**Note:** You should always justify your answers. Whenever you are asked to describe an algorithm, you should present three things: the algorithm, a proof of its correctness, and a derivation of its running time.

In [1]:

```
def something(A):
""" The input is a python 'list'. For our purposes you may assume it is an array of size n = len(A) """
n = len(A)
k = 0
for i in range(0, n): # In pseudo-code: for i=0,...,n-1
if A[i]%2 == 0: # In pseudo-code: If A[i] is even
k = k + 1
return n - k
```

(a) Prove by induction on $n \ge 1$ that $\sum_{i=1}^n (i-\frac{1}{2}) = \frac{n^2}{2}$ by using the induction hypothesis: *Assume that $\sum_{i=1}^n (i-\frac{1}{2}) = \frac{n^2}{2}$ holds.*

(b) Consider a $2^n \times 2^n$ chessboard with one (arbitrarily chosen) square removed. We want to tile the chessboard without gaps (except for the removed square) or overlaps by L-shaped pieces, each composed of 3 squares. The figure below shows an L-shaped piece, a chessbord with a square removed, and a tiling of this chessboard.

Give a high-level description (i.e., it is enough to convey the idea) of a recursive algorithm to compute such a tiling. Prove the correctness of your algorithm by a (weak) induction on $n$. You do not need to analyze the running time.

Are the following statements true or false? Briefy explain your answers (a-c). Provide a formal proof for your answer for (d-e).

(a) $n^2 \log n = O(n^2)$

(b) $n + \log n = \Omega(n)$

(c) $10 n^2 + 4n -17 = O(n^2 - 5n)$

(d) $n \log n = \Theta(n)$

(e) If $f(n) = O(g(n))$, then $g(n) = \Omega(f(n))$.

You are given an array $A$ of size $n$ that contains distinct (i.e. no number occurs more than once) numbers with the following property: First the numbers in $A$ increase and then they decrease. Stated differently, there is an (unknown) index $i$ such that $A[j] < A[j+1]$ for all $0 \le j < i$, and $A[j] > A[j+1]$ for all $i \le j < n-1$.

(a) Give an iterative worst-case $O(\log n)$-time algorithm that determines the index of the maximum element of $A$. *Don't forget to prove correctness and analyze the running time.*

(b) As (a), but now give a recursive algorithm. *Note: For the running time it is sufficient to give an informal argument.*

In [ ]:

```
```