AIME Inequalities and Optimization

Question 1

For a positive real number x, what is the minimum value of x + 4/x?

Accepted Answer

4

Answer

2

Answer

3

Answer

6

Question 2

If a and b are positive reals with a + b = 10, what is the maximum value of ab?

Accepted Answer

25

Answer

20

Answer

24

Answer

30

Question 3

For real numbers x and y satisfying x + y = 6, what is the minimum value of x² + y²?

Accepted Answer

18

Answer

16

Answer

20

Answer

24

Question 4

For positive reals x, y, z with x + y + z = 9, what is the maximum value of xy + yz + xz?

Accepted Answer

27

Answer

21

Answer

24

Answer

30

Question 5

For positive reals a, b, c with abc = 1, what is the minimum value of a + b + c?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 6

For positive reals a and b, what is the minimum value of (a² + b²) / (ab)?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 7

What is the maximum value of 3x − 4y for real numbers x and y satisfying x² + y² = 25?

Accepted Answer

25

Answer

15

Answer

20

Answer

30