Single-Sentence Challenges

Obviously, discard zeros and other trivial answers for these.

1. What number increases by one when squared?

2. What number doubles when cubed?

3. What number's cube root is half its square root?

4. Find two integers whose quotient is their sum.

5. Find two integers whose mean is the square of their difference.

6. Find two numbers whose difference is their harmonic mean.

7. Find two numbers whose product is their mean absolute value.

Trigonometry Challenge

The problems below require only right-angle trigonometry and circle geometry. Try to find answers in exact form!

#1. Two Gears

#2. A Rowling Stone

#3. Threaded Bolt

#4. Five-Pointed Star

#5. Pac-Mountain

#6. Toothpicks

#7. An Ice Problem