Mathematics Interview Questions (III)
11. What is the integral of ?
Solution. Consider the triple-angle formula . That gives . Therefore,
12. How many squares can be made from a grid of ten by ten dots (ignore diagonal squares)?
Solution. For squares, there are of them. For squares, there are of them. Following the pattern, there are squares in total. Recall that , the number of squares is .
13. Integrate .
Solution. Using ,
14. Prove that every number in the Pascal's triangle is the sum of the two above it.
Solution. This is just the definition of it. Algebraically, this is equivalent to . We can use two methods to prove that:
Combinatoric proof (a.k.a. story proof).
Consider choosing people out of . On the one hand, it is equal to . On the other hand, consider the tallest person. If he/she is not chosen, there are methods. Otherwise, there are methods.
15. Integrate .
Solution. Cannot be expressed in terms of elementary functions.