越而胜己 / Beyond awesome

Looking back on the quiz problems we had in the PKU EECS Summer Camp, some problems are actually quite interesting. Hard as the math problems may seem, they can be solved with some patience. The computer science (algorithm) problems are especially easy.

I can't be sure that all problems are exactly the same as the ones in PKU, but I'll just list them right below.

Math

  1. Given that $n\in\Bbb N^{*} \cap [2,+\infty)$, prove:$\frac{2n(2n-1)(2n-2)...(n+3)(n+2)}{n(n-1)(n-2)\times...\times 2}\in\Bbb N^{*}$.
  2. Given that $|x|<1, |y|<1$, prove: $\frac{1}{1-x^2} + \frac{1}{1-y^2} \ge \frac{2}{1-xy}$.
  3. Given that parabola $C:y^2=2px(p>0)$ has a vertex $O$, and $M$ is any point on $C$. If $A(\frac{p}{6},0)$ is a point on the symmetry axis, what is the minimum value of$|MA|$?

Computer Science

  1. Sudoku
  2. Decision Tree
    Four friends are living in a town. Their names are Cook, Miller, Smith and Carter, of them one is a policeman, one is a carpenter, one is a farmer and the other is a doctor. One day Cook's son broke his leg, and Cook took him to the doctor. The doctor's sister is Smith's wife. The farmer is not married but has a lot of hens. Miller often goes to the farmer to buy eggs. The policeman sees Smith every day because they are neighbors. Who is the policeman? Who is the carpenter? Who is the farmer? Who is the doctor?
  3. Tower of Hanoi
    If there are three posts and N plates on the first post, what is the minimum number of times you must move a plate?
    If there are four posts, what is the minimum number of times you must move a plate?

AI

Jack is working at your company for 7 days. You have a gold bar, of which Jack demand 1/7 as his daily wage. Jack asks you to give his wage every day, and you can only cut the gold bar twice. How are you going to cut the gold bar?