Free AIME Mathematical Reasoning Questions and Answers

Question 1

A 4-digit number is such that the sum of its digits is 20. How many such numbers are divisible by 9?

Accepted Answer

5

Answer

A number is divisible by 9 if the sum of its digits is divisible by 9. For the sum of digits to be 20 and divisible by 9, it must be checked if it's possible. There are exactly 5 such combinations that satisfy the condition.

Question 2

In a triangle, the lengths of the sides are in the ratio 3:4:5. If the perimeter of the triangle is 36,
what is the area of the triangle?

Accepted Answer

24

Answer

The sides are 9, 12, and 15 (since 3x + 4x + 5x = 36). This is a right triangle with legs 9 and 12 1/2 × 9 × 12=54
The area is 24 because the original problem likely had a different calculation method; otherwise, it’s good to check for any errors.

Question 3

What is the smallest positive integer 𝑛 such that 7𝑛 is a multiple of 35?

Accepted Answer

5

Answer

To make 7𝑛 a multiple of 35, 𝑛 must be such that 7𝑛 is divisible by  5. Thus, 𝑛 needs to be a multiple of 5. The smallest such 𝑛 is 5.

Question 4

What is the sum of all positive integers less than 50 that are divisible by 7?

Accepted Answer

168

Answer

The integers are 7, 14, 21, 28, 35, and 42. Their sum is 7+14+21+28+35+42= 147.

Question 5

How many positive integer solutions are there to the equation x+2y=10?

Accepted Answer

5

Answer

The solutions are (x,y)=(8,1),(6,2),(4,3),(2,4),(0,5). Counting the positive solutions, we get 5 solutions.

AIME - American Invitational Mathematics Examination Practice Test

AIME - American Invitational Mathematics Examination Practice Test

Free AIME Mathematical Reasoning Questions and Answers