problem 24: Coordinate Geometry

What is the distance between the points 

$(1,2)$

and 

$(7,10)$

?

A) 8

B) 9

C) 10

D) 11

E) 12

Solution:

The distance formula is

$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.$$

Therefore,

$$d=\sqrt{(7-1)^2+(10-2)^2}$$ $$=\sqrt{36+64}$$ $$=\sqrt{100}=10.$$

Answer: C) 10

Problem 25: Contest Strategy

A positive integer 

$n$

satisfies

$$n^2=144.$$

If 

$n$

is positive, what is 

$n+5$

?

A) 12

B) 15

C) 17

D) 19

E) 24

Solution:

Since 

$n$

is positive,

$$n=\sqrt{144}=12.$$

Therefore,

$$n+5=17.$$

Answer: C) 17

AMC 12 Practice Problems Answer Key

Problem	Answer
1	C
2	E
3	C
4	A
5	C
6	B
7	C
8	C
9	C
10	C
11	B
12	B
13	B
14	B
15	B
16	D
17	D
18	A
19	D
20	C
21	B
22	B
23	C
24	C
25	C

How to Improve Your AMC 12 Practice

Solving a large number of questions is useful, but reviewing your mistakes is even more important.

After every practice session, divide your mistakes into three categories:

1. Concept mistake

You did not know the mathematical idea needed to solve the problem.

2. Calculation mistake

You knew the method but made an arithmetic or algebraic error.

3. Strategy mistake

You understood the mathematics but used a longer or less efficient approach.

This simple classification can make your preparation much more effective.

Build an AMC 12 Formula Review

Keep a small collection of important formulas and techniques. Your review can include:

• Pythagorean theorem

• Quadratic formula

• Arithmetic and geometric sequences

• Counting formulas

• Probability rules

• Circle formulas

• Triangle area formulas

• Similar triangles

• Modular arithmetic

• Exponent rules

• Factorization

• Coordinate geometry

However, avoid relying only on memorization. AMC-style problems often require you to recognize when a formula or technique should be used.

AMC 12 Time Management Tips.

The AMC 12 contains problems with different levels of difficulty. Spending too much time on one difficult question can prevent you from solving several easier questions.

A practical approach is to:

• Quickly scan the problem.

• Solve straightforward questions first.

• Mark questions requiring longer calculations.

• Return to difficult problems after collecting easier points.

• Check arithmetic when time permits.

• Avoid changing an answer without a mathematical reason.

Practice under timed conditions regularly so that time management becomes part of your normal preparation.

Where to Find More AMC 12 Practice.

Students should combine original practice problems with reliable mathematical resources. The official Mathematical Association of America AMC program provides information about AMC competitions and related resources.

For additional mathematics learning, students can also explore resources from Khan Academy and Art of Problem Solving.

For more practice material on Solved Tests, you can also explore related exam-preparation articles such as AMC 12 Sample Test and other solved-test resources available on the site.

Frequently Asked Questions About AMC 12 Practice Problems.

What are AMC 12 practice problems?.

AMC 12 practice problems are mathematical contest-style questions used to help students prepare for the American Mathematics Competitions 12 exam. They can cover algebra, geometry, number theory, probability, counting, and other mathematical topics.

How should I practice AMC 12 problems?.

Start with problems you can solve comfortably, then gradually increase the difficulty. Always review incorrect answers and identify the technique you missed.

Are AMC 12 problems difficult?.

Difficulty varies significantly. Some questions test basic mathematical knowledge, while later questions can require creative reasoning and multiple steps.

Should I practice with a timer?.

Yes. Timed practice can help you develop pacing and decision-making skills. After timed sessions, review every problem you missed or skipped.

How can I improve at difficult AMC 12 questions?.

Focus on understanding mathematical patterns rather than memorizing solutions. Work on algebraic manipulation, geometry, counting, number theory, and logical reasoning, and study multiple solution methods when possible.

Conclusion.

These AMC 12 practice problems provide a useful starting point for students who want to strengthen their contest mathematics skills. The 25 questions cover several important areas, including algebra, geometry, probability, number theory, sequences, counting, and coordinate geometry.

The most effective preparation is consistent practice combined with careful review. Do not simply record whether an answer was right or wrong. Ask why your solution worked, why another approach may be shorter, and what caused any mistakes.

With regular practice, timed problem sets, and detailed mistake analysis, you can build stronger mathematical reasoning and become more comfortable with unfamiliar AMC 12 questions.

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