AIME Inequalities and Optimization 2

Question 1

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

Accepted Answer

4

Answer

2

Answer

3

Answer

6

Question 2

What is the maximum value of sin(θ) + cos(θ) over all real θ?

Accepted Answer

√2

Answer

1

Answer

√3

Answer

2

Question 3

For positive reals a, b, c with a + b + c = 1, what is the minimum value of a² + b² + c²?

Accepted Answer

1/3

Answer

1/6

Answer

1/4

Answer

1/2

Question 4

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

Accepted Answer

4

Answer

2

Answer

3

Answer

5

Question 5

If x > 0 and x² + x⁻² = 3, what is the value of x⁴ + x⁻⁴?

Accepted Answer

7

Answer

5

Answer

6

Answer

9

Question 6

What is the maximum product of positive integers that sum to 10?

Accepted Answer

36

Answer

24

Answer

40

Answer

45

Question 7

For real numbers a and b satisfying a² + b² = 4, what is the maximum value of a + 2b?

Accepted Answer

2√5

Answer

4

Answer

2√6

Answer

6