AIME Inequalities and Optimization 3

Question 1

For positive reals x and y with xy = 1, what is the minimum value of (x + 1/x)² + (y + 1/y)²?

Accepted Answer

8

Answer

4

Answer

6

Answer

10

Question 2

Among all triangles with perimeter 12, what is the maximum area?

Accepted Answer

4√3

Answer

6√3

Answer

8√3

Answer

12

Question 3

For positive reals a and b satisfying a + b = 1, what is the minimum value of a³ + b³?

Accepted Answer

1/4

Answer

1/8

Answer

1/2

Answer

3/4

Question 4

What is the maximum value of 2x + y for real x, y satisfying x² + y² ≤ 5?

Accepted Answer

5

Answer

4

Answer

2√5

Answer

5√2

Question 5

For non-negative integers m and n with m + n = 10, what is the maximum value of m·n?

Accepted Answer

25

Answer

20

Answer

24

Answer

30

Question 6

Which of the following correctly states the AM-GM inequality for three positive reals x, y, z?

Accepted Answer

(x+y+z)/3 ≥ (xyz)^(1/3)

Answer

x + y + z ≥ xyz

Answer

xyz ≥ (x+y+z)/3

Answer

x²+y²+z² ≤ xyz

Question 7

What is the minimum value of x² + y² for real numbers x and y satisfying 2x + 3y = 13?

Accepted Answer

13

Answer

9

Answer

10

Answer

12